Waves in fluid and solid media - Building Acoustics - page 59

At a temperature of 20 °C the characteristic impedance of air will be 415 Pa⋅s/m,
which inserted into Equation (3.17) will give ≈ 2.5⋅10 -3 m/s = 2.5 mm/s.

3.2.2 Spherical waves

Assuming spherical symmetry we arrive at the second idealized type of wave, the
spherical wave. The wave equation may then be expressed as

( ) ( )

2 2

∂⋅ rp ∂⋅ rp

1

− = 0. (3.19)

2 2 2

∂ r c ∂ t

0

Analogous to the plane wave we may then express a partial spherical wave propagating
from a centre (coordinate r = 0) as

ˆ

p

( )

j

ω− tkr

prt (,) =⋅ e . (3.20)

r

In this case, the coordinate vector has the same direction as the wave number vector and
we may omit the vector notion. In contrast to the case of plane waves, the specific
impedance will not be constant but will depend on the ratio of wavelength and distance
from the source point. Using Equation (3.3) , with the gradient expressed in spherical
coordinates, in Equation (3.20) we get

j kr

Z = ρ c . (3.21)

s 0 0

1j + kr

As seen from the equation, there will be a phase difference between sound pressure and
velocity. Only in the case where the distance r is much larger than the wavelength, i.e.
when kr >> 1, we may set Z s ≈ ρ 0 c 0 .

3.2.3 Energy loss during propagation

In the expressions for the sound pressure, given in Equation (3.8) for a plane wave and in
Equation (3.20) for a spherical wave, we presupposed that the wave number k was a real
quantity. In the physical sense this implies that the wave suffers no energy loss; the wave
is not attenuated during propagation through the medium. However, in real media there
will always be some energy losses caused by various mechanisms. Furthermore, in many
cases one does try to optimize such losses; e.g. by the design of sound absorbers to be
applied in rooms or to be used in various types of silencer. In other cases, e.g. during
outdoor sound propagation over large distances natural losses will occur due to so-called
relaxation phenomena. These losses, which are strongly frequency dependent, will be
treated in Chapter 4.
It must be stressed that the attenuation we are concerned with here represents a real
energy loss as opposed to a purely spherical spreading of sound energy over an
increasing volume. We shall, whenever necessary, use the term excess attenuation to
distinguish such losses from the latter type. Formally, we shall introduce such losses