Point source above rigid framed porous layers - Propagation of Sound in Porous Media - page 156

The ratio [ ∂p pole /∂z 2 ] /p pole has the same expression as [ ∂p t /∂z 2 ] /p t in Equation
(7.56). The difference is that the impedance condition concerns the total field and must
be satisfied close to the reflecting surface only.

7.8 The modified version of the Chien and Soroka model

In Section 7.1 an exact integral expression for the reflected field is given. An approximate
expression obtained with the passage path method and valid for large k 0 R 1 is given in
Section 7.4. Several equivalent expressions valid for θ 0 close to π /2 when one pole is
located at an angle of incidence θ p close to grazing incidence are given in Section 7.6.
The validity of a modified version of the Chien and Soroka model suggested by Nicolas
et al . (1985) and Li et al . (1998), currently used to predict the reflected monopole field
above porous layers in the context of long-range sound propagation and also for the
evaluation of acoustic surface impedance, is studied in this section. The initial work by
Chien and Soroka (1975) concerns the monopole field reflected by an impedance plane
Z s . In the initial formulation, the reflected field is given by

exp (

− jk 0 R 1 )

R 1

V L ( sin θ 0 ))( 1 + √ πu exp (u 2 ) erfc ( p r = [ V L ( sin θ 0 ) + ( 1 − − u)) ] (7.60)

where V L is the reflection coefficient

cos θ Z 0 /Z s

cos θ + Z 0 /Z s

−

V L ( sin θ) = (7.61)

and u is given by

= exp −

k 0 R 1 / 2 cos θ 0 +

j 3 π
4

Z 0
Z s

u (7.62)

Z 0 /Z s . This expression
can be obtained from Equation (1.4.10) in Brekhovskikh and Godin (1992) for θ p and
θ 0 close to π /2. For nonlocally reacting media it has been suggested to use Z s ( sin θ 0 )
instead of the constant impedance Z s in Equations (7.60) - (7.62). In the modified Chien
and Soroka formulation Z s ( sin θ 0 ) is substituted for Z s which does not depend on the
angle of incidence. The reflection coefficient V L of the impedance plane becomes the
reflection coefficient V of the porous layer at an angle of incidence θ 0 . In the modified
formulation the reflected pressure is given by

There is one pole of the reflection coefficient and cos θ p =−

exp ( − jk 0 R 1 )

R 1

+ √ πu exp (u 2 ) erfc ( p r = [ V( sin θ 0 ) + ( 1 − V( sin θ 0 ))( 1 − u)) ] (7.63)

where V is the reflection coefficient

cos θ − Z 0 /Z s ( sin θ)

V( sin θ) = (7.64)

cos θ + Z 0 /Z s ( sin θ)

and u is now given by

exp −

k 0 R 1 / 2 cos θ 0 +

j 3 π
4

Z 0
Z s ( sin θ 0 )

u = (7.65)