Abstract:
The invention as disclosed is a fiber optic interferometric directional acoustic density sensor that increases the directionality of a vector sensor that is much smaller in size than the wave length of an acoustic wave. This is accomplished through the use of second order directionality by measuring the acoustic fluctuations of fluid density at a point, wherein the acoustic density fluctuations are determined according to the principles of fluid compressibility and conservation of mass using a density fluctuation measuring apparatus that restricts two of the three vector components of the particle velocity of the acoustic wave and that employs a laser interferometer to measure the fluid density fluctuation along the remaining vector component.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     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 REFERENCES TO OTHER PATENT APPLICATIONS 
     None. 
     BACKGROUND OF THE INVENTION 
     (1) Field of the Invention 
     The present invention is directed to acoustic vector sensors. In particular, the present invention is directed to increasing the directionality of a sensor having a size much less than the wave length of an acoustic wave, by measuring the acoustic fluctuations of fluid density at a point. 
     (2) Description of the Prior Art 
     Conventional vector sensors measure particle velocity, v (v x , v y , v z ), associated with an acoustic wave. The measurements of a vector quantity (velocity) instead of a scalar quantity (pressure) allows for directional sensing even if the size of the vector sensor is much smaller than the size of the wavelength of the acoustic wave. The directionality pattern of a vector sensor is proportional to cos(θ), where θ is the directional angle of the vector sensor. Vector sensor directionality is equivalent to the dipole-type or first order sensor that is realized by measuring particle velocity at a point, (which is the vector sensor sensing approach for underwater sensors), or by measuring the gradient of the acoustic pressure at two closely spaced (less than the wavelength of an acoustic wave) points as it is commonly done in air acoustics where this type of sensor is called an acoustic intensity probe. The two approaches of obtaining vector sensor directionality as described above are in fact mathematically equivalent according to a linearized equation of momentum conservation as expressed in Eq. (1) below: 
                     ρ   0     =           ∂   v       ∂   t       +     ▽   ⁢           ⁢   p       =   0             (   1   )               
Here ρ 0  is the undisturbed fluid density and is the ∇p gradient of the acoustic pressure. For a plane wave propagating in the x-direction Eq. (1) can be re-written as Eq. (2) below:
 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       0 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         v 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∂ 
                         p 
                       
                       
                         ∂ 
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     There continues to be a need to increase the directionality of a vector sensor. One approach to further increase the directionality of a vector sensor while still maintaining a sensor size that is much smaller than the wavelength of an acoustic wave, is to utilize a second order or quadruple-type sensor arrangement measuring the diversion of the velocity, div v. This can be accomplished by measuring the components of the velocity, v (v x , v y , v z ), at closely spaced points (at least two points) along the corresponding axis yielding the directionality, which is proportional to cos 2 (θ). The benefit of the increased directionality, however, when measured using this approach comes at a cost of the number of vector sensors necessary and at a cost of a reduction in sensitivity. Two vector sensors for each axis (a minimum of six sensors) are required to measure the diversion of the velocity and there is a significant reduction in sensor sensitivity proportional to kd&lt;&lt;1, where k is the wave-number and d is the spacing between the two vector sensors. 
     A more advantageous means to achieve a second order directionality is to measure the acoustic fluctuations of fluid density at a point rather than directly measuring the diversion of the velocity. 
     SUMMARY OF THE INVENTION 
     It is a general purpose and object of the present invention to increase the directionality of a vector sensor having a size much less than the wave length of an acoustic wave. 
     The above object is accomplished with the present invention through the use of second order directionality by measuring the acoustic fluctuations of fluid density at a point, wherein the acoustic density fluctuations are determined according to the principles of fluid compressibility and conservation of mass using a density fluctuation measuring apparatus that restricts two of the three vector components of the particle velocity of the acoustic wave and that employs a laser interferometer to measure the fluid density fluctuation along the remaining vector component. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the invention and many of the attendant advantages thereto will be more readily appreciated by referring to the following detailed description when considered in conjunction with the accompanying drawings, wherein like reference numerals refer to like parts and wherein: 
         FIG. 1  illustrates a measuring cell that restricts two of the three vector components of velocity; and 
         FIG. 2  illustrates a density fluctuation measuring apparatus that employs a laser interferometer. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Acoustic density fluctuations are determined by fluid compressibility and conservation of mass and are respectively described by the following two linearized equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ρ 
                         
                           ρ 
                           0 
                         
                       
                       = 
                       
                         
                           1 
                           B 
                         
                         ⁢ 
                         p 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     and 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       1 
                       
                         ρ 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         ρ 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           ∂ 
                           
                             v 
                             x 
                           
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                     - 
                     
                       
                         ∂ 
                         
                           v 
                           y 
                         
                       
                       
                         ∂ 
                         y 
                       
                     
                     - 
                     
                       
                         ∂ 
                         
                           v 
                           z 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where ρ=ρ total −ρ 0  is the density disturbance due to an acoustic wave and B is the fluid bulk modulus. Equation (4), above, provides the basis for a second order vector sensor utilizing density acoustic fluctuation measurements provided that the measured fluctuations are only associated with one component of 
               div   ⁢           ⁢     v   →       =         ∂     v   x         ∂   x       +       ∂     v   y         ∂   y       +       ∂     v   z         ∂   z               
(i.e., the right hand side of Equation (4)). This is achieved by using a measuring cell  10  that restricts two of the spatial vector components of particle velocity associated with the acoustic wave. Such a measuring cell  10  is illustrated in the  FIG. 1 .
 
     In the measuring cell  10  illustrated in  FIG. 1 , the fluid flow is restricted in y and z directions, while fluid flow is permitted along the x axis. Correspondingly, in terms of the fluid flow within this cell: 
                   ∂     v   y         ∂   y       =         ∂     v   z         ∂   z       =   0       ,         
leaving Eq. (4) in a form similar to Eq. (2) and similarly forming the basis for a new (second order) vector sensor equation:
 
                       1     ρ   0       ⁢       ∂   ρ       ∂   t         =     -       ∂     v   x         ∂   x                 (   5   )               
Taking the time derivative of the Eq. (3) and combining it with Eq. (5) yields the equation for the total density fluctuations:
 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       ρ 
                     
                     
                       ∂ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         
                           ρ 
                           0 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     - 
                     
                       
                         
                           ρ 
                           0 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             x 
                           
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     For a plane harmonic wave propagating along an arbitrary direction, r: 
                     p   =       P   0     ⁢     ⅇ     j   ⁡     (       ω   ⁢           ⁢   t     -       k   →     *     r   →         )             ,       and   ⁢           ⁢     v   x       =         P   0         ρ   0     ⁢     c   0         ⁢   cos   ⁢           ⁢   θ   ⁢           ⁢     ⅇ     j   ⁡     (       ω   ⁢           ⁢   t     -     kx   ⁢           ⁢   cos   ⁢           ⁢   θ       )             ,           (   7   )               
where P 0  is the acoustic pressure amplitude, c 0 =√(B/ρ 0 ) is the speed of the acoustic wave in fluid, ω is angular frequency of the acoustic plane-wave, t is unit of time, vector k is the acoustic wave-vector, vector r is the positional vector in the Cartesian coordinates of the reference frame, x is the spatial coordinate along the x-axis, and j is the square root of negative one.
 
     Substituting Eq. (7) into Eq. (6) a solution for the total density fluctuation is obtained: 
                        ρ        =         P   0       2   ⁢     c   0   2         +         P   0       2   ⁢     c   0   2         ⁢     cos   2     ⁢   θ               (   8   )               
Here the first and second terms on the right hand side of Eq. (8) are determined by the Eq. (3) and (5), respectively.
 
     In order to achieve a desirable second order directionality while preserving the high sensitivity of the measurements the directly measured density fluctuation is determined by only the second term in Eq. (8): 
                 P   0       2   ⁢     c   0   2         ⁢         cos   2     ⁡     (   θ   )       .           
In a preferred embodiment an optical interferometer scheme is implemented to directly measure the density fluctuation.
 
     An example of the optical interferometer apparatus  50  of the present invention is illustrated in the  FIG. 2 . In a preferred embodiment there is a measuring cell  10  (the measuring cell  10  is rectangular, as previously illustrated in  FIG. 1 ) with acoustically rigid walls (i.e., the walls do not move or flex in any way due to the incident acoustic wave). Measuring cell  10  is filled with water or another fluid medium. In an alternative embodiment the measuring cell  10  has a circular cross-section. Inside of measuring cell  10  there are two laser beams a and b that originate from the same coherent optical light source  12 . The light from light source  12  is split with beam splitter  13  into two perpendicular beams: beam a is along the cell axis x (measuring beam) and beam b, perpendicular to the axis x (reference beam). The in-fluid lengths of both beams are equal. Both beams are collected into one end of separate optical fiber cables  14  with beam to fiber couplers  15  at the ends of the cables  14  that collect the beams. The optical fiber cables  14  are connected at their other end to a mixing cell  16  that is equipped with a photodiode  18 . The two laser beams a and b are mixed in a mixing cell  16 . The lengths of fibers  14  are tuned to create constructive or destructive interference inside the mixing cell  16 . The mixing cell  16  is joined to a data acquisition and digital processing unit  20  that is programmable. Because the light index of refraction, n, is highly sensitive to the density fluctuations of the medium, any difference in density fluctuations along beam a (as opposed to beam b) can be detected by the data acquisition and digital processing unit  20  with high precision. 
     If the incident acoustic wave travels perpendicular to axis x, that is cos(θ)=0, the density change along both beams a and b will be the same and determined only by the first (scalar) term of Eq. (8). Consequently, there is no difference in the density fluctuations along the beams. This results in zero output of the interferometer. The maximum difference will be achieved if the acoustic wave travels along axis x, that is, cos(θ)=1. In this scenario, the density change along beam a will be twice as high as along beam b. For an arbitrary direction of the incident acoustic wave, the change in density will be proportional to the second term in Eq. (8). 
     The interferometer apparatus  50  will measure phase difference, Δφ, between the reference beam b and measuring beam a:
 
Δφ≅(2π/λ light ) L (Δ C   ab   /C   b ),  (9)
 
where λ light  is the light wavelength in the fluid, L is the length of the either beam in the fluid, C b  is the speed of light in the fluid along beam b, ΔC ab =C a −C b  is the difference in speeds of lights along beams a and b determined by the difference in the refraction indexes, n a  and n b , along the respective beams:
 
Δ C   a,b   =C   V   /n   a,b   (10)
 
The density dependence of the refraction index for light in water is determined by the following formula, (assuming ρ/ρ 0 &lt;&lt;1, here ρ is the acoustic density disturbance);
 
                         n   2     -   1         n   2     +   2       ≅       a   0     ⁡     (       ρ     ρ   0       +   1     )               (   11   )               
where the index of refraction n=C V /C F , C F  is the speed of light in the fluid, C V ≅3·10 8  m/s is the speed of light in vacuum, and a 0  is a constant determined by temperature, light wavelength, and static pressure. Solving (11) for n:
 
                     n     a   ,   b       =           2   ⁢     a   0       +   1   +     2   ⁢       a   0     ⁡     (       ρ     a   ,   b       /     ρ   0       )             1   -     a   0     -       a   0     ⁡     (       ρ     a   ,   b       /     ρ   0       )                     (   12   )               
Here ρ a  and ρ b  are the acoustic density disturbances along the respective beams a and b. The acoustic density disturbances ρ a  and ρ b  when represented by Eq. (8) yield the following equations:
 
                     ρ   a     =         P   0       2   ⁢     c   0   2         +         P   0       2   ⁢     c   0   2         ⁢     cos   2     ⁢   θ               (     13   ⁢           ⁢   a     )                 ρ   b     =       P   0       2   ⁢     c   0   2                 (     13   ⁢           ⁢   b     )               
Taking into account that ρ a,b /ρ 0 &lt;&lt;1 and using the Taylor&#39;s expansion to transform the formulas (12) and (10) we obtain:
 
     
       
         
           
             
               
                 
                   
                     n 
                     
                       a 
                       , 
                       b 
                     
                   
                   = 
                   
                     
                       
                         
                           1 
                           + 
                           
                             2 
                             ⁢ 
                             
                               a 
                               0 
                             
                           
                         
                         
                           1 
                           - 
                           
                             a 
                             0 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           3 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             a 
                             0 
                           
                         
                         
                           2 
                           ⁢ 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             1 
                             - 
                             
                               a 
                               0 
                             
                           
                           
                             1 
                             + 
                             
                               2 
                               ⁢ 
                               
                                 a 
                                 0 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           ρ 
                           
                             a 
                             , 
                             b 
                           
                         
                         
                           ρ 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       C 
                       
                         a 
                         , 
                         b 
                       
                     
                   
                   = 
                   
                     
                       C 
                       V 
                     
                     ⁢ 
                     
                       
                         1.5 
                         ⁢ 
                         
                           a 
                           0 
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 a 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     a 
                                     0 
                                   
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         P 
                         0 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ρ 
                           0 
                         
                         ⁢ 
                         
                           c 
                           0 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       cos 
                       2 
                     
                     ⁢ 
                     θ 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Expressions (9) and (15) can be used to evaluate the sensitivity of a second order vector sensor of the present invention according to the following equation: 
                     P     0   ⁢           ⁢   min       =           (     1   +     2   ⁢     a   0         )     ⁢     (     1   -     a   0       )         3   ⁢     a   0     ⁢   π       ⁢     ρ   0     ⁢     c   0   2     ⁢       λ   light     L     ⁢     Δφ   min               (   16   )               
Assuming the following parameters:
 
interferometer phase measurement accuracy Δφ min =0.25 μrad;
 
L=the beam length (L a =L b =L) 25.4 mm (1 inch);
 
c 0 =1500 m/s, ρ 0 =1000 kg/m 3 ;
 
λ light  (in water)=1 μm (typical for a laser diode);
 
a 0 =0.202 (fresh water, 20° C., atm. pressure);
 
For these parameters and for θ=0 the formula (16) yields sensitivity estimate for the proposed scheme:
 
 P   0 min ≅0.026 Pa or approximately  SPL≅ 28 dB re 1 μPa.  (17)
 
     The advantage of the present invention is that it is immune to non-acoustic density variations due to temperature and hydrostatic pressure fluctuations as both laser beams (a and b) are equally exposed to these variations. 
     While it is apparent that the illustrative embodiments of the invention disclosed herein fulfill the objectives of the present invention, it is appreciated that numerous modifications and other embodiments may be devised by those skilled in the art. Additionally, feature(s) and/or element(s) from any embodiment may be used singly or in combination with other embodiment(s). Therefore, it will be understood that the appended claims are intended to cover all such modifications and embodiments, which would come within the spirit and scope of the present invention.