3.3. PERTURBATION THEORY 31
terms, i.e. terms containing only one second-order variable g2, or terms containing a product
of two rst-order variables, such as g2
1,
0 = T @tp2 �� p @tT2 +r v2 + v1 (Trp1 �� prT1); (3.18a)
0 = 0@tv2 + 0(T @tp1 �� p@tT1)v1 + 0(T p1 �� pT1)@tv1
��r
2 �� p2 1 �� 0v1v1
; (3.18b)
0 = 0v1@tv1 + 0cp
@tT2 + T p1@tT1 �� pT1@tT1
�� p
T0@tp2 + T1@tp1
+ 0cp(v1 r)T1 + (1 �� pT0)(v1 r)p1
��r
h
v1 1 + kth
0 rT2 + kth
1 rT1
i
; (3.18c)
where 2 is given by
2 = 0
rv2 + (rv2)T
+
b
0 ��
2
3
0
(r v2) 1
+ 1
rv1+ (rv1)T
+
b
1��
2
3
1
(rv1) 1: (3.18d)
In Eqs. (3.18b) and (3.18c) we have applied the substitution 1 = 0(T p1 �� pT1), according
to Eq. (3.4b), and we have included perturbations in the transport coecients due
to their temperature and density dependences, as introduced in Ref. 29 Appendix D,
(T; ) = 0(T0; 0) + 1(T0; T1; 0; 1); (3.19a)
1 =
@
@T
T=T0
T1 +
@
@
=0
1; (3.19b)
b(T; ) = b
0 (T0; 0) + b
1 (T0; T1; 0; 1); (3.19c)
b
1 =
@b
@T
T=T0
T1 +
@b
@
=0
1; (3.19d)
kth(T; ) = kth
0 (T0; 0) + kth
1 (T0; T1; 0; 1); (3.19e)
kth
1 =
@kth
@T
T=T0
T1 +
@kth
@
=0
1: (3.19f)
The full thermoviscous second-order acoustic equations (3.18) are quite elaborate and
it is not straight forward to understand the physical meaning of each term. That is why
we limited the treatment in Chapter 2 to the adiabatic case, which greatly simplies the
equations. Moving on to the treatment of a steady periodic state, we will not treat the
secondary oscillatory components v2!
2 , p2!
2 , and T2!
2 in the general thermodynamic case,
but limit the presentation to the time-averaged components
v2
,
p2
, and
T2
.