Building Acoustics

204 Building acoustics

(

jj

jj c

()

1

and ( ).

kx kx

px Ae B e

vx Ae B e Z

−

=⋅ +⋅

=⋅−⋅)

(5.78)

The quantities A and B will be determined by the boundary conditions on each side of the
layer. On the left-hand side, where x is equal zero, we get

(

1

1 c

1

and.

pAB

vA Z

=+

=−B)

.

(5.79)

On the right-hand side, where x is equal to d, the pressure is given by

p 2 =⋅ +⋅ = + ⋅ −⋅ − ⋅A e−jjkd B ekd ()cosj()sinA B kd A B kd (5.80)

Hence, using the Equations (5.79)

p 21 =⋅ −⋅p coskd jZ vc1coskd. (5.81)

Correspondingly, the particle velocity on the output side will be

21 1

c

cos j sin.

p v v kd kd Z

=⋅ −⋅ (5.82)

We may now cast the Equations (5.81) and (5.82) into the form sought after

c 1 12 c

cos j sin sin. jcos

kd Z kd pp kd vvkd Z

⎡⎤⋅

⎡⎤⎢⎡⎤

⎢⎥=⎢⎢⎥

⎣⎦ ⋅ ⎣⎦

⎢⎥⎣⎦

⎥ 2

⎥ (5.83)

As an example, we shall assume that the layer is placed on to an infinitely hard wall,
which implies that v 2 is equal to zero or that the load impedance ZL in Equation (5.76) is
infinite. The input impedance will then be

(^) 1c^1
1
jcotg()
p
Z Zkd,
v

==−⋅ (5.84)

which was expected from our formerly derived result (see Equation (5.50)).
Equation (5.83) may, however, be put into a general form. First, it is common to
substitute the wave number by the propagation coefficient Γ, which is given by j⋅k.
Second, we shall not assume normal incidence but introduce oblique incidence giving the
wave vector propagating through the material an angle φ with the normal. The result will
be