Waves in fluid and solid media - Building Acoustics - page 79

3.5.1.1 Sound pressure in front of a boundary surface
Determination of the sound absorption factor of small specimens and for normal
incidence is performed in a so-called standing wave tube . This will be treated in detail in
section 5.3. In the “classical” method of performing such measurements one needs an
expression for the total pressure in front of the specimen surface. At an arbitrary distance
x in front of the surface we get

pxt (,) = p xt (,) + p xt (,), .e.

i r

(3.71)

j( ω tkx − ) j( ω tkx + + δ )

⎡ ⎤

pxt (,) = p ˆ e + R ⋅ e .

⎣ ⎦

i p

When carrying out measurements one determines the RMS-value of the sound pressure,
by definition given by

T

1

} 2

∫

{

px () = ⎡ Re (,) pxt ⎤ d. t

(3.72)

⎣ ⎦

T

0

As before, we shall indicate the RMS-value by using a curly over-bar to distinguish it
from the amplitude value, the latter is indicated by a “hat” on top of the symbol. Inserting
Equation (3.71) into (3.72) we get

1

ˆ

p

⎡

2

⎤

2

i

px () = 1 + R + 2 R cos(2 kx δ + ) . (3.73)

⎢

p p

⎥

⎣ ⎦

2

The pressure will therefore exhibit maximum and minimum values given by the
equations

p ˆ

()

⎡ ⎤

i

p = 1 + R

⎣

p

⎦

max

2
ˆ

(3.74)

p

()

i

⎡ ⎤

and p = 1 − R .

⎣

p

⎦

min

2

From the measurements of these pressure values one determines the absolute value of the
reflection factor and thereby the absorption factor. (How do we determine the impedance
Z g ?)

3.5.2 Oblique sound incidence

We shall extend the above calculations by giving the incident wave (see Figure 3.13) an
angle ϕ with the normal to the surface. We may then rotate the coordinate system and
obtain a new x coordinate given as x '= x ⋅cosϕ + y ⋅sinϕ. The sound pressure and the
normal component of the particle velocity may then be expressed as