14 CHAPTER 2. ADIABATIC THEORY
−w/2 0 +w/2
y
−w/2 0 +w/2
v1 = vbc cos(!t) v1 = vbc cos(!t)
y
cos( :
w y)
y sin( :
w y)
sin( :
w y)
y cos( :
w y)
v1
components
;1
components
Figure 2.1: (a) Sketch of two parallel plates oscillation in phase in the direction of their surface normal
vectors. The one-dimensional domain is dened by ��w
y w
and the boundary condition for the
2 2 uid velocity eld is v1(w
2 ) = vbc cos(!t). (b) Graphs showing the structure of the rst-order velocity
v1 and density elds, for actuation frequency matched to the fundamental resonance frequency of the
1 parallel plate system, ! = cs=w. The full lines show the spatial structure of the primary components of
the ��
resonant
elds Eq. (2.31), whereas the dashed lines show the structure of the small component of order
O
��
in Eq. (2.31).
which represents the in-phase oscillations of the two plates. The boundary conditions
for the secondary oscillatory velocity v2!
2 and the steady non-oscillatory velocity
v2
are
determined by conservation of mass, i.e. there should be no oscillatory mass-current and
no steady mass current through the walls,
v2!
2
y =
w
2
= ��
1
0
1v1
2!
y=w=2
; (2.21)
v2
y =
w
2
= ��
1
0
1v1
y=w=2 : (2.22)
2.3.1 First-order elds
We consider a velocity eld in one dimension oscillating in steady state
v1(r; t) = Re
h
v1(y)e��i!t
i
(2.23)
The rst-order equations (2.10) in one dimension becomes
��i!0v1 = ��c2s
@y1 +
4
3 + b
@2
yv1; (2.24a)
��i!1 = ��0@yv1: (2.24b)
Inserting Eq. (2.24b) in Eq. (2.24a) yields
v1 +
c2s
!2
h
1 �� i
!
0c2s
4
3 + b
i
@2
yv1 = 0: (2.25)