Waves in fluid and solid media - Building Acoustics - page 60

either by using a complex wave number k ' or a complex propagation coefficient Γ. 1 We
shall introduce these quantities by writing Γ = j⋅ k ' = α + j⋅β. For the sound pressure in a
plane wave, propagating in the positive x-direction, we may then write

−Γ x j ω t − + ( αβ ω j ) x j t

ˆ ˆ

pxt (,) =⋅ p e ⋅ e =⋅ p e ⋅ e , (3.22)

0 0

where the components α and β are the attenuation coefficient and the phase coefficient ,
respectively. Comparing with Equation (3.9) we immediately see that the phase
coefficient β is equal to our real wave number k x , whereas the attenuation coefficient
represents the energy losses. The latter is often specified by the number of decibels per
metre, which by using Equation (3.22) is given by

Attenuation (dB/m) ≈ 8.69 ⋅ α (3.23)

In this topic we shall reserve the symbol α for the absorption factor . Therefore later on
we will replace the attenuation coefficient α with the quantity m /2, where m is called
power attenuation coefficient.
Figure 3.1 may be used as an illustration of the sound pressure amplitude of an
ideal plane wave and a wave being attenuated during propagation, respectively. These
may be regarded as sections of the wave fronts. One must, however, be aware that the
actual physical waves are compressional and not transverse types of waves, the former
exhibiting alternating condensation and rarefaction.

a) b)

Figure 3.1 Sketch of the sound pressure in a wave front. a) Ideal plane wave. b) Attenuated plane wave.

3.2.3.1 Wave propagation with viscous losses
In our illustrations using the simple mass-spring system, we introduced viscous losses in
the equation of force. In an analogous way we will do the same with Equation (3.3) , the
so-called Euler equation. For simplicity, we shall assume that the wave is a plane one,
and we shall add a loss term r ⋅ v x where v x is the particle velocity in the x-direction. The
quantity r is the airflow resistivity of the medium of propagation having a dimension of
Pa⋅s/m 2 . As we shall see later, this is an important parameter when characterizing porous
materials. Equation (3.16) will then be modified to

1 In ISO 80000 Part 8, the small Greek letter γ is used.