2.3. EXAMPLE: ORTHOGONALLY OSCILLATING PLATES 13
where A and B could be any rst-order elds. This can be used to decompose the secondorder
equations into one set of equations governing the steady component and one set of
equations governing the oscillatory component of the second-order elds. The second-order
continuity equation (2.11b) is thus in steady-state separated into
0 = ��0r
v2
��r
1v1
; (2.18a)
��i2!2!
2 = ��0r v2!
2 ��r
��
1v1
2!; (2.18b)
where we have utilized that
@tg2
= 0 for any steady-state second-order eld, and @tg2!
2 =
��i2!g2!
2 . Similarly, the second-order momentum equation (2.11a) separates into
1(��i!v1)
+ 0
(v1 r) v1
= ��c2s
r
2
+ r2
v2
+
1
3 + b
r
��
r
v2
; (2.19a)
��i2!0v2!
2 +
��
1(��i!v1)
2! + 0
��
(v1 r) v1
2! = ��c2s
r2!
2 + r2v2!
2
+
1
3 + b
r
��
r v2!
2
: (2.19b)
Equations (2.18) and (2.19) together with a set of boundary conditions constitute a steadystate
second-order acoustic problem, under the assumptions of adiabatic thermodynamics
and single frequency vibrations of the boundaries.
2.2.3 Summary
Following the perturbation approach we have transformed the two coupled nonlinear equations
(2.5) governing and v into six partially coupled equations, governing the primary
oscillatory modes 1 and v1 Eq. (2.10), the secondary oscillatory modes 2!
2 and
v2!
2 Eqs. (2.18b) and (2.19b), and the non-oscillatory modes
2
and
v2
Eqs. (2.18a)
and (2.19a). In the next section, these equations will be solved in the case of two orthogonally
oscillating plates.
2.3 Example: orthogonally oscillating plates
In this section we study a system consisting of two innite parallel plates separated by
a distance w and oscillating in the direction of their normal vectors. This is one of the
most simple, yet still meaningful examples of an acoustic cavity. This section is inspired
by the similar example in Ref. 41, which treats the rst-order problem for the two plates
oscillating in anti-phase. We will treat both the rst-order and second-order problem for
the two plates oscillating in phase. The system is sketched in Fig. 2.1. The boundary
conditions for the primary oscillatory velocity are
v1
y =
w
2
= vbc cos(!t) = Re
h
vbce��i!t
i
; (2.20)