Introduction to Acoustics

Although the simple result of (3.319) predicts that
the fraction of incident power transmitted is unity at
a frequency of fc/ sin 2 θI, the presence of damping processes
in the plate causes the fraction of incident power
transmitted always to be less than unity. A simple way of
taking this into account makes use of a loss factor η (assumed
to be much less than unity) for the plate, which
corresponds to the fraction of stored elastic energy that
is dissipated through damping processes during one radian
(a cycle period divided by 2π). Because twice the
loss factor times the natural frequency is the time coefficient
for exponential time decay of the amplitude when
the system is vibrating in any natural mode, the former
can be regarded as the negative of the imaginary
part of a complex frequency. Then, because the natural
frequency squared is always proportional to the elastic
modulus, and because the loss factor is invariably small,
the loss factor can be formally introduced into the math-
3.11 Spherical Waves
In many circumstances of interest, applicable idealizations
are waves that locally resemble waves spreading
out radially from sources or from scatterers. The mathematical
description of such waves has some similarities
to plane waves, but important distinctions arise. The
present section is concerned with a number of important
situations where the appropriate coordinates are
spherical coordinates.
3.11.1 Spherically Symmetric Outgoing
Waves
For a spherically symmetric wave spreading out radially
from a source in an unbounded medium, the symmetry
implies that the acoustic field variables are a function of
only the radial coordinate r and of time t. The Laplacian
reduces then to
∇ 2 p = ∂2 p 2 ∂p 1 ∂
+ =
∂r2 r ∂r r
2 (rp)
∂r2 , (3.323)
so the wave equation of (3.74) takes the form
∂2 (rp) 1
−
∂r2 c2 ∂2 (rp)
= 0 . (3.324)
∂t2 The solution of this is
f (r − ct)
p(r, t) = +
r
g(r + ct)
. (3.325)
r
Causality considerations (no sound before the source
is turned on) lead to the conclusion that the second term
Basic Linear Acoustics 3.11 Spherical Waves 65
ematical model by the replacement of the real elastic
modulus by the complex number (1 − iη)E. When this
is done, one finds
�
f
Zsl = ωηmpl
fc
� � �2 f
× 1 − sin 4 �
θI
fc
� 2
sin 4 θI − iωmsl
(3.321)
for the slab impedance that is to be inserted into (3.314).
The extra term ordinarily has very little effect on the fraction
of incident power that is transmitted except when
the (normally dominant) imaginary term is close to zero.
When the frequency f is fc/ sin2 θI, one finds the value
of τ to be
�
τ = 1 + 1
�−2 ωηmpl
cos θI , (3.322)
2 ρc
rather than identically unity.
x
Source
yL
�
z
θ
r
zL
v
Listener
Fig. 3.22 Spherical coordinates. The common situation is
when the source is at the origin and the listener (sound
receiver) has coordinates (r, θ, φ)
on the right side of (3.325) is not an appropriate solution
of the wave equation when the source is concentrated
near the origin. The expression
p(r, t) =
f (r − ct)
r
xL
, (3.326)
y
Part A 3.11