Abstract:
A method for calculating material properties of a material includes conducting two insertion loss tests of the material having a single thickness and a double thickness. These tests are conducted at a zero wavenumber. Utilizing these insertion loss tests, a dilatational wavespeed is computed. The method continues by calculating a shear wavespeed by performing three insertion loss tests of the material at single, double and triple thicknesses. These tests are conducted at a non-zero wavenumber. A shear wavespeed can be calculated from the dilatational wavespeed and these insertion loss tests. Lamé constants, Young&#39;s modulus, Poisson&#39;s ratio, and the shear modulus for the material of interest can then be calculated using the dilatational and shear wavespeeds.

Description:
STATEMENT OF GOVERNMENT INTEREST 
       [0001]    The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefore. 
     
    
     CROSS REFERENCE TO OTHER RELATED APPLICATIONS 
       [0002]    None. 
       BACKGROUND OF THE INVENTION 
       [0003]    (1) Field of the Invention 
         [0004]    The present invention relates to material properties measurement and, more particularly, to a method for measuring material properties using wall displacement measurements recorded during an insertion loss experiment. 
         [0005]    (2) Description of the Prior Art 
         [0006]    Insertion loss is a common measurement that is used to determine how effective a piece of material attenuates acoustic energy at a specific frequency. Insertion loss is calculated by projecting acoustic energy at piece of material and measuring the pressure on the projector side and the opposite side of the material, normally with hydrophones. 
         [0007]      FIG. 1  depicts a typical setup for insertion loss. Sound pressure is transmitted to a test sample  10  by an acoustic projector  12 . Acoustic projector  12  can transmit an acoustic wave at a preset frequency. Using a 1 m by 1 m specimen, the minimum frequency is about 10 kHz. A first hydrophone  14  is positioned on the opposite side of the sample  10  to measure the transmitted pressure. The ratio of the source pressure to the transmitted pressure expressed in decibels is the insertion loss of the material. A second hydrophone  16  is positioned on the same side of sample  10  as projector  12  to measure the reflected acoustic pressure. 
         [0008]    Insertion loss can also be determined by measuring the motion of the test sample with either accelerometers or laser velocimeters and calculating the pressure field based on conservation of linear momentum. In the test setup shown here, a first laser velocimeter  18  is used to measure the acceleration and position of a first side  20  of sample  10 . A second laser velocimeter  22  is used to measure the acceleration and position of a second side  24  of sample  10 . Laser velocimeters  18 ,  22  are preferred because accelerometers must be positioned on sample  10  and might interfere with the measurements. The projector  12  angle θ relative to the test material can be changed so that the effects of acoustic energy at varying angles can be studied. Changing the excitation angle θ is equivalent to changing the excitation wavenumber. Thus, the two parameters that are typically varied during this test are frequency and wavenumber. 
         [0009]    For underwater applications, the material is submerged in a fluid (normally water), and an underwater speaker or projector transmits energy at the material; however, a gaseous environment could be used. Because this test is only interested in acoustic attenuation of the material, the height and width of the test specimen are large compared to its thickness. In view of this, the test specimen should have a thickness between 10 mm and 100 mm. This prevents acoustic energy from moving around the specimen and contaminating the transmitted pressure field and interacting with the opposite side to the test specimen. The test is also dependent on the environment where it is conducted. Small test tanks prevent low frequency measurements due to reflection and reverberation of the acoustic energy. These are, however, practical limitations and do not enter into this theoretical analysis. 
       SUMMARY OF THE INVENTION 
       [0010]    One object of this invention is to accurately determine the material properties of a sample in an insertion loss experiment. 
         [0011]    Another object of the present invention is to determine the material properties of dilatational wavespeed, shear wavespeed, Lamé constants, Young&#39;s modulus, and shear modulus of a material of interest. 
         [0012]    The present invention features an inverse method where normal wall movement measurements obtained during an insertion loss test are combined to equal material properties. This allows for the calculation of Young&#39;s modulus, shear modulus, and Poisson&#39;s ratio from an insertion loss test. Alternatively, Lame constants and Poisson&#39;s ratio or complex dilatational and shear wavespeeds are also obtainable from this method. For dilatational wave energy, the test requires two material samples, one being twice as thick as the other. For shear wave energy, the test requires three material samples, one being twice as thick as the first and the second being three times as thick as the first. Measurements of these multiple samples allow the governing equations and test data be combined in a manner that results in an inverse method in which the material properties are closed form solutions of the measurement data. This is sometimes referred to as a linear inverse method. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]    These and other features and advantages of the present invention will be better understood in view of the following description of the invention taken together with the drawings wherein: 
           [0014]      FIG. 1  is a diagram of the test setup for the current invention; 
           [0015]      FIG. 2  is a diagram showing the coordinate system used by the current invention; 
           [0016]      FIGS. 3A and 3B  are graphs of the transfer frequency magnitude and phase angle at a zero degree excitation angle; 
           [0017]      FIG. 4  is a graph of the function s versus frequency; 
           [0018]      FIGS. 5A and 5B  are graphs of the real and imaginary portions of the actual and estimated wavenumber alpha versus frequency; 
           [0019]      FIGS. 6A and 6B  are graphs of the real and imaginary portions of the actual and estimated dilatational wavespeed versus frequency; 
           [0020]      FIGS. 7A and 7B  are graphs of the transfer frequency magnitude and phase angle at a fifteen degree excitation angle; 
           [0021]      FIG. 8  is a graph of the function r and the angle of the discriminant versus frequency; 
           [0022]      FIGS. 9A and 9B  are graphs of the real and imaginary portions of the actual and estimated wavenumber beta versus frequency; 
           [0023]      FIGS. 10A and 10B  are graphs of the real and imaginary portions of the actual and estimated shear wavespeed versus frequency; 
           [0024]      FIGS. 11A and 11B  are graphs of the real and imaginary portions of the actual and estimated Lamé constant μ versus frequency; 
           [0025]      FIGS. 12A and 12B  are graphs of the real and imaginary portions of the actual and estimated Lamé constant λ versus frequency; and 
           [0026]      FIGS. 13A and 13B  are graphs of the real and imaginary portions of the actual and estimated Young&#39;s modulus versus frequency. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0027]    The coordinate system of the test configuration is shown in  FIG. 2 . Projector  12  is oriented at an angle θ with respect to sample  10 . A first measurement location  28  is located on the far side of sample  10  from projector  12 . This is the position where the beam from laser velocimeter  18  shown in  FIG. 1  contacts surface  20 . A second measurement location  26  corresponds to where second laser velocimeter  22  beam contacts surface  24 . Under the coordinate system, the z axis is orthogonal to the second surface of sample  10  with the origin at this surface. Note that using this orientation results in b=0 and a having a value less than zero (−h). The thickness of the sample, h, is a positive value. The y axis is oriented into the page. 
         [0028]    The system model has three governing differential equations that are coupled at their interfaces using conservation of linear momentum. The acoustic pressure in the fluid on the projector side of the test specimen is governed by the wave equation and is written in Cartesian coordinates as [1] 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             
                               p 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             
                               p 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       - 
                       
                         
                           1 
                           
                             c 
                             f 
                             2 
                           
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 p 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               t 
                               2 
                             
                           
                         
                       
                     
                     = 
                     0 
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 1 (x,z,t) is the pressure (N/m 2 ), z is the spatial location (m) normal to the plate, x is spatial location along the plate (m), c f  is the compressional wavespeed of the fluid (m/s), t is time (s), and the subscript one denotes the area on the projector side of the test material. The motion of the material is governed by the equation [2] 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         μ 
                          
                         
                           
                             ∇ 
                             2 
                           
                            
                           u 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             λ 
                             + 
                             μ 
                           
                           ) 
                         
                          
                         
                           
                             ∇ 
                             ∇ 
                           
                           · 
                           u 
                         
                       
                     
                     = 
                     
                       ρ 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           u 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ρ is the density (kg/m 3 ), λ and μ are the complex Lamé constants (N/m 2 ), • denotes a vector dot product; u is the Cartesian coordinate displacement vector of the material. The acoustic pressure in the fluid on opposite the projector side of the test specimen is governed by the wave equation and is written in Cartesian coordinates as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             
                               p 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             
                               p 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       - 
                       
                         
                           1 
                           
                             c 
                             f 
                             2 
                           
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 p 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               t 
                               2 
                             
                           
                         
                       
                     
                     = 
                     0 
                   
                   , 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 2 (x,z,t) is the pressure (N/m 2 ) and the subscript two denotes the area opposite the projector side of the test material. The interface between the first fluid and solid surface of the material at z=b satisfies the linear momentum equation, which is [3] 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ρ 
                         f 
                       
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             
                               u 
                               z 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 b 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                     = 
                     
                       - 
                       
                         
                           ∂ 
                           
                             
                               p 
                               1 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 b 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ρ f  is the density of the fluid (kg/m 3 ). The interface between the second fluid and solid surface of the material at z=a also satisfies the linear momentum equation, and is written as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ρ 
                       f 
                     
                      
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           
                             u 
                             z 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               a 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           ∂ 
                           
                             
                               p 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 a 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The above five equations are the governing partial differential equations of the insertion loss experiment. 
         [0029]    Equations (1)-(3) are now transformed from partial differential equations into ordinary differential equations and then into algebraic expressions. The acoustic pressure in equation (1) is modeled as a function at definite wavenumber and frequency as 
         [0000]        p   1 ( x,z,t )= P   1 ( z,k   x ,ω)exp( ik   x   x )exp( iωt ),   (6) 
         [0000]    where ω is frequency (rad/s), k x  is the spatial wavenumber in the x direction (rad/m), and i is the square root of −1. The spatial wavenumber is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       k 
                       x 
                     
                     = 
                     
                       
                         ω 
                         
                           c 
                           f 
                         
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where θ is the angle of incidence (rad) of the incoming acoustic wave with θ=0 corresponding to excitation normal to the sample (or broadside excitation). Inserting equation (6) into equation (1) and solving the resulting ordinary differential equation yields 
         [0000]        P   1 ( z,k   x ,ω)= H ( k   x ,ω)exp( iγz )+ P   S (ω)exp(− iγz ).   (8) 
         [0000]    In equation (8), the first term on the right hand side represents the reradiated (or reflected) pressure field and the second term represents the applied incident pressure field (the forcing function) acting on the structure. The term H(k x ,ω) is the wave propagation coefficient of the reflected pressure field and the term P S (ω) is the source (or excitation) level. Additionally, 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     = 
                     
                       
                         
                           
                             ( 
                             
                               ω 
                               
                                 c 
                                 f 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           k 
                           x 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where γ is the wavenumber of the acoustic pressure in the fluid. 
         [0030]    Equation (2) is manipulated by writing the Cartesian coordinate displacement vector u as 
         [0000]    
       
         
           
             
               
                 
                   
                     u 
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 u 
                                 x 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 u 
                                 y 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 u 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with y denoting the direction into the material in  FIG. 2 . The symbol ∇ is the gradient vector differential operator written in three-dimensional Cartesian coordinates as [4] 
         [0000]    
       
         
           
             
               
                 
                   
                     ∇ 
                     
                       = 
                       
                         
                           
                             ∂ 
                             
                               ∂ 
                               x 
                             
                           
                            
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               ∂ 
                               y 
                             
                           
                            
                           
                             i 
                             y 
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               ∂ 
                               z 
                             
                           
                            
                           
                             i 
                             z 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with i x  denoting the unit vector in the x-direction, i y  denoting the unit vector in the y-direction, and i z  denoting the unit vector in the z-direction; ∇ 2  is the three-dimensional Laplace operator operating on vector u as 
         [0000]      ∇ 2   u=∇   2   u   x   i   x +∇ 2   u   y   i   y +∇ 2   u   z   i   z ′  (12) 
         [0000]    with ∇ 2  operating on scalar u as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∇ 
                         2 
                       
                        
                       
                         u 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                       
                     
                     = 
                     
                       
                         ∇ 
                         
                           · 
                           
                             ∇ 
                             
                                 
                             
                              
                             
                               u 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               u 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                             
                           
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         + 
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               u 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                             
                           
                           
                             ∂ 
                             
                               y 
                               2 
                             
                           
                         
                         + 
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               u 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                             
                           
                           
                             ∂ 
                             
                               z 
                               2 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the term ∇•u is called the divergence and is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     ∇ 
                     
                       · 
                       u 
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           u 
                           x 
                         
                       
                       
                         ∂ 
                         x 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           u 
                           y 
                         
                       
                       
                         ∂ 
                         y 
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             u 
                             z 
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0031]    The displacement vector u is written as 
         [0000]        u=∇φ+∇×{right arrow over (ψ)},    (15) 
         [0000]    where φ is a dilatational scalar potential, × denotes a vector cross product, and {right arrow over (ψ)} is an equivoluminal vector potential expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     Ψ 
                     -&gt; 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 Ψ 
                                 x 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 Ψ 
                                 y 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 Ψ 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The structural problem is formulated as a two-dimensional response (y≡0 and ∂(·)/∂y≡0) problem. Expanding equation (15) and breaking the displacement vector into its individual nonzero terms yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         x 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           z 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             φ 
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           x 
                         
                       
                       - 
                       
                         
                           ∂ 
                           
                             
                               ψ 
                               y 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       z 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         z 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           φ 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             
                               ψ 
                               y 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           x 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0032]    Equations (17) and (18) are next inserted into equation (2), which results in 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         c 
                         d 
                         2 
                       
                        
                       
                         
                           ∇ 
                           2 
                         
                          
                         
                           φ 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           φ 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       c 
                       s 
                       2 
                     
                      
                     
                       
                         ∇ 
                         2 
                       
                        
                       
                         
                           ψ 
                           y 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             z 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         
                           ψ 
                           y 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             z 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     
                       ∂ 
                       
                         t 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where equation (19) corresponds to the dilatational component and equation (20) corresponds to the shear component of the displacement field [5]. Correspondingly, the constants c d  and c s  are the complex dilatational and shear wave speeds, respectively, and are determined by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       d 
                     
                     = 
                     
                       
                         
                           λ 
                           + 
                           
                             2 
                              
                             μ 
                           
                         
                         ρ 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     s 
                   
                   = 
                   
                     
                       
                         μ 
                         ρ 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The relationship of the Lamé constants to the compressional and shear moduli is shown as 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     = 
                     
                       
                         E 
                          
                         
                             
                         
                          
                         υ 
                       
                       
                         
                           ( 
                           
                             1 
                             + 
                             υ 
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               2 
                                
                               υ 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     μ 
                     = 
                     
                       G 
                       = 
                       
                         E 
                         
                           2 
                            
                           
                             ( 
                             
                               1 
                               + 
                               υ 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E is the complex Young&#39;s (compressional) modulus (N/m 2 ), G is the complex shear modulus (N/m 2 ), and υ is the Poisson&#39;s ratio of the material (dimensionless). 
         [0033]    The conditions of infinite length and steady-state response are now imposed, allowing the scalar and vector potential to be written as 
         [0000]      φ( x,z,t )=Φ( z )exp( ik   x   x )exp( iωt ),   (25) 
         [0000]      and 
         [0000]      ψ y ( x,z,t )=Ψ( z )exp( ik   x   x )exp( iωt ).   (26) 
         [0000]    Inserting equation (25) into equation (19) yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             2 
                           
                            
                           
                             Φ 
                              
                             
                               ( 
                               z 
                               ) 
                             
                           
                         
                         
                            
                           
                             z 
                             2 
                           
                         
                       
                       + 
                       
                         
                           α 
                           2 
                         
                          
                         
                           Φ 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     α 
                     = 
                     
                       
                         
                           k 
                           d 
                           2 
                         
                         - 
                         
                           k 
                           x 
                           2 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     k 
                     d 
                   
                   = 
                   
                     
                       ω 
                       
                         c 
                         d 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Inserting equation (26) into equation (20) yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             2 
                           
                            
                           
                             Ψ 
                              
                             
                               ( 
                               z 
                               ) 
                             
                           
                         
                         
                            
                           
                             z 
                             2 
                           
                         
                       
                       + 
                       
                         
                           β 
                           2 
                         
                          
                         
                           Ψ 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     β 
                     = 
                     
                       
                         
                           k 
                           s 
                           2 
                         
                         - 
                         
                           k 
                           x 
                           2 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     k 
                     s 
                   
                   = 
                   
                     
                       ω 
                       
                         c 
                         s 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0034]    The solution to equation (27) is 
         [0000]      Φ( z )= A ( k   x ω)exp( iαz )+ B ( k   x ,ω)exp(− iαz ),   (33) 
         [0000]    and the solution to equation (30) is 
         [0000]      Ψ( z )= C ( k   x ,ω)exp( iβz )+ D ( k   x ,ω)exp(− iβz ),   (34) 
         [0000]    where A(k x ,ω), B(k x ,ω), C(k x ,ω), and D(k x ,ω) are wave propagation constants that are determined below. The displacements can now be written as functions of the unknown constants using the expressions in equations (17) and (18). They are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               u 
                               z 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 z 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   z 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     x 
                                   
                                    
                                   x 
                                 
                                 ) 
                               
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   ω 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             { 
                             
                               
                                 α 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         A 
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               x 
                                             
                                             , 
                                             ω 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             z 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               x 
                                             
                                             , 
                                             ω 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               α 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             z 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               + 
                             
                           
                         
                       
                     
                     
                       
                         
                             
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               
                                 k 
                                 x 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       C 
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             x 
                                           
                                           , 
                                           ω 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       exp 
                                        
                                       
                                         ( 
                                         
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           z 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             x 
                                           
                                           , 
                                           ω 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       exp 
                                        
                                       
                                         ( 
                                         
                                           
                                             - 
                                             β 
                                           
                                            
                                           
                                               
                                           
                                            
                                           z 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                       
                     
                     
                       
                         
                             
                            
                           
                             
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       x 
                                     
                                      
                                     x 
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             x 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               U 
                               x 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 z 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   x 
                                 
                                  
                                 x 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                 ω 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           { 
                           
                             
                                
                                
                               
                                   
                               
                                
                               
                                 
                                   k 
                                   x 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         A 
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               x 
                                             
                                             , 
                                             ω 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             z 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               x 
                                             
                                             , 
                                             ω 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               α 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             z 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             - 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           β 
                            
                           
                             [ 
                             
                               
                                 
                                   C 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         x 
                                       
                                       , 
                                       ω 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   exp 
                                    
                                   
                                     ( 
                                     
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       z 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         x 
                                       
                                       , 
                                       ω 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   exp 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         β 
                                       
                                        
                                       
                                           
                                       
                                        
                                       z 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         } 
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             exp 
                              
                             
                               ( 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   x 
                                 
                                  
                                 x 
                               
                               ) 
                             
                           
                            
                           
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   ω 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The normal stress the top of the plate (z=b) is equal to opposite the pressure in the fluid. This expression is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         τ 
                         zz 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           b 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               λ 
                               + 
                               
                                 2 
                                  
                                 μ 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   z 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     b 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                         + 
                         
                           λ 
                            
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   x 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     b 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                         
                       
                       = 
                       
                         - 
                         
                           
                             p 
                             1 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               b 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the tangential stress at the top of the plate is zero and this equation is written as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       τ 
                       zx 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         b 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                        
                       
                         [ 
                         
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   x 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     b 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   z 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     b 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The normal stress the bottom of the plate (z=a) is equal to opposite the pressure in the fluid. This expression is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         τ 
                         zz 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           a 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               λ 
                               + 
                               
                                 2 
                                  
                                 μ 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   z 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     a 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                         + 
                         
                           λ 
                            
                           
                             
                               ∂ 
                               
                                 
                                   u 
                                   x 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     a 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                         
                       
                       = 
                       
                         - 
                         
                           
                             p 
                             2 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               a 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the tangential stress at the bottom of the plate is zero and this equation is written as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         τ 
                         zx 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           a 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         μ 
                          
                         
                           [ 
                           
                             
                               
                                 ∂ 
                                 
                                   
                                     u 
                                     x 
                                   
                                    
                                   
                                     ( 
                                     
                                       x 
                                       , 
                                       a 
                                       , 
                                       t 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 z 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   
                                     u 
                                     z 
                                   
                                    
                                   
                                     ( 
                                     
                                       x 
                                       , 
                                       a 
                                       , 
                                       t 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 x 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       0 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 2 (x,b,t) in equation (39) represents the radiated acoustic pressure in the fluid load on the opposite side of the acoustic projector. 
         [0035]    The acoustic pressure in equation (3) is modeled as a function at definite wavenumber and frequency as 
         [0000]        p   2 ( x,z,t )= P   2 ( z,k   x ,ω)exp( ik   x   x )exp( iωt ),   (41) 
         [0000]    Inserting equation (41) into equation (3) and solving the resulting ordinary differential equation yields 
         [0000]        P   2 ( z,k   x ,ω)= K ( k   x ,ω)exp(− iγz ),   (42) 
         [0000]    which is the outgoing (or transmitted) acoustic energy in the second fluid. The term K(k x ,ω) is the wave propagation coefficient of the transmitted pressure field. Note that there is no incoming wave energy on this side of the test specimen and thus only one exponential term is present. 
         [0036]    Assembling equations (1)-(42) and letting b=0 yields the four-by-four system of linear equations that model the system. They are 
         [0000]        Ax=b,    (43) 
         [0000]    where the entries of equation (43) are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       
                         11 
                          
                         s 
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             α 
                             2 
                           
                         
                          
                         λ 
                       
                       - 
                       
                         2 
                          
                         
                           α 
                           2 
                         
                          
                         μ 
                       
                       - 
                       
                         λ 
                          
                         
                             
                         
                          
                         
                           k 
                           x 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       
                         11 
                          
                         f 
                       
                     
                     = 
                     
                       
                         
                           ρ 
                           f 
                         
                          
                         
                           ω 
                           2 
                         
                          
                         α 
                       
                       γ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       11 
                     
                     = 
                     
                       
                         A 
                         
                           11 
                            
                           s 
                         
                       
                       + 
                       
                         A 
                         
                           11 
                            
                           f 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       12 
                     
                     = 
                     
                       
                         A 
                         
                           11 
                            
                           s 
                         
                       
                       - 
                       
                         A 
                         
                           11 
                            
                           f 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       
                         13 
                          
                         s 
                       
                     
                     = 
                     
                       2 
                        
                       
                         k 
                         x 
                       
                        
                       βμ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       
                         13 
                          
                         f 
                       
                     
                     = 
                     
                       
                         
                           ρ 
                           f 
                         
                          
                         
                           ω 
                           2 
                         
                          
                         
                           k 
                           x 
                         
                       
                       γ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       13 
                     
                     = 
                     
                       
                         - 
                         
                           A 
                           
                             13 
                              
                             s 
                           
                         
                       
                       + 
                       
                         A 
                         
                           13 
                            
                           f 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       14 
                     
                     = 
                     
                       
                         A 
                         
                           13 
                            
                           s 
                         
                       
                       + 
                       
                         A 
                         
                           13 
                            
                           f 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       21 
                     
                     = 
                     
                       
                         - 
                         2 
                       
                        
                       μ 
                        
                       
                           
                       
                        
                       
                         k 
                         x 
                       
                        
                       α 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       22 
                     
                     = 
                     
                       - 
                       
                         A 
                         21 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       23 
                     
                     = 
                     
                       
                         μβ 
                         2 
                       
                       - 
                       
                         μ 
                          
                         
                             
                         
                          
                         
                           k 
                           x 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       24 
                     
                     = 
                     
                       A 
                       23 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       31 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             A 
                             
                               11 
                                
                               s 
                             
                           
                           - 
                           
                             A 
                             
                               11 
                                
                               f 
                             
                           
                         
                         ) 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             α 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       32 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             A 
                             
                               11 
                                
                               s 
                             
                           
                           + 
                           
                             A 
                             
                               11 
                                
                               f 
                             
                           
                         
                         ) 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               α 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       33 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             - 
                             
                               A 
                               
                                 13 
                                  
                                 s 
                               
                             
                           
                           - 
                           
                             A 
                             
                               13 
                                
                               f 
                             
                           
                         
                         ) 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             β 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       34 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             A 
                             
                               13 
                                
                               s 
                             
                           
                           - 
                           
                             A 
                             
                               13 
                                
                               f 
                             
                           
                         
                         ) 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               β 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       41 
                     
                     = 
                     
                       
                         A 
                         21 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             α 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       42 
                     
                     = 
                     
                       
                         - 
                         
                           A 
                           21 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             α 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       43 
                     
                     = 
                     
                       
                         A 
                         23 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             β 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       44 
                     
                     = 
                     
                       
                         A 
                         23 
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               β 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       11 
                     
                     = 
                     
                       A 
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       21 
                     
                     = 
                     
                       B 
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       31 
                     
                     = 
                     
                       C 
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       41 
                     
                     = 
                     
                       D 
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       b 
                       11 
                     
                     = 
                     
                       
                         - 
                         2 
                       
                        
                       
                         
                           P 
                           S 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       b 
                       21 
                     
                     = 
                     0 
                   
                   , 
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       b 
                       31 
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     41 
                   
                   = 
                   0. 
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It is noted that the subscript s corresponds to terms related to the structure and the subscript f corresponds to terms related to the fluid. Using equations (43)-(71) the solution to the constants A(k x ,ω), B(k x ,ω), C(k x ,ω) and D(k x ,ω) can be calculated at each specific wavenumber and frequency. Written in transfer function form with reference to the source excitation level, they are 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           A 
                            
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               ω 
                             
                             ) 
                           
                         
                         
                           
                             P 
                             S 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           { 
                           
                             
                               4 
                                
                               
                                 A 
                                 
                                   13 
                                    
                                   s 
                                 
                               
                                
                               
                                 A 
                                 22 
                               
                                
                               
                                 
                                   A 
                                   23 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           βα 
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             - 
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         A 
                                         11 
                                       
                                        
                                       
                                         A 
                                         23 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         A 
                                         
                                           13 
                                            
                                           f 
                                         
                                       
                                        
                                       
                                         A 
                                         22 
                                       
                                        
                                       
                                         A 
                                         23 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     β 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       α 
                                     
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                          
                         
                           Δ 
                           
                             - 
                             1 
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           B 
                            
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               ω 
                             
                             ) 
                           
                         
                         
                           
                             P 
                             S 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           { 
                           
                             
                               4 
                                
                               
                                 A 
                                 
                                   13 
                                    
                                   s 
                                 
                               
                                
                               
                                 A 
                                 22 
                               
                                
                               
                                 
                                   A 
                                   23 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             + 
                             
                               4 
                                
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         A 
                                         12 
                                       
                                        
                                       
                                         A 
                                         23 
                                         2 
                                       
                                     
                                     - 
                                     
                                       
                                         A 
                                         
                                           13 
                                            
                                           f 
                                         
                                       
                                        
                                       
                                         A 
                                         22 
                                       
                                        
                                       
                                         A 
                                         23 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     β 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                          
                         
                           Δ 
                           
                             - 
                             1 
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           C 
                            
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               ω 
                             
                             ) 
                           
                         
                         
                           
                             P 
                             S 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           { 
                           
                             
                               4 
                                
                               
                                 A 
                                 
                                   11 
                                    
                                   s 
                                 
                               
                                
                               
                                 A 
                                 22 
                               
                                
                               
                                 
                                   A 
                                   23 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               β 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             - 
                             
                               4 
                                
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         A 
                                         13 
                                       
                                        
                                       
                                         A 
                                         22 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         A 
                                         
                                           11 
                                            
                                           f 
                                         
                                       
                                        
                                       
                                         A 
                                         22 
                                       
                                        
                                       
                                         A 
                                         23 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       β 
                                     
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                          
                         
                           Δ 
                           
                             - 
                             1 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     and 
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           D 
                            
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               ω 
                             
                             ) 
                           
                         
                         
                           
                             P 
                             S 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           { 
                           
                             
                               4 
                                
                               
                                 A 
                                 
                                   11 
                                    
                                   s 
                                 
                               
                                
                               
                                 A 
                                 22 
                               
                                
                               
                                 
                                   A 
                                   23 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         exp 
                                          
                                         
                                           ( 
                                           
                                              
                                              
                                             
                                                 
                                             
                                              
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             + 
                             
                               4 
                                
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         A 
                                         14 
                                       
                                        
                                       
                                         A 
                                         22 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         A 
                                         
                                           11 
                                            
                                           f 
                                         
                                       
                                        
                                       
                                         A 
                                         22 
                                       
                                        
                                       
                                         A 
                                         23 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   ( 
                                   
                                     β 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                          
                         
                           Δ 
                           
                             - 
                             1 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     where 
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     = 
                     
                       
                         Δ 
                         1 
                       
                       + 
                       
                         Δ 
                         2 
                       
                       + 
                       
                         Δ 
                         3 
                       
                       + 
                       
                         Δ 
                         4 
                       
                       + 
                       
                         Δ 
                         5 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       1 
                     
                     = 
                     
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               α 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             β 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 11 
                               
                                
                               
                                 A 
                                 23 
                               
                             
                             + 
                             
                               
                                 A 
                                 14 
                               
                                
                               
                                 A 
                                 22 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       2 
                     
                     = 
                     
                       
                         - 
                         
                           exp 
                            
                           
                             ( 
                             
                               
                                 - 
                                 α 
                               
                                
                               
                                   
                               
                                
                               a 
                             
                             ) 
                           
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               β 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 11 
                               
                                
                               
                                 A 
                                 23 
                               
                             
                             + 
                             
                               
                                 A 
                                 13 
                               
                                
                               
                                 A 
                                 22 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       3 
                     
                     = 
                     
                       
                         - 
                         
                           exp 
                            
                           
                             ( 
                             
                               α 
                                
                               
                                   
                               
                                
                               a 
                             
                             ) 
                           
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                              
                              
                             
                                 
                             
                              
                             β 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 12 
                               
                                
                               
                                 A 
                                 23 
                               
                             
                             - 
                             
                               
                                 A 
                                 14 
                               
                                
                               
                                 A 
                                 22 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       4 
                     
                     = 
                     
                       
                         exp 
                          
                         
                           ( 
                           
                             α 
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               β 
                             
                              
                             
                                 
                             
                              
                             a 
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 12 
                               
                                
                               
                                 A 
                                 23 
                               
                             
                             - 
                             
                               
                                 A 
                                 13 
                               
                                
                               
                                 A 
                                 22 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     S 
                   
                   = 
                   
                     
                       - 
                       8 
                     
                      
                     
                       A 
                       
                         11 
                          
                         s 
                       
                     
                      
                     
                       A 
                       
                         13 
                          
                         s 
                       
                     
                      
                     
                       A 
                       22 
                     
                      
                     
                       
                         A 
                         23 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
         [0037]    The transfer function between the wall motion in the z direction at z=a and the wall motion in the z direction at z=b (=0) is now written using equations (35), (72), (73), (74), and (75). Additionally, the individual terms from the matrix A are inserted into the expression resulting in 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               T 
                               ba 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   b 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   a 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             = 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             κ 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 k 
                                                 x 
                                               
                                               , 
                                               ω 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               α 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               β 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 κ 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     x 
                                                   
                                                   , 
                                                   ω 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   α 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 κ 
                                                 3 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     x 
                                                   
                                                   , 
                                                   ω 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   α 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       κ 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           x 
                                         
                                         , 
                                         ω 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         α 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       κ 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           x 
                                         
                                         , 
                                         ω 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         β 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         κ 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                     = 
                     
                       γ 
                        
                       
                         ( 
                         
                           4 
                            
                           ρβα 
                            
                           
                               
                           
                            
                           
                             k 
                             x 
                             2 
                           
                            
                           
                             ω 
                             2 
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         κ 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         γ 
                          
                         
                           ( 
                           
                             
                               
                                 ω 
                                 2 
                               
                                
                               ρ 
                             
                             - 
                             
                               2 
                                
                               μ 
                                
                               
                                   
                               
                                
                               
                                 k 
                                 x 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             β 
                             4 
                           
                           - 
                           
                             k 
                             x 
                             4 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       κ 
                       3 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         αρ 
                         r 
                       
                     
                      
                     
                       
                         
                           ω 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               β 
                               4 
                             
                             + 
                             
                               2 
                                
                               
                                 β 
                                 2 
                               
                                
                               
                                 k 
                                 x 
                                 2 
                               
                             
                             + 
                             
                               k 
                               x 
                               4 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Further manipulation of equation (82) results in 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               T 
                               ba 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   b 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   a 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             = 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               α 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               β 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         [ 
                                         
                                           
                                             
                                               M 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     x 
                                                   
                                                   , 
                                                   ω 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   α 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             
                                               N 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     x 
                                                   
                                                   , 
                                                   ω 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   α 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       α 
                                        
                                       
                                           
                                       
                                        
                                       a 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     M 
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           x 
                                         
                                         , 
                                         ω 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         β 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       M 
                        
                       
                         ( 
                         
                           
                             k 
                             x 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           κ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               x 
                             
                             , 
                             ω 
                           
                           ) 
                         
                       
                       
                         
                           κ 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               x 
                             
                             , 
                             ω 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                      
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           κ 
                           3 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               x 
                             
                             , 
                             ω 
                           
                           ) 
                         
                       
                       
                         
                           κ 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               x 
                             
                             , 
                             ω 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equations (86), (87), and (88) are a mathematical model of the ratio of wall motion of the test specimen. These equations are written so that the transfer function (or experimental data) is a function of material properties. They will be combined in such a manner that the material properties become functions of the experimental data. This process is explained in the next section. 
         [0038]    For completeness, it is noted that the reflected acoustic field on the projector side is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       R 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   2 
                                 
                                  
                                 
                                   ρ 
                                   f 
                                 
                               
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 γ 
                               
                             
                             ) 
                           
                            
                           
                             
                               U 
                               z 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 b 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                         
                         + 
                         1 
                       
                       ] 
                     
                      
                     
                       
                         exp 
                          
                         
                           ( 
                           
                             γ 
                              
                             
                                 
                             
                              
                             
                               z 
                               b 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where z b  is the position where the field is evaluated (m). The total pressure field on the projector side is a sum of the reflected field and the phase shifted source level written as 
         [0000]        P   Total ( k   x ,ω)= P   R ( k   x ,ω)+ P   S (ω)exp(− iγz   b ).   (90) 
         [0000]    The transmitted pressure field on the opposite side of the projector is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 
                                   ω 
                                   2 
                                 
                               
                                
                               
                                 ρ 
                                 f 
                               
                             
                             γ 
                           
                           ) 
                         
                          
                         
                           
                             U 
                             z 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               a 
                               , 
                               ω 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                      
                     
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               γ 
                             
                              
                             
                                 
                             
                              
                             
                               z 
                               a 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where z a  is the position where the field is evaluated (m). The insertion loss is then calculated using 
         [0000]    
       
         
           
             
               
                 
                   
                     IL 
                      
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     20 
                      
                     
                         
                     
                      
                     
                       
                         
                           log 
                           10 
                         
                          
                         
                           [ 
                           
                             
                               
                                 P 
                                 S 
                               
                                
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                             
                               
                                 P 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where IL(k x ,ω) is in units of decibels. These measurements are not necessary for the calculation of material properties according to the invention. z a  and z b  are the positions of hydrophones  14  and  16 . 
         [0039]    Applicant&#39;s measurement method is a two step method. In the first step, projector  12  provides acoustic waves to the sample at zero wavenumber. In view of equation (7), this means that the projector is oriented to provide acoustic waves at an angle θ of 0. In the second step, projector  12  provides acoustic waves to the sample at a non-zero wavenumber. This means that the projector is oriented to project acoustic waves at any angle θ other than 0. 
         [0040]    The first part of the measurement method involves insonifying two separate pieces of the material at zero wavenumber. The second piece of material is twice as thick as the first piece of material. For zero wavenumber, equation (82) reduces to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         ba 
                       
                        
                       
                         ( 
                         
                           0 
                           , 
                           ω 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             U 
                             z 
                           
                            
                           
                             ( 
                             
                               0 
                               , 
                               b 
                               , 
                               ω 
                             
                             ) 
                           
                         
                         
                           
                             U 
                             z 
                           
                            
                           
                             ( 
                             
                               0 
                               , 
                               a 
                               , 
                               ω 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               α 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       ρ 
                                       f 
                                     
                                      
                                     
                                       c 
                                       f 
                                     
                                   
                                   ωρ 
                                 
                                 ] 
                               
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   a 
                                 
                                 ) 
                               
                             
                           
                         
                         = 
                         
                           
                             T 
                             1 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and, written to correspond to the to the test piece that is twice as thick, becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       
                         b 
                          
                         
                             
                         
                          
                         2 
                          
                         a 
                       
                     
                      
                     
                       ( 
                       
                         0 
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           U 
                           z 
                         
                          
                         
                           ( 
                           
                             0 
                             , 
                             b 
                             , 
                             ω 
                           
                           ) 
                         
                       
                       
                         
                           U 
                           z 
                         
                          
                         
                           ( 
                           
                             0 
                             , 
                             
                               2 
                                
                               a 
                             
                             , 
                             ω 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               2 
                                
                               α 
                                
                               
                                   
                               
                                
                               a 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             α 
                              
                             
                               [ 
                               
                                 
                                   
                                     ρ 
                                     f 
                                   
                                    
                                   
                                     c 
                                     f 
                                   
                                 
                                 ωρ 
                               
                               ] 
                             
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             T 
                             2 
                           
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T 1 (ω) and T 2 (ω) are the transfer function data from the experiment. It is noted, based on examination of equations (93) and (94), that no shear energy is excited in the structure when excitation is at zero wavenumber. Equations (93) and (94) can be combined and reduced using a double angle trigonometric expression to yield 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         
                           α 
                            
                           
                               
                           
                            
                           h 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               T 
                               2 
                             
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                           + 
                           1 
                         
                         
                           2 
                            
                           
                             
                               T 
                               1 
                             
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                       = 
                       φ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ is typically a complex valued number and h is the thickness of the first specimen (m). Equation (95) can be expanded into real and imaginary parts and solved, resulting in a value for α at every frequency in which a measurement is made. The solution to the real part of α is 
         [0000]    
       
         
           
             
               
                 
                   
                     Re 
                      
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 Arc 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 neven 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 Arc 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       - 
                                       s 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 nodd 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       where 
                     
                   
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
             
               
                 
                   
                     s 
                     = 
                     
                       
                         
                           [ 
                           
                             Re 
                              
                             
                               ( 
                               φ 
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                       
                         
                           [ 
                           
                             Im 
                              
                             
                               ( 
                               φ 
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       - 
                       
                         
                           
                             
                               { 
                               
                                 
                                   
                                     [ 
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     [ 
                                     
                                       Im 
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                               
                               } 
                             
                             2 
                           
                           - 
                           
                             { 
                             
                               
                                 
                                   2 
                                    
                                   
                                     [ 
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 2 
                               
                               - 
                               
                                 
                                   2 
                                    
                                   
                                     [ 
                                     
                                       Im 
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 2 
                               
                               - 
                               1 
                             
                             } 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   97 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and n is a non-negative integer and the capital A denotes the principal value of the inverse cosine function. The value of n is determined from the function s, which is a periodically varying cosine function with respect to frequency. At zero frequency, n is 0. Every time s cycles through π radians (180 degrees), n is increased by 1. When the solution to the real part of α is found, the solution to the imaginary part of α is then written as 
         [0000]    
       
         
           
             
               
                 
                   
                     Im 
                      
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       h 
                     
                      
                     
                       log 
                       e 
                     
                      
                     
                       
                         { 
                         
                           
                             
                               Re 
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                             
                               cos 
                                
                               
                                 [ 
                                 
                                   
                                     Re 
                                      
                                     
                                       ( 
                                       α 
                                       ) 
                                     
                                   
                                    
                                   h 
                                 
                                 ] 
                               
                             
                           
                           - 
                           
                             
                               Im 
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     Re 
                                      
                                     
                                       ( 
                                       α 
                                       ) 
                                     
                                   
                                    
                                   h 
                                 
                                 ] 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   98 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The real and imaginary parts of a from equations (96) and (98) respectively are combined to yield the complex wavenumber. Because this measurement is made at zero wavenumber (k x ≡0), this is equal to the dilatational wavenumber. Thus, the dilatational wavespeed is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     d 
                   
                   = 
                   
                     
                       ω 
                       
                         [ 
                         
                           
                             Re 
                              
                             
                               ( 
                               α 
                               ) 
                             
                           
                           + 
                           
                              
                              
                             
                                 
                             
                              
                             
                               Im 
                                
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   99 
                   ) 
                 
               
             
           
         
       
     
         [0000]    To solve for the shear wavespeed, the specimen must be excited at a nonzero wavenumber. This is done in the next section. 
         [0041]    The second part of the measurement method involves insonifying three separate pieces of the material at nonzero wavenumber. The second piece of material is twice as thick as the first piece of material, and the third piece of material is three times as thick as the first piece of material. For nonzero wavenumber, the equations corresponding to the three pieces is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               T 
                               ba 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   b 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   a 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             M 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             N 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   M 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         x 
                                       
                                       , 
                                       ω 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               R 
                               1 
                             
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   100 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               T 
                               
                                 b 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 a 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   b 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   
                                     2 
                                      
                                     a 
                                   
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             2 
                                              
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             2 
                                              
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             M 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 2 
                                                  
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             N 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 2 
                                                  
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           2 
                                            
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   M 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         x 
                                       
                                       , 
                                       ω 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       2 
                                        
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   101 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               T 
                               
                                 b 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 a 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   x 
                                 
                                 , 
                                 ω 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   b 
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                             
                               
                                 U 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   
                                     3 
                                      
                                     a 
                                   
                                   , 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             3 
                                              
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             3 
                                              
                                             β 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             M 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 3 
                                                  
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             N 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 ω 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 3 
                                                  
                                                 α 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           3 
                                            
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     3 
                                      
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   M 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         x 
                                       
                                       , 
                                       ω 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       3 
                                        
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               R 
                               3 
                             
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   102 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It is noted that the α and β wavenumbers have different values when compared to the previous section due to their modification by the nonzero spatial wavenumber k x . This dependency is shown in equations (28) and (31). Equations (100), (101), and (102) are now combined, the constants M and N are condensed out, and the sine and cosine terms are reduced using multiple angle trigonometric expressions. Additionally, it is noted that 
         [0000]      cos(β a )=cos(α a )   (103) 
         [0000]    is one of the solutions to the resulting expression and this term is factored out because it is extraneous. This results in 
         [0000]        U ( k   x ,ω)cos 2 (β h )+ V ( k   x ,ω)cos(β h )+ W ( k   x ,ω)=0,   (104) 
         [0000]    where the constants U, V, and W, are, written with the wavenumber and frequency dependence suppressed, equal to 
         [0000]        U= 4 R   1 └4 R   2  cos 2 (α a )−2 R   3  cos(α a )− R   2 −1┘,   (105) 
         [0000]        V =2[−2 R   1  cos(α a )+R 2 +1][2 R   3  cos(α a )+1],   (106) 
         [0000]      and 
         [0000]        W =( R   2 +1)└−4 R   1  cos 2 (α a )+2 cos(α a )+ R   1   +R   3 ].   (107) 
         [0000]    where α was determined with equation (28) using the values of c d  calculated in the previous section. Equation (104) can be solved as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         
                           β 
                            
                           
                               
                           
                            
                           h 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             - 
                             V 
                           
                           + 
                           
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 4 
                                  
                                 UW 
                               
                             
                           
                         
                         
                           2 
                            
                           U 
                         
                       
                       = 
                       
                         ϕ 
                         + 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   108 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         
                           β 
                            
                           
                               
                           
                            
                           h 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             - 
                             V 
                           
                           - 
                           
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 4 
                                  
                                 UW 
                               
                             
                           
                         
                         
                           2 
                            
                           U 
                         
                       
                       = 
                       
                         ϕ 
                         - 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   109 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ +  and φ −  are typically a complex valued numbers. Two values of φ are present but only one is the correct number. At zero (and very low) frequency, the φ value closest to unity is the correct one to use. As frequency increases, every time the angle of the discriminant in equation (108) passes through π radians, the value of φ changes from equation (108) to equation (109) or vice versa. Once the correct value of φ is known, equation (108) or (109) can be expanded into real and imaginary parts and solved, resulting in a value for β at every frequency in which a measurement is made. The solution to the real part of β is 
         [0000]    
       
         
           
             
               
                 
                   
                     Re 
                      
                     
                       ( 
                       β 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 Arc 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     r 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 meven 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 Arc 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       - 
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     2 
                                      
                                     h 
                                   
                                 
                                  
                                 modd 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       where 
                     
                   
                 
               
               
                 
                   ( 
                   110 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     = 
                     
                       
                         
                           [ 
                           
                             Re 
                              
                             
                               ( 
                               ϕ 
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                       
                         
                           [ 
                           
                             Im 
                              
                             
                               ( 
                               ϕ 
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       - 
                       
                         
                           
                             
                               { 
                               
                                 
                                   
                                     [ 
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     [ 
                                     
                                       Im 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                               
                               } 
                             
                             2 
                           
                           - 
                           
                             { 
                             
                               
                                 
                                   2 
                                    
                                   
                                     [ 
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 2 
                               
                               - 
                               
                                 
                                   2 
                                    
                                   
                                     [ 
                                     
                                       Im 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 2 
                               
                               - 
                               1 
                             
                             } 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   111 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and m is a non-negative integer and the capital A denotes the principal value of the inverse cosine function. The value of m is determined from the function r, which is a periodically varying cosine function with respect to frequency. At zero frequency, m is 0. Every time r cycles through π radians (180 degrees), m is increased by 1. When the solution to the real part of β is found, the solution to the imaginary part of β is then written as 
         [0000]    
       
         
           
             
               
                 
                   
                     Im 
                      
                     
                       ( 
                       β 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       h 
                     
                      
                     
                       log 
                       e 
                     
                      
                     
                       
                         { 
                         
                           
                             
                               Re 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                             
                               cos 
                                
                               
                                 [ 
                                 
                                   
                                     Re 
                                      
                                     
                                       ( 
                                       β 
                                       ) 
                                     
                                   
                                    
                                   h 
                                 
                                 ] 
                               
                             
                           
                           - 
                           
                             
                               Im 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     Re 
                                      
                                     
                                       ( 
                                       β 
                                       ) 
                                     
                                   
                                    
                                   h 
                                 
                                 ] 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   112 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The real and imaginary parts of β from equations (110) and (112) respectively are combined to yield the complex wavenumber. Because this measurement is made at nonzero wavenumber, this has to be modified by the spatial wavenumber k x  to calculate the shear wavenumber. This equation is 
         [0000]        k   s =√{square root over (β 2   +k   x   2 )}.   (113) 
         [0000]    The shear wavespeed is then calculated using 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     s 
                   
                   = 
                   
                     
                       ω 
                       
                         k 
                         s 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   114 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Once the dilatational and shear wavespeeds are known, the Lamé constants or Young&#39;s modulus, shear modulus, and Poisson&#39;s ratio can also be calculated. A numerical example of all these calculations is included below. 
         [0042]    The above measurement method can be simulated by means of a numerical example. Soft rubber-like material properties are used in this simulation. The material has a Young&#39;s modulus E of {1e7(1−0.20i)[1+(1e−4)f]}N/m 2  where f is frequency in Hz, Poisson&#39;s ratio υ equal to 0.45 (dimensionless), and a density of ρ equal to 1200 kg/m 3 . The base thickness of the material h is 0.01 m, the other transfer functions (subscripts 2 and 3) are calculated using two and three times this value. The water has a density ρ f  of 1025 kg/m 3  and a compressional (acoustic) wave velocity of c f  of 1500 m/s. All other parameters can be calculated from these values. 
         [0043]      FIGS. 3A and 3B  are plots of transfer function of normal wall motion at z=b divided by normal wall motion at z=a versus frequency at zero wavenumber (θ=0°). The x&#39;s correspond to h=0.01 m thickness and the +&#39;s correspond to h=0.02 m thickness.  FIG. 3A  is the magnitude, and  FIG. 3B  is the phase angle. These functions are listed above as equations (93) and (94), respectively.  FIG. 4  is a plot of the function s versus frequency and corresponds to equation (97). The values of n in equation (96) can be determined from inspection of  FIG. 4  and are listed in Table 1, below. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                   
                 Minimum 
                 Maximum 
               
               
                   
                 Frequency 
                 Frequency 
               
               
                 n 
                 (Hz) 
                 (Hz) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 5660 
               
               
                 1 
                 5660 
                 10000 
               
               
                   
               
             
          
         
       
     
         [0044]      FIGS. 5A and 5B  are plots of the actual and estimated values of wavenumber a versus frequency.  FIG. 5A  is the real part, and  FIG. 5B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers.  FIGS. 6A and 6B  are plots of the actual and estimated values of dilatational wavespeed versus frequency.  FIG. 6A  is the real part, and  FIG. 6B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers. 
         [0045]      FIGS. 7A and 7B  are plots of transfer function of normal wall motion at z=b divided by normal wall motion at z=a versus frequency at wavenumbers corresponding to an insonifcation angle of 15 degrees (θ=15+). The x&#39;s correspond to h=0.01 m thickness, the +&#39;s correspond to h=0.02 m thickness, and the o&#39;s correspond to h=0.03 m.  FIG. 7A  is the magnitude, and  FIG. 7B  is the phase angle. These functions are listed above as equations (100), (101), and (102), respectively.  FIG. 8  is a plot of the function r (solid line with markers) and the angle of the discriminant (dashed line) versus frequency and corresponds to equation (111) and (108) respectively. Also included in this plot is the function r calculated using φ, (equation 108) and .φ −  (equation 109) so that the interchange relationship between these two functions and the discriminant can be illustrated. The values of m in equation (110) can be determined from inspection of  FIG. 8  and are listed in Table 2, below. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                   
                 Minimum 
                 Maximum 
               
               
                   
                 Frequency 
                 Frequency 
               
               
                 m 
                 (Hz) 
                 (Hz) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 1460 
               
               
                 1 
                 1460 
                 3110 
               
               
                 2 
                 3110 
                 5000 
               
               
                 3 
                 5000 
                 7120 
               
               
                 4 
                 7120 
                 9500 
               
               
                 5 
                 9500 
                 10000 
               
               
                   
               
             
          
         
       
     
         [0046]      FIGS. 9A and 9B  are plots of the actual and estimated values of wavenumber β versus frequency.  FIG. 9A  is the real part, and  FIG. 9B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers.  FIGS. 10A and 10B  are plots of the actual and estimated values of shear wavespeed versus frequency.  FIG. 10A  is the real part, and  FIG. 10B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers. 
         [0047]    Finally, the material properties can be determined from the wavespeeds. The Lamé constants are calculated with equations (21) and (22) written as 
         [0000]      μ=ρc s   2    (115) 
         [0000]      and 
         [0000]      λ=ρ c   d   2 −2ρ c   s   2 .   (116) 
         [0000]    Alternatively, shear modulus, Poisson&#39;s ratio, and Young&#39;s modulus and can be calculated using equations (23), (24), and (115) which results in 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       ≡ 
                       μ 
                     
                     = 
                     
                       ρ 
                        
                       
                           
                       
                        
                       
                         c 
                         s 
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   117 
                   ) 
                 
               
             
             
               
                 
                   
                     υ 
                     = 
                     
                       λ 
                       
                         2 
                          
                         
                           ( 
                           
                             μ 
                             + 
                             λ 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   118 
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     = 
                     
                       
                         2 
                          
                         
                           μ 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 μ 
                               
                               + 
                               
                                 3 
                                  
                                 λ 
                               
                             
                             ) 
                           
                         
                       
                       
                         2 
                          
                         
                           ( 
                           
                             μ 
                             + 
                             λ 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   119 
                   ) 
                 
               
             
           
         
       
     
         [0000]    respectively.  FIGS. 11A and 11B  are plots of the actual and estimated values of Lamé constant μ versus frequency.  FIG. 11A  is the real part, and  FIG. 11B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers. This corresponds to equation (115).  FIGS. 12A and 12B  are plots of the actual and estimated values of Lamé constant λ versus frequency.  FIG. 12A  is the real part, and  FIG. 12B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers. This corresponds to equation (116). The shear modulus G is identical to the Lamé constant μ and therefore is not plotted. Estimation of Poisson&#39;s ratio υ yields a value of 0.45 (dimensionless). Because this is a constant with respect to frequency, it is not shown as a figure.  FIGS. 13A and 13B  are plots of the actual and estimated values of Young&#39;s modulus E versus frequency.  FIG. 13A  is the real part, and  FIG. 13B  is the imaginary part. The actual values are shown with a solid line and the estimated values are depicted with square markers. This corresponds to equation (119). 
         [0048]    In light of the above, it is therefore understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described.