Introduction to Acoustics

280 Part B Physical and Nonlinear Acoustics
Part B 8.10
dispersion relation (8.162) is first considered:
ω = Ck + iα1k 2 − α2k 3 , (8.163)
where
1 1
=
C2 C2 l
+ 1
C2 , α1 =
b
δeffC4 ω2 bC2 , α2 =
b
C5
2C2 bω2 .
b
(8.164)
Noting that
ω =−i ∂ ∂
, k = i , (8.165)
∂t ∂x
(8.163) can be treated as the Fourier-space equivalent of
an operator equation which when operating on ˜p, yields
∂ ˜p ∂ ˜p ∂
+ C − α1
∂t ∂x 2 ˜p ∂
+ α2
∂x2 3 ˜p
= 0 . (8.166)
∂x3 Equation (8.166) should be corrected to account for
weak nonlinearity. A systematic derivation is normally
based on the multiple-scales technique [8.101]. Here we
show how this derivation can be done less rigorously,
but more simply. We assume that the nonlinearity in
bubbly liquids comes only from bubble dynamics. Then
(8.155) for weakly nonlinear bubble oscillations has an
additional nonlinear term and becomes:
Π (p ~ )
0
˜p =−B1 ˜R + B2 ˜R 2 − 4µeff
R0
∂ ˜R
∂t − ρl0
∂
R0
2 ˜R
.
∂t2 2W
α 0
A
B
3W
α 0
(8.167)
Fig. 8.15 The potential well used to illustrate the “soliton”
solution for the KdV equation
p ~
�
B1 = p0 + 2σ
�
3κ
R0 R0
�
B2 = p0 + 2σ
R0
− 2σ
,
R 2 0
� 3κ(3κ + 1)
2R 2 0
− 2σ
R3 . (8.168)
0
Equation (8.167) has to be combined with the linear
wave equation (8.159), which in the case of plane waves
can be written:
C −2 ∂
l
2 ˜p
∂t2 − ∂2 ˜p
∂x2 = 4π R2 0n0ρ0 ∂2 ˜R
. (8.169)
∂t2 Taking into account that all the terms except the
first one on the right-hand side of (8.167) are small,
i. e., of the second order of magnitude, one can derive
the following nonlinear wave equation for pressure
perturbations in a bubbly liquid:
C −2 ∂2 ˜p
∂t2 − ∂2 ˜p
∂x2 = 4π R 2 0n0ρ0 ×
�
B2
B 3 1
∂2 ˜p 2
4µeff
+
∂t2 R0 B2 ∂
1
3 ˜p
∂t3 + ρl0 R0
B2 1
∂4 ˜p
∂t4 �
.
(8.170)
Equation (8.170) is derived for plane weakly nonlinear
pressure waves traveling in a bubbly liquid in both
directions. Namely, the left part of this equation contains
a classic wave operator when applied to a pressure perturbation
function describes waves traveling left to right
and right to left. The right part of this equation contain
terms of second order of smallness and is therefore
responsible for a slight change of these waves. Thus,
(8.170) may be structured as follows:
�
��
�
−1 ∂ ∂ −1 ∂ ∂
�
C + C − ˜p = O ε
∂t ∂x ∂t ∂x
2�
.
(8.171)
If one considers only waves traveling left to right
then
−1 ∂ ∂
C + = O (ε) , (8.172)
∂t ∂x
and we can use the following derivative substitution
(see [8.10])
∂ ∂
≈−C . (8.173)
∂t ∂x
Then, after one time integration over space, and
translation to the frame moving left to right with