For thick panels under certain conditions, continually low values of transmission loss can be shown in the high frequency range. This effect has been thought t o be due to the shear type wave coincidence effect. In this work, the dispersion properties of thick panels are precisely investigated to get a right explanation for this phenomenon and to determine conditions in which this phenomenon occurs. It is also sh own that lowering the stiffness of the panels is not always an effective approach to improving the transmission loss for thick panels though the approach is well -known to be effective for thin panels and has been also proposed for thick panels. I. INTRODUCTION The sound transmission of various kinds of panels has been extensively st udied. In previous studies, the wave speed in a panel is often treated as one of the key factors to determine the transmission behaviour of panels. The bending wave of a panel is well known as a cause of the coincidence phenomenon. In addition, for certain types of panels, such as sandwich panels, the shear wave of a panel plays an important role in determining the transmission behaviour. A simple impedance model for san dwic h panels was prepared by Kurtze & Watters[1], and this model suggests that if the sh ear wave speed in a panel coincides with the speed of sound in air, it causes a drastic decrease in the transmiss ion loss (TL). Many researchers have treated this type of coincidence [2 -6]. A similar type of phenomenon exists in thick single layer panels. Sharp presented a simple impedance model for thick panels to explain this phenomenon[7]. In this model, shear wav e impedance has been treated as a key factor for explai ning for the transmission characteristics of thick panels, but this model cannot provide a proper explanation for characteristics peculiar to the high frequency range transmission loss of thick panels and this model does not provide proper measures for imp roving the noise reduction capability of panels. In this work, the aim is to understand properly the transmission characteristics of thick panels and to provide the a proper noise control measure for them. For an infinite elastic panel, the analytical solution for TL exist [8 -10]. Based on this solution, the dispersion properties of thick pa nels are precisely investigated, and the exact conditions under which the high frequency range TL decreases most drastically are shown. And It is also shown that loweri ng the stiffness of the panels is not an effective approach to improving the transmissio n loss for thick panels though the approach is well -known to be effective for thin panels and it has been also proposed for thick panels. II. SHARP’S IMPEDANCE MODEL F OR THICK PANELS Sharp provided the following impedance model for thick panels [7]. (1) ZB and Z S are respectively called the bending wave impedance and the shear wave impedance and are described as follows. (2) where E is the Young’s modulus, ν is the Poisson ratio , ω is the angular frequency, θ is the sound incident angle, c0 is the speed of sound in air, G is the second Lame coefficient, namely, the shear modulus, and h is the panel thickness. The forcing in -plane wave sp eed c is given by c=c0/sinθ Using this relation and letting the i mpedance Z, shown as equation (1), be 0, the dispersion curve, which shows the dependence of the generated wave spe ed in a panel on the frequency of an input force, can b e obtained. Sharp discussed the TL characteristics of three hypothetical panels with d ifferent properties [7]. Thus, the TL characteristics of panels A, B and C, shown in Table 1, will be discussed. Pa nels A, B and C have the shear wave speed S BSB mZ ZZZhωρ Z++j 2 024 04 3 23 sinjsin )- 12(1j cθhG- Zcθω νEh - Z SB =  = ρG respectively higher than, the same as and lower than the forcing in -plane wa ve speed caused by sound incidence at the angle of 78 °. An angle of 78° is the upper limit angle for estimating the field incident TL, and the angle that characterises th e field incidence TL has physical meaning in a sense [11,12] and was therefore selecte d for the incident angle in this investigation. The dispersion curves obtained for these panels are shown in Figur e 1. In the high frequency range, the shear wave is do minant, and the wave propagating speed in the panel is constant. If the wave propagati ng speed coincides with the forcing wave speed by the sound incidence, as in the case of panel B, it is expected th at TL reduction in the high frequency range occurs broa dly, as suggested by Sharp. 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) FIG.1. Dispersion curves of panels estimated based on Sharp’s impedance model. , Panel A ; , Panel B ; ,Panel C ; ,forced wave speed by 78°incidence sound. The TL c haracteristics can be directly evaluated from the panel impedance based on the method [13] described in Appendix A. The TL curves thus obt ained for A, B and C are shown in Figure 2. In this estimation, the value 1.21 kg/m3 is used for the air density ρ0 . The loss factor η is assumed to be 0.02 using complex elasticity E(1+jη), G(1+jη) in equation (2). These results are similar to the schema tic graph prepared by Sharp [7]. For panel B, in which the wave propagating speed coincides with the forcing wave speed due to the sound incidence, TL reduction in the high frequency range is observed. On the other hand, for panel C, in which the shear sti ffness is sufficiently low, there is no dip in the TL curve. 20304050607080 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.2. TL properties estimated based on Sharp’s im pedance model. , Panel A ; , Panel B ; , Panel C. III. EVALUATION OF SHARP’S MODEL BASED ON ANALYTICAL SOLUTIONS For an infinite elastic pa nel, an analytical solution for the TL exists. The details of the analytical model are shown in Appendix B. This model is equivalent to the existing approaches [8 -10]. Thus, Sharp’s impedance model can be evaluated by comparison with this analytical model. Firstly, the dispersion characteristics will be investigated. The dispersion curve can be obtained for a panel by letting Z , the (1,2) component of the (B -6) matrix in Appendix B, be 0. The dispersion curve obtained for panel B with such a procedure is s hown in Figure 3. Precisely, the graph shows the counter of Z/1 . The branches in the dispersion curve can be observed in it. The overall aspects of the branches are similar to the results obtained previously for a thick panel [14]. In t he high frequency range, part of the constant wave speed is observed in this curve. In the figure, the dispersion curve based on Sharp’s impedance model is overwritten. The velocity of the flat part in this dispersion curve is slightly lower than that obta ined from Sharp’s impedance. In the high velocity range, there are many branches that cannot be observed in the dispersion curve given by Sharp’s impedance. These features are common for dispersion curves of single -layer thick panels that have the flat par t of the dispersion curves in the audible frequency range. 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz)The velosity of the flat part esdimated based on Sharp's impedance model The velosity of the flat part esdimated based on the analytical model FIG.3. Dispersion curve of panel B estimated based on the analytical solution shown as the inverse of the impedance and the estimation based on Sharp’s impedance model. . A. DISPERSION CURV E CHARACTERISTICS The velocity of flat portion in the dispersion curve coincides with the Rayleigh velocity[14]. This is also shown as follows. In Appendix B, the detailed expression of the (1,2) component of the (B -6) matrix is shown. The following K0 part determines the sign of this component. The sign of the other parts are constant in the condition in this investigation. (3) where K1 and K2 are as follows. ()  ()() 22 2 2 1 222 2 2 1 0 - 8 - - 4 βξαβξ KKβξαβξ K K + + = ()()hβ hα K sin sin1= ()()hβ hα K cos cos-12= K0 mainly consists of the real part with a negative sign. In the velocity range in whic h the flat part in the dispersion curve exists, the sign of K0 changes to be positive only in a very local velocity range. The first term in K0 usually has a minus sign, and it becomes 0 only at the Rayleigh velocity [14]. The second term is always a very small, positive value. Thus, at the two points of velocity near the Rayleigh velocity, the sign of K0 changes, and the (1,2) component becomes 0. In Fig. 3 the two separated branches are not observed clearly. However, it can been seen that the zone in whic h the impedance is zero exists near the Rayleigh velocity. When the Poisson’s ratio is 0.25, the analytic solution of the Rayleigh velocity exists. The velocity of the flat part in the dispersion curve, which implies the zero impedance region, coincides wi th the Rayleigh velocity ρG3/22− obtained analytically. When the velocity of the flat part in the dispersion curve coincides with the forcing velocity due to the incident sound, the TL reduction in the high frequency range occurs. Thus, the v elocity of the flat part is the important factor that determines the TL characteristics. In this investigation, the velocity is shown to be near the Rayleigh velocity ρG3/22− rather than near the shear wave speed ρG Sharp s uggested. In general, the Rayleigh velocity is a little less than the speed of the shear wave, and it is well -known to be approximated as ρGνν )1/()12.187.0( + + . For example, in the case that the Poisson’s ratio is 0.2, the Rayleigh velocity is about 10% less than the shear wave speed. Estimating this velocity base on Sharp’s model results in an erroneous estimation. When the ρG value with which this type of coincidence occurs is estimated, the estimated value based on Sharp’s model can be about 20% lower than true value that is based on the analytical solution. B. PLANE WAVE TRANSMISSION To show the relation between the dispersion curve in Fig. 3 and the TL characteristics, the transmission of plane waves at a single incident angle is estimated. The angle of the incident wave is arbitrary, but 78° is the angle that characterises the field incidence TL in the next section, so it was selected for the incident angle in this investigation. Five hypothetical types of panels whose properties are shown in Table 1 were selected for evaluating TL characteristics. The dispersion curves for these panels based on the analytical model are shown in Figure 4. The forcing wave velocity caused by incident sound at the angle of 78° is over -written. The fi ve types of panels have different types of relationships between the forcing wave velocity and the flat part of the velocity in the dispersion curve. The TL of each panel was calculated for 78° incident sound with a 0 loss factor condition and 0.02 loss fa ctor condition. These results are shown in Figure 5. In each case, a matrix that characterises wave transmission was given by equation (B -6) in Appendix B. Then, TL is calculated with equation (A -3) in Appendix A. It is clearer in the 0 loss factor conditi on that Type B’, which has a Rayleigh wave speed coincident with the forcing wave speed, has a wider frequency range of the dip than type B, which has a shear wave speed that is the same as the forcing wave speed. Nearly the same tendency is confirmed in t he case with the 0.02 loss factor. In contrast with Sharp’s results, shown in Figure 2, many dips in TL are observed for panel C, which is in the condition where the velocity of the flat part of the dispersion curve is sufficiently lower than the forcing w ave speed because of the many branches in the higher velocity range over the flat part in the dispersion curve. TABLE I. Panel Properties Type G(N/m2) Wave Speed Condition A 3.491e8 )78sin(/0 c ρG B’ 1.746e8 )78sin(/ ) 322(0 =− c ρG B” 1.602e 8 )78sin(/2) 322( 1 0 =−+ c ρG B 1.476e8 )78sin(/0 =c ρG C 7.378e7 )78sin(/0 c ρG Note: ρ=1200kg/m2 , h=0.15m , ν=0.25 , c 0=343m/s 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) (a) Estimation for panel A 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) (b) Estimation for panel B’ 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) (c) Estimation for panel B” 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) (d) Estimation for panel B 0100200300400500600 0 2000 4000 6000 8000 10000Wave Speed (m/s) Frequency (Hz) (e) Estimation for panel C FIG.4. Dispersion curves of panels estimated based on t he anal ytical solution shown as the inverse of the impedance and forced wave speed with 78°incidence sound. . 0102030405060708090100 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.5. TL properties estimated based on the analytical solution for 78°sound incidence with the panel loss factor η=0. , Panel A ; , Panel B ’ ; , Panel B” ; , Panel B ; , Panel C. 0102030405060708090100 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.6. TL properties estimated based on the analytical solution for 78°sound incidence with the panel loss factor η=0.02. , Panel A ; , Panel B’ ; , Panel B” ; , Panel B ; , Panel C. C.TRANSMISSION IN DI FFUSE FIE LD TL characteristics in the diffuse field, which are more important for practical problems, should be considered. The TL characteristics of the five kinds of panels shown in Table 1 were calculated in the field incidence condition based on the me thod show n in Appendix A. The loss factor was assumed to be 0.02, and the increment of incident angles for evaluating the average transmission ratio was 0.1°. The results are shown in Figure 6. Similar to the results of the previous section, in the TL of t ype B’, t he most visible TL reduction is shown. In contrast to Sharp’s results, many dips exist in the TL for type C, where the shear stiffness is sufficiently low. For sound transmission in the diffuse field, the influence of shallow incident angle condit ions, in which the transmission coefficients are higher, is more dominant, and thus the TL in the field incidence condition has the same tendency as that in the plane wave incident condition at the incident angle of 78°, which is the upper limit angle for evaluatin g the field incidence condition. 20304050607080 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.7. TL properties estimated based on the analytical solution in the field incidence condition with a panel loss factor η of 0.02. , Panel A ; , Panel B’ ; , Panel B” ; , Panel B ; , Panel C. C. THE OTHE R INVESTI GATIONS To confirm whether the approximated formula ρGνν )1/()12.187.0( + + can estimate the velocity of flat part in the dispersion curve practically for panels with an arbitrary value of the Poisson ratio, the dispersion properties and TL characte ristics w ere calculated for panels that assumed to have the properties shown in Table 2. The Poisson ratio of panel B -1, B’-1 is 0.2 and that of panel B -2, B’ -2 is 0.4. The results are shown in Figure 8. The B’-1， B’-2 panels, where the velocity based on the approxi mated formula coincides with the incident sound speed in air, have a wider frequency range dip than B -1, B-2. Therefore, it is shown that the value ρGνν )1/()12.187.0( + + is a good approximation to th e velocity of the flat part in a dispersion curve and that the TL of the thick panel decreases most drastically in the high frequency range when it coincides with the forcing wave speed. TABLE II. Properties of panel s with poison ratio of 0.2 and 0.4. Type G(N/m2) ν Wave Speed Condition B-1 1.476e8 0.2 )78sin(/0 =c ρG B’-1 1.776e8 0.2 )78sin(/112.1 87.0 0 =++c ρG νν B-2 1.476e8 0.4 )78sin(/0 =c ρG B’-2 1.665e8 0.4 )78sin(/112.1 87.0 0 =++c ρG νν Note: ρ=1200kg/m2 , h=0.15m , c 0=343m/s 0102030405060 100 1000 10000Transmission Loss (dB) Frequency (Hz) (a) Comparison between B -1 and B’ -1: , Panel B -1 ; , Panel B’ -1. 0102030405060 100 1000 10000Transmission Loss (dB) Frequency (Hz) (b) Comparison between B -2 and B’ -2: , Panel B -2 ; , Panel B’ -2. FIG.8. TL properties comparison for Poisson ratios of 0.2 and 0.4. For thin panels, i t is well -known that no dips in the TL curve appear at the 0° sound incidence condition. However, for thick panels, the TL curve in that condition has many dips, as shown in Figure 9 because of the many branches over the flat part in the dispersion curve. This characteristic does not depend on the panel material property, which is common for the thick panels used in this investigation. Sharp’s model cannot show the TL dips in the 0° incidence condition, as in Figure 10. To improve the TL for thick panels i n the high frequency range, the approach to increase the stiffness of panels is effective, as shown in Figure 7, though the opposite approach is commonly effective for thin panels. Sharp’s model cannot show this characteristic. Although the stiffest panel A has the highest TL in the high frequency range of more than 2 kHz in Figure 7, the softest panel C shows the highest TL in Figure 2, which shows the predictions based on Sharp’s model. 0102030405060708090 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.9. TL properties at 0°, 30° and 60° sound incidence condit ions for panel A estimated based on the analytical solution: , 0°; , 30°; , 60°. 0102030405060708090 100 1000 10000Transmission Loss (dB) Frequency (Hz) FIG.10. TL properties at 0° , 30° and 60° sound incidence conditions for panel A estimated based on Sharp’s impedance model: , 0°; , 30°; , 60°. IV. CONCLUSIONS Invest igation of the detailed dispersion characteristics of thick panels leads to a proper understanding for the TL behaviour for them and the following conclusions. ・When the velocity of the flat part in the dispersion curve, which is approximated as ρGνν )1/()12.187.0( + + , coincides with the forcing wave speed caused by the incident sound, the TL of the thick panel decreases most drastically in the high frequency ran ge. ・The approach to increasing the stiffness of thick panels is effective for increasing the TL for the panel, and the TL for thick panels at a 0° sound incidence does not have good properties, which are different than the common characteristics of thin p anels because of the many branches existing in the high velocity range over the flat part in the dispersion curve of a thick panel. APPENDIX A : TRANSMISSION LOSS CALCULATION FIG.A-1. Sound transmission through a panel at an incident angle θ. Fig. A -1 illustrates a plane acoustic wave impinging upon a panel of thickness h at an incidence angle θ. The relation between the pressure P1, velocity perpendicular to the surface ν1 on the incident side and corresponding variables on the tran smitted side , P2,ν2 is described as follows[13]. (A-1) Assuming that the characteristic impedance of each side is Z0=ρ0c0, the transmission coefficient at a given angle of incidence θ is shown as follows. ()() ()()   =  22 22 2112 11 11 νP θFθFθFθF νPh/2 h/2 θ θ x y ( Panel ) 0 θ (A-2) where From these expressio ns, the tran smission loss at the incident angle θ is shown as follows. (A-3) In the case of a diffuse field excitation with an incidence angle θ varying from 0 to θf , the transmission coefficient τd is defined by (A-4) The field incide nce transmis sion loss is obtained by letting θf be 78°, so the expression is as follows. (A-5) When the deformation of a whole plate is the main mechanism for transmission, ν1 is almost equal to ν2, and each element of the matrix (A -1) is approximated as follows based on the reciprocity and the symmetric property. ()() ()()θFθF Z ZθFθFτ θθθ 22 21 0 0 12 11 / /2 + + += = ff θθ θ d θdθθθdθθτ τ 00 sincossincos     =   78 078 0 sincossincos -10log TL θdθθθdθθτθ f θ θ τ -10log TL= θ Z Zθ /cos0 0= where Z is the panel impedance defined by ν1 (=ν2)/( P1- P2). () () () ()101 22211211 ==== θFθFZθFθF APPENDIX B: ANALYTICAL SOLUTION FOR (A -1) MATRIX In the case shown in Fig. A -1, the wave in the panel is described by the following displacement potentials [14]. (B-1) where ξ is the wave number in the x direction, ω is the angle frequency, A ～D are unfixed constants and α ,β are given as follows. c1,c2 are defined as where λ , G indicate t he Lamé const ants and ρ indicates the density of the material of the panel. λ is expressed as follows.    yβ Dyβ C Ηyα Byα A Φ t ωx ξ Zt ωx ξ cos sin e j cos sin e )-( j)-( j + =+ = 2 2 222 2 12 -- ξ cωβξ cωα == ()()νννEλ21 1−+= ρGcρGλc =+= 212 Here, the loss factor η can be introduced using the complex elasticity E(1+jη),G(1+jη). The displacements in the x and y directions are described base d on the pote ntials as follows. (B-2) The stresses are expressed based on these displacements as follows. (B-3) From the above expression, the following relations are obtained. (B-4) Where yΗ xΦuyΗ xΦu Z yZ x  =+= - ()     +=++= yu xuGτxuλyuGλτ x y yxx y yy 2 () T tω-xξ TWM ν =je      =tu tuττy x yx yyν  DCBA=W From (B -4), the following relations are obtained. (B-5) From (B -5) and the following conditions, the relation (B -6) is given. (B-6) where N ij are the elements of the matrix ()()-12 2 h h- M MN  = By using the relation ξ = k sin θ , (B-6) correspond to (A -1). Accordingly, based on (B -6), the TL can be obtained by the process shown in Appendix A. ()() ()()     = yβωξ yβωξ yαωα yαωαyβωβξ yβωβξ yαωξ yαωξyββξGyββξG yααξG yαξGyββξG yββξG yαβξGyαβξG cosj- sinj- sinj cosj-sin - cos cos sincos - j sin - j sin 2j- cos 2jsin 2- cos 2 cos - sin - 2 2 2 22 2 2 2 M ()()()()T Thx, h h- h-x, 2 2 2 2-1ν M M ν   = 2 12 1 /2)( , /2)-()2/(, )2/-(0)2/( )2/-( ν htuν htup hτp hτhτ hτ y yyy yyyx yx === == =          =  22 2324 43 44 2321 43 412324 13 14 2321 13 11 11 - -- - νp NNN NNNN NNNN NNNN N νp As shown in Appendix A, the (1,2) element of the matrix corresponds to the panel impedance, and thus, by setting the (1,2) element to 0, the wave propagating velocity can be obtained using the relation c=ωξ. The expression for the (1,2) element is as follows. where where K1 and K2 are as follo ws. ()()()hββξ hααβξK αωGcNNN NZ sin - sin 4j- 22 2 20 32 22324 13 14 += = ()  ()() 22 2 2 1 222 2 2 1 0 - 8 - - 4 βξαβξ KKβξαβξ K K + + = ()()hβ hα K sin sin1= ()()hβ hα K cos cos-12=