12 CHAPTER 2. ADIABATIC THEORY
2.2.2 Second-order equations
Inserting the expansions Eq. (2.7) for and v in the governing equations (2.5) and keeping
only second-order terms, i.e. second-order perturbations of single variables or products of
two rst-order perturbations, yields
0@tv2 + 1@tv1 + 0 (v1 r) v1 = ��c2s
r2 + r2v2 +
1
3 + b
r(r v2) ; (2.11a)
@t2 = ��0r v2 ��r (1v1): (2.11b)
These equations for the second-order velocity and density can be further split into two
sets of equation. In steady state the second-order variables consist of a steady component
and an oscillatory component oscillating at 2!, similar to the product of two sines,
sin(!t) sin(!t) = 1
2 �� 1
2 cos(2!t). The steady component is denoted by superscript 0 and
the oscillatory second-order component is denoted by superscript 2!,
g2(r; t) =
g2(r; t)
0
+
g2(r; t)
2!
=
g2(r; t)
+ Re
g2!
2 (r) e��i2!t
: (2.12)
This decomposition is valid only when considering a steady periodic state, and it is essentially
a temporal Fourier decomposition of the second-order elds. In the transient regime
a continues distribution of frequency components exist, and this is treated in Ref. 30
Appendix E.
g2
denotes time-averaging over one oscillation period t0 = 2
! and in steady
state it equals the zero-order temporal Fourier component of the eld
g2(r; t)
0
=
g2(r; t)
=
1
t0
Z t+t0=2
t��t0=2
g2(r; t0) dt0: (2.13)
g2!
2 (r) is the complex amplitude of the secondary oscillatory mode, equivalent to g1(r) in
Eq. (2.9a), and is given by the second-order Fourier component
g2!
2 (r) =
1
t0
Z t+t0=2
t��t0=2
g2(r; t0)e��i2!t0 dt0: (2.14)
The real part of any complex number Z can be written as ReZ = 1
2 (Z + Z), where
the asterisk denotes complex conjugation. The product A(r; t)B(r; t) of two oscillating
elds A(r; t) = Re
Ae��i!t
and B(r; t) = Re
Be��i!t
can be decomposed into a steady
component and an oscillatory component,
A(t)B(t) = 1
2
Ae��i!t + Aei!t
1
2
Be��i!t + Bei!t
= 1
2 Re
h
AB
i
+ 1
2 Re
h
ABe��i2!t
i
;
(2.15)
from which we introduce the following notation
AB
1
2 Re
h
AB
i
; (2.16)
��
AB
2! 1
2AB; (2.17)