3.4. DISCUSSION OF THERMODYNAMIC THEORY 33
get the following relation for the orders of magnitude of the terms,
��i!0 v1 = r
(
0
rv1 + (rv1)T
+
b
0 ��
2
3
0
(r v1) 1 �� p11
)
) (3.21a)
!0jv1j 0jr2v1j + jrp1j; (3.21b)
where indicates same order of magnitude, which implies removal of numeric prefactors,
disregarding signs, and assuming b . If we evaluate jr2v1j and jrp1j far from boundaries
we obtain jrj k = !=cs. If we furthermore substitute s = 1=(0c2s
) Eq. (2.3) and
2
s = 20
0! Eq. (2.64), we can rewrite Eq. (3.21b) into
jv1j
cs sjp1j + 2
s k2 jv1j
cs ) (3.21c)
sjp1j jv1j
cs
+ O
��
2
s k2
; (3.21d)
where we have utilized that the viscous diusion length s is much shorter than the acoustic
wavelength = 2=k. Equation (3.21d) relates the scale of the pressure eld p1 to the
scale of the velocity eld v1.
We can do a similar analysis of the rst-order energy equation in the frequency domain
Eq. (3.16c). Substituting the thermal diusion length 2
t = 2kth
0
Eq. (3.17) and the
0cp! thermodynamic identity ��1 =
2
pt0
s0cp
Eq. (3.7b), the dimensional analysis of Eq. (3.16c)
becomes
��i!0cp T1 + i!pT0 p1 = kth
0 r2T1 ) (3.22a)
jT1j
pT0
0cp jp1j +
kth
0
!0cp jr2T1j ) (3.22b)
jT1j
��1
p
sjp1j + 2
t k2jT1j ) (3.22c)
pjT1j
��1 sjp1j jv1j
cs
; (3.22d)
where we have utilized that t . This relates the scale of the temperature eld T1 to
the scales of the pressure eld p1 and the velocity eld v1.
3.4.2 The adiabatic approximation
To discuss the validity of the adiabatic approximation, we rst consider a standing bulk
acoustic wave with the rst-order temperature eld T1 = Ta sin(k0y), far from any boundaries.
The Laplace term in the rst-order frequency-domain energy equation (3.16c) can