Building Acoustics

Waves in fluid and solid media 57

3.2.1 Plane waves

Using the notion plane wave implies a wave where the sound pressure varies in one
direction only. We have not yet introduced any energy losses in the Equations (3.2)
through (3.4), which means that the pressure amplitude will be constant, and we may
write

pt p(,)r =⋅ˆ ej(ωt−kr⋅), (3.8)

where k = n · 2 π/λ is the wave number vector and λ is the wavelength. It should be
observed that the last term in the exponent is the vector product of the wave number
vector and the coordinate vector r. For a plane wave in the positive x-direction we obtain

pxt(,)=⋅ˆpej(ωtkx−x), (3.9)

where kx is the component of the wave number in the x-direction. The equation tells us
that either we observe the pressure as a function of time or as a function of location at a
given time, it represents a simple oscillatory motion. As for other types of oscillation we
shall use the RMS-value as a characteristic quantity,

22

0

1

(,)d,

T

ppxtt

T

 = ∫ (3.10)

which further may be expressed in decibels (dB) by the sound pressure level Lp

2

2

0

p 10 lg (dB).

p

L

p

⎛⎞

=⋅⎜⎟⎜⎟

⎝⎠



(3.11)

The reference value p 0 is equal to 2⋅ 10 -5 Pa, an international standard value for sound in
air.
In practice, one may generate a plane wave in a duct or tube under the condition
that the diameter is much less than the wavelength. In addition, the wave must not be
reflected back from the end of the duct or tube; it must be equipped with a so-called
anechoic termination. This technique is especially used for determination of sound power
emitted by sources in duct systems, e.g. air conditioning fans as specified in ISO 5136.

3.2.1.1 Phase speed and particle velocity

The most common form expressing the pressure in a plane wave is given by Equation
(3.8). For a plane wave, however, the wave Equation (3.5) will be satisfied by any
function having the argument (t – x/c) or (t + x/c). The sound pressure will be constant as
long as this argument has a constant value, which makes us realize that the quantity c
really represents the propagation speed. As mentioned above, when travelling along with
the wave at a speed dx/dt = ± c one will always “see” the same phase of the wave and the
pressure will be constant. An analogous example is when surfing in the sea. This is the
reason for calling c the phase speed.
According to the relationship we have used concerning changes in pressure and
density, Equation (3.4), we have implicitly assumed that the changes take place