Modelling multilayered systems with porous materials using the transfer matrix method - Propagation of Sound in Porous Media - page 279

p 1 C 1 (C 2 + 2 Nβ 2 ) p 2 C 2 (C 1 + 2 Nβ 2 ) − 2 Nβ 2 p 3 (C 1 −

− C 2 )

T 46 = (11.A.24)

4 Nα 1 q 1 D 2 − 4 Nα 2 q 2 D 1 −

C 2 D 1 )

q 3 α 3 −

β 2
β 2 α 3 (C 1 D 2 −

jNβ 2
ω

T 51 = (11.A.25)

2 Nβp 1 [ µ 2 (α 3 − β 2 ) 2 β 2 µ 3 ] 2 Nβp 2 [ µ 1 (α 3 − β 2 ) 2 β 2 µ 3 ]

+ − +

+ 2 Nβp 3 (α 3 − β 2 )(µ 1 − µ 2 )

T 52 = (11.A.26)

α 3 ) ω(µ 1 − µ 2 )(β 2 +

− 2 Nβ
ω(µ 1 − µ 2 ) (p 1 −

T 53 = p 2 ) (11.A.27)

α 1 q 1 D 2 −

D 1 )

α 3 − β 2

2 jNβ q 3

2

T 54 = α 2 q 2 D 1 − (D 2 − (11.A.28)

α 3

p 1 µ 2

2 β 2
β 2

p 3 α 3 − β 2
2 β 2

− µ 3

p 2 µ 1

− µ 3

T 55 =

µ 1 + µ 2 +

(11.A.29)

α 3 + µ 2 − µ 1 −

α 1 q 1 (C 2 + 2 Nβ 2 ) − α 2 q 2 (C 1 + 2 Nβ 2 )

T 56 = − 2 jNβ

C 2 )

α 3 − β 2

q 3
2

+ (C 1 − (11.A.30)

α 3

2 NβD 1 D 2
ω

T 61 = (p 1 − p 2 )

(11.A.31)

α 2 q 1 D 1 [ µ 2 (α 3 − β 2 ) + 2 β 2 µ 3 ] − α 1 q 2 D 2 [ µ 1 (α 3 − β 2 ) + 2 β 2 µ 3 ]

j
ω

T 62 =− (11.A.32)

+ α 3 )

α 1 α 2 (µ 1 − µ 2 )(β 2

q 1 D 1
α 1

j q 2 D 2

α 2

T 63 = − (11.A.33)

ω(µ 1 − µ 2 )

T 64 = −

D 1 D 2

(p 1 − p 2 ) (11.A.34)

q 1 D 1
α 1

jβ
N(β 2

µ 2 − µ 3 q 2 D 2

α 2

µ 1 − µ 3

T 65 =−

µ 1 +

(11.A.35)

α 3 )

µ 2 − µ 1 − µ 2

+

2 Nβ 2 ) 2 Nβ 2 )

p 1 D 1 (C 2 + − p 2 D 2 (C 1 +

T 66 = (11.A.36)

In these expressions, the quantities α i ,β,C i ,D i ,p i ,q i and are equal to, respectively,

α i = k i 3 i = 1 , 2 , 3 (11.A.37)

β = k t (11.A.38)

Qµ i )(β 2 α i ) 2 Nβ 2

C i = (P + + − i = 1 , 2 (11.A.39)

Q)(β 2 α i ) D i = (Rµ i + + i = 1 , 2 (11.A.40)