2.3. EXAMPLE: ORTHOGONALLY OSCILLATING PLATES 17
−w/2 0 +w/2
w y + sin( 2:
2.2: Graphs showing the structure of the solutions for the second-order time-averaged velocity
(top graph) and density
(bottom graph), and the second-order harmonic velocity v2!
graph), for actuation frequency matched to the fundamental resonance frequency of the parallel plate
and assuming the system is at the fundamental resonance k0
2 , n = 1 Eq. (2.30), the
solutions (2.36) becomes
sin(2k0y) + O
2k0y + sin(2k0y)
This concludes our treatment of the time-averaged second-order elds. The spatial dependences
of the solutions Eq. (2.38) are shown in Fig. 2.2. Since only the gradient of
the time-averaged density @y
enters the governing equations (2.34), there is no extra
information gained from integrating Eq. (2.38a).
It may seem contradictory to have a steady velocity in this one-dimensional system.
However, the conserved quantities are mass and momentum, not velocity. The mass is
indeed conserved because the steady mass current
, from the walls towards the
center, is balanced by the steady mass current 0
, from the center towards the walls.
There is thus zero net steady mass current in the system.
The leading-order components of 1 and v1, Eq. (2.31), are oscillating out of phase,
and thus we nd that
j1jjv1j, which is an order of smaller than one would
expect from dimensional analysis.
2.3.3 Second-order oscillatory elds
The governing equations for the socond-order oscillatory component 2!
2 and v2!
2 , Eqs. (2.18b)
and (2.19b), respectively, in one dimension become
2 = 0@yv2!
2! + 0
2! = c2s
3 + b
2 : (2.39b)