Biot theory of sound propagation in porous materials having an elastic frame - Propagation of Sound in Porous Media - page 128

This force tends to zero with ε , and a boundary condition for the stress at x =− l is

σ xx ( σ xx ( p( − l) + − l) + − l) = 0 (6.97)

Another boundary condition is derived from the continuity of pressure and can be
expressed

σ xx ( − l) =− φp( − l)

(6.98)

φ being the porosity of the material. The use of Equations (6.97) and (6.98) yields
σ xx (

− l) =− ( 1 − φ)p( − l) (6.99)

The conservation of the volume of air and frame through the plane x =− l yields
φ u f ( φ)u s ( u a ( − l) + ( 1 − − l) = − l) (6.100)

u a ( l) being the velocity of the free air at the boundary. The surface impedance Z of

the material is given by

−

l)/u a ( Z = p( − − l) (6.101)

This surface impedance can be evaluated in the following way. At first, it can easily
be shown that Equations (6.91), (6.92) and (6.95) together yield
V i =− V r , i =− V r (6.102)

Equations (6.98) - (6.102) yield

φ)u a ( Z 1 V i [exp (jδ 1 l) − ( 1 − − l)Z =− jδ 1 l) ]

− Z 2 V i [exp (jδ 2 l) + exp ( − jδ 2 l) ]

+ exp ( −

(6.103)

− φ u a ( − l)Z =− Z 1 φµ 1 V i [exp (jδ 1 l) + exp ( − jδ 1 l) ]
−

(6.104)

Z 2 φµ 2 V i [exp (jδ 2 l) + exp ( − jδ 2 l) ]

φ) ] V i [exp (jδ 1 l)

[ φµ 1 + ( 1 − − exp ( − jδ 1 l) ]

(6.105)

φ) ] V i [exp (jδ 2 l) u a (

+ [ φµ 2 + ( 1 − − exp ( − jδ 2 l) ] = − l)

This system of three equations (6.103) - (6.105) has a solution ( V i , V i )if

− 2 Z 1 cos δ 1 l − 2 Z 2 cos δ 2 l

− ( 1 − φ)Z

2 Z 1 µ 1 cos δ 1 l 2 Z 2 µ 2 cos δ 2 l

− Z − − = 0 (6.106)

1 2 j sin δ 1 l(φµ 1 + 1 − φ) 2 j sin δ 2 l(φµ 2 + 1 − φ)

and Z is given by

j (Z 1 Z 2 µ 2 −

Z 2 Z 1 µ 1 )

Z =− (6.107)

D

where D is given by
D = ( 1 − φ + φµ 2 ) [ Z 1 − ( 1 − φ)Z 1 µ 1 ]tg δ 2 l
+

φµ 1 ) [ Z 2 µ 2 ( 1 −

Z 2 ]tg δ 1 l ( 1 − φ + φ) − (6.108)