Porous frame excitation by point sources in air and by stress circular and line sources –modes of air saturated porous frames - Propagation of Sound in Porous Media - page 170

σ zz =−

φp e (8.11)

σ zz =− ( 1 − φ)p e

(8.12)

σ xz =

0 (8.13)

where the summation is performed on the six Biot waves. If the excitation is created by
a normal unit stress field τ zz applied on the frame, Equation (8.12) must be replaced by
σ zz =− τ zz ( 1 − φ)p + (8.14)

At the upper face, the velocity components and the stress components related to

λ j b j can be written, the x dependence being discarded

σ xz =

M 1 , 1 λ 1 + M 1 , 2 λ 2 + M 1 , 3 λ 3 (8.15)

u s z =

M 2 , 1 λ 1 + M 2 , 2 λ 2 + M 2 , 3 λ 3 (8.16)

u z =

M 3 , 1 λ 1 + M 3 , 2 λ 2 + M 3 , 3 λ 3 (8.17)

σ zz =

M 4 , 1 λ 1 + M 4 , 2 λ 2 + M 4 , 3 λ 3 (8.18)

σ zz = M 5 , 1 λ 1 + M 5 , 2 λ 2 + M 5 , 3 λ 3 (8.19)

u s x = M 6 , 1 λ 1 + M 6 , 2 λ 2 + M 6 , 3 λ 3 (8.20)

The coefficients M i,j are given in Appendix 8.A. For the case of the impinging plane
wave (Figure 8.1a), the coefficients λ j related to the unit total pressure field p with
p e

1 can be obtained from Equations (8.11) - (8.13). Inserting these values in Equation
(8.10) gives the velocity υ z , the surface impedance Z s (ξ/k 0 )

=

p e /v z , and the reflection =

coefficient V(ξ/k 0 )

Z s (ξ/k 0 ) − Z 0 / cos θ

V(ξ/k 0 ) = (8.21)

Z s (ξ/k 0 ) + Z 0 / cos θ

(ξ/k 0 ) 2 ) 1 / 2 .The

where Z 0 is the characteristic impedance of the free air and cos θ = ( 1 −

normal total velocity components U z = u s z , U z = u z , and the reflection coefficient
can be obtained with the parameters T ij given by

M 1 ,j
M 1 , 3 M i + 1 , 3

T ij = M i + 1 ,j −

(8.22)

i =

1 ,..., 5 ,j

=

1 , 2

Using Equations (8.13) and (8.15), Equations (8.18) - (8.19) can be rewritten
σ zz =

T 31 λ 1 + T 32 λ 2 (8.23)

σ zz = T 41 λ 1 + T 42 λ 2

(8.24)

From Equations (8.11) - (8.12) the ratio λ 2 /λ 1 is given by
λ 2
λ 1 = s 1 =−

φT 41 − ( 1 − φ)T 31

(8.25)

φT 42 − ( 1 − φ)T 32