Introduction to Acoustics

Let us consider a carrier wave of wave number k0
and frequency ω0 = ω(k0). A Taylor expansion of the
dispersion relation (8.191) around k0 gives
� �
∂ω
Ω = K +
∂k
1
�
∂2ω 2 ∂k2 �
K 2 +
0
0
� ∂ω
∂(|A| 2 )
�
|A| 2 ,
0
(8.192)
where Ω = ω−ω0 and K = k −k0 are the frequency and
wave number of the variation of the wave-train amplitude.
Equation (8.192) represents a dispersion relation
for the complex amplitude modulation. Noting that
Ω =−i ∂ ∂
, K = i , (8.193)
∂t ∂x
(8.192) can be treated as the Fourier-space equivalent of
an operator equation that when operating on A yields
�
∂
i
∂t +
� � �
∂ω ∂
A +
∂k 0 ∂x
1
�
∂2ω 2 ∂k2 �
∂
0
2 A
∂x2 �
∂ω
−
∂(|A| 2 �
|A|
)
2 A = 0 . (8.194)
0
In the frame moving with group velocity
ξ = x − Cgt, Cg = (∂ω/∂k)0 , (8.195)
(8.194) represent the classical nonlinear Schrödinger
(NLS) equation
i ∂A
∂t
βd =− 1
2
∂
= βd
2 A
∂ξ2 + γn|A| 2 A , (8.196)
�
∂2ω ∂k2 � �
∂ω
, γn =
∂(|A| 2 �
, (8.197)
)
0
which describes the evolution of wave-train envelopes.
Here A is a complex amplitude that can be represented
as follows:
A(t,ξ) = a(t,ξ)exp[iϕ(t,ξ)] , (8.198)
where a and ϕ are the (real) amplitude and phase of
the wave train. A spatially uniform solution of (8.196)
corresponds to an unperturbed wave train. To analyze the
stability of this uniform solution let us represent (8.196)
as a set of two scalar equations:
− a ∂ϕ
�
∂
= βd
∂t 2 � � �
2
a ∂ϕ
− a + γna
∂ξ2 ∂ξ
3 , (8.199)
�
∂a
= βd 2
∂t ∂a ∂ϕ
∂ξ ∂ξ + a ∂2ϕ ∂ξ2 �
. (8.200)
It is easy to verify that a spatially uniform solution
(∂/∂ξ ≡ 0) of (8.199)and(8.200)is:
a = a0, ϕ=−γna 2 0t . (8.201)
0
Nonlinear Acoustics in Fluids 8.10 Bubbly Liquids 283
The evolution of a small perturbation of this uniform
solution
a = a0 + ã, ϕ=−γna 2 0t + ˜ϕ, (8.202)
is given by the following linearized equations:
∂ ˜ϕ
a0
∂ã
∂t
∂t
∂
+ βd
2ã ∂ξ2 + 2γna 2 0ã = 0 . (8.203)
− βda0
∂2 ˜ϕ
= 0 . (8.204)
∂ξ2 Now let us consider the evolution of a periodic perturbation
with wavelength L = 2π/K that can be written
as
� � � �
ã a1
= exp (σt + iKξ) . (8.205)
˜ϕ ϕ1
The stability of the uniform solution depends on the
sign of the real part of the growth-rate coefficient σ. To
compute σ we substitute the perturbation (8.205) into
the linearized equations (8.203) and(8.204) and obtain
the following formula for σ
σ 2 = β 2 �
2 2γn
d K a 2 �
2
0 − K . (8.206)
βd
This shows that the sign of the βdγn product is crucial
for wave-train stability. If βdγn < 0thenσis always an
imaginary number, and a uniform wave train is stable to
small perturbations of any wavelength; if βdγn > 0then
in the case that
�
2γn
K < Kcr =
(8.207)
a0
βd
σ is real, and a long-wavelength instability occurs.
This heuristic derivation shows how the equation
for the evolution of the wave-train envelope arises. The
NLS equation has two parameters: βd,γn. Eventually,
the parameter βd can be calculated from the linear dispersion
relation discussed in detail earlier. However, the
parameter γn has to be calculated from more-systematic,
nonlinear arguments.
The general method of derivation is often given the
name of the method of multiple scales [8.101]. A specific
multiscale technique for weakly nonlinear oscillations of
bubbles was developed in [8.105, 106]. This technique
has been applied to analyze pressure-wave propagation
in bubbly liquids.
In [8.107] the NLS equation describing the propagation
of weakly nonlinear modulation waves in bubbly
liquids was obtained for the first time. It was derived
Part B 8.10