Introduction to Acoustics

ubble natural acoustic resonance ω2 0 , and the expression
for the scattered field from an acoustic bubble under an
incident pressure field at frequency ω.
The acoustic resonance in the linear regime of a sin-
gle bubble of radius a is:
ω 2 0 =
�
3γ p0
2T
+ (3γ − 1)
ρwa2 ρwa3 �
, (5.14)
where p0 is the ambient pressure outside the bubble,
T is surface tension (tensile force/length in units
N/m), ρw is the density of water and γ is the ratio
of specific heats. Neglecting surface tension in (5.14)
for bubble sizes larger than 1 µm and considering the
acoustic expansion/compression process to be adiabatic
(c 2 air = γ p0/ρair), we obtain the approximate expression
ω0 = 1
a
�
3c 2 air ρair
ρw
, (5.15)
and for cair � 340 m/s, we get f0 � 3 a with a in m and
f0 in Hz.
We now consider an incident plane wave at frequency
ω in the regime ka = ωa/c = 2πa/λ ≪ 1. The
far-field expression for the spatial part of the radiated
acoustic field pr is
pr(r) =− a
r pi
�
exp − iω
�
(r − a)
c
�
× 1 − ω2 0
ω2 �
1 − iωa
�
c
�−1 . (5.16)
First, we note that in the high-frequency limit, we recover
pr(r, t) =− a r pi
� � r−a
t − c that was given by the
boundary conditions at the bubble surface.
To understand the effect of the resonance frequency,
consider two cases. The first case is ω ≫ ω0 for which
we obtain:
pr(r) =− a
r pi e − iω c (r−a) , (5.17)
whereas for the case ω = ω0 we get:
pr(r) =− ic
ωr pi e − iω c (r−a) =− iλ
2πr pi e − iω c (r−a) .
(5.18)
Comparing the two equations, (5.18) appears to be the
field radiating from a sphere of radius λ/2π which is
much larger than a. For example, neglecting surface
tension, at 1 atm, ω0a ≈ 20 so that λ ≈ 500a. This resonance
effect is also quite apparent when considering the
scattering cross section of the bubble.
a)
0
1
2
3
4
5
6
1480
Depth (km)
c)
80
70
Underwater Acoustics 5.2 Physical Mechanisms 163
Critical depth
= 4420m
Bottom = 5322m
b)
250Hz
500Hz 150Hz 100Hz
50Hz
25Hz
18Hz
100Hz
1500 1520 1540 50 60 70 80 90
Sound velocity (m/s) Spectrum level (dB re 1Pa)
Spectrum level (dB/Pa/sterad)
3781m
713m
–20 –10 0 10 20
Down Up
Elevation angle (deg)
Fig. 5.16a–c Noise in the deep ocean. (a) Sound-speed profile and
(b) noise level as a function of depth in the Pacific (after [5.30]).
(c) The vertical directionality of noise at the axis of the deep sound
channel and at the critical depth in the Pacific (after [5.31])
The scattering cross section σs is the ratio of the total
scattered power (intensity × area = pressure × velocity
× enclosing area) to the incident plane-wave intensity
(given by p 2 i /2ρwc, with the factor of 1/2 coming from
averaging over a cycle) and therefore has the units of
area. We perform this calculation in the far field using
(5.16) to obtain
σs =
�
1 − ω2 0
ω 2
4πa 2
� � iωa
1 − c
�2 ω=ω0
−→ λ2
, (5.19)
π
which is consistent with a surface area associated with
the discussion below (5.18). The resonance makes the
bubble appear larger in surface area than its dimension.
On the other hand, for the case ω ≪ ω0,wehave
σs = 4πa 2
� �4 ω
, (5.20)
ω0
Part A 5.2