20 CHAPTER 2. ADIABATIC THEORY
In terms of the quality factor Q the perturbation criteria becomes
vbc
cs
2
Q2 : (2.55)
The criterion Eq. (2.55) from jv2!
2 j jv1j is more restrictive than the criterion Eq. (2.33)
2 j 1
from jv1j cs. This is because the amplitude jv2!
��j1jjv1j Eq. (2.53) is a factor of 1
��
larger than what is expected from dimensional analysis of the governing equations.
2.3.4 Summary of orthogonally oscillating plates
In summary we have derived the following solution for the velocity eld for the case of
two innite parallel plates separate by a distance w and oscillating in-phase in the direction
of their normal vectors with a velocity amplitude vbc and frequency matched to the
fundamental resonance f = cs=2w,
v = Re
v1e��i!t
+
v2
+ Re
v2!
2 e��i2!t
; (2.56a)
v1 = i
2
vbc
��
h
cos(k0y) �� i��k0y sin(k0y)
i
+ O
��
��2
; (2.56b)
v2
=
1
2
v2b
c
��cs
2k0y + sin(2k0y)
+ O
��
��
; (2.56c)
v2!
2 = ��
1
42
1
��3
v2b
c
cs
sin(2k0y) + O
��
��
; (2.56d)
originally stated in Eqs. (2.31a), (2.38b), and (2.53), and where k0 =
w. In terms of the
Q-factor the magnitude of the three velocity components are
v1 Qvbc;
v2
Qv2b
c
cs
; v2!
2
Q3v2b
c
cs
: (2.57)
The new result of this analysis is that the criterion for the validity of the perturbation
expansion is Q2vbc cs, based on the assumption of jv2j jv1j, and not the criterion
Qvbc cs, based on the assumption of jv1j cs, which is usually stated in the literature.
In the present case of orthogonally oscillating plates there is no boundary-driven acoustic
streaming, and the weak steady velocity
v2
merely ensures zero net steady mass
current, as discussed in Section 2.3.2. In the more physical relevant case of a rectangular
channel with vibrating sidewalls, the acoustic streaming ow is driven at the top
and bottom walls and its magnitude becomes
v2
Q2v2b
c
j1jjv1j, as expected from
cs dimensional analysis of the governing equations. This will be discussed in the following
section.
2.4 The acoustic boundary layer and acoustic streaming
In Section 2.3 we treated an example of wave propagation orthogonal to a wall, and the
only source of damping was the bulk absorption. In this section we study acoustic wave