2.2. PERTURBATION EQUATIONS 11
2.2 Perturbation equations
The governing equations (2.5) are non-linear and thus complicated to solved directly. The
standard approach within acoustics is to limit the theoretical treatment to cases where
the acoustic amplitudes are small, i.e. the amplitude of the density oscillations are much
smaller than the equilibrium density 0. This is known as the perturbation approach. Each
eld g is written as
g = g0 + g1 + g2; (2.6)
for which jg2j jg1j jg0j. The perturbation approach transforms the pair of coupled
nonlinear governing equations into pairs of coupled linear equations for each order of the
perturbation expansion. The homogeneous, isotropic quiescent state at thermal equilibrium
is taken to be the zeroth-order state of the uid, and thus v0 = 0,
v = v1 + v2 (2.7a)
= 0 + 1 + 2 (2.7b)
2.2.1 First-order equations
Inserting the expansions Eq. (2.7) for and v in the governing equations (2.5) and keeping
only rst-order terms yields
0@tv1 = ��c2s
r1 + r2v1 +
1
3 + b
r(r v1) ; (2.8a)
@t1 = ��0r v1: (2.8b)
Usually, the acoustic elds are excited by a single-frequency vibration of the boundaries.
When the system has stabilized in a steady oscillatory state, the rst-order elds can be
described by pure harmonics, oscillating with the excitation frequency !. The solution can
then be expressed in the frequency domain instead of the time domain, and we use the
complex notation
g1(r; t) = Re
h
g1(r)e��i!t
i
; (2.9a)
@tg1 = ��i!g1: (2.9b)
Considering only the steady-state solution, the rst-order equations (2.8) can be transformed
from the time-domain to the frequency domain,
��i!0v1 = ��c2s
r1 + r2v1 +
1
3 + b
r(r v1) ; (2.10a)
��i!1 = ��0r v1: (2.10b)
Equation (2.10) together with a set of boundary conditions constitute a steady-state rstorder
acoustic problem under the assumptions of adiabatic thermodynamics and single
frequency vibrations of the boundaries.