2.3. EXAMPLE: ORTHOGONALLY OSCILLATING PLATES 19
The A and B coecients of the homogeneous solution is determined by the boundary
conditions Eq. (2.21), which to lowest order in �� becomes
v2!
2 (w=2) = ��
1
0
1v1
2!
y=w=2
= ��i
1
4cs
v2b
c
cos2(k w
2 )
sin(2k(w=2)) )
2 ) + B sin(2q w
Acos(2q w
2 ) + C sin(2k w
2 ) = ��i
1
4cs
v2b
c
cos2(k w
2 )
sin(2k(w=2)) )
Acos(qw)
B sin(qw) + C sin(kw)
= i
1
4cs
v2b
c
cos2(k w
2 )
sin(kw); (2.49)
which has the solution
1
A = 0 ^ B = ��
8��
+ i
1
4
1
cs
v2b
c
cos2(k w
2 )
sin(2k w
2 )
sin(2q w
2 )
= ��
1
8��cs
v2b
c
cos2(k w
2 )
sin(2k w
2 )
sin(2q w
2 )
+ O
��
��
:
(2.50)
Since we are only interested in the solution to v2!
2 to leading order in �� and the boundary
value, right hand side of Eq. (2.49), is an order of �� smaller than the amplitude C of the
inhomogeneous solution Eq. (2.48), the boundary value turns out to have no inuence on
the solution to leading order in ��, expressed in Eq. (2.50). The solution for the second-order
oscillatory velocity thus becomes
v2!
2 =
1
8
1
��cs
v2b
c
cos2(k w
2 )
sin(2ky) ��
sin(2k w
2 )
sin(2q w
2 )
: (2.51)
sin(2qy)
With the expansion of the trigonometric function to rst order in �� and applying the
resonance condition k0
w
2 =
2 , n = 1 Eq. (2.30), we obtain
cos2(k w
2 )
h
cos(k0
w
2 ) �� i��k0
w
2 sin(k0
i2
w
2 )
2 )2 = ��
= (��i��
2
4
��2; (2.52a)
sin(2k w
2 )
2 )
sin(2q w
sin(2k0
w
2 ) + i��2k0
w
2 cos(2k0
w
2 )
sin(2k0
w
2 ) + i2��2k0
w
2 cos(2k0
w
2 )
= ��i��
��i2��
=
1
2
; (2.52b)
sin(2ky) �� 1
2 sin(2qy) sin(2k0y) + i��2k0y cos(2k0y) �� 1
2
h
sin(2k0y) + i2��2k0y cos(2k0y)
i
= 1
2 sin(2k0y) (2.52c)
and v2!
2 thus becomes
v2!
2 = ��
1
42
1
��3
v2b
c
cs
sin(2k0y) + O
��
��
: (2.53)
The spatial dependences of the solution Eq. (2.53) are shown in Fig. 2.2. In order for the
perturbation expansion to be valid, we need to check that the amplitude of v2!
2 is much
smaller than the amplitude of v1,
jv2!
2 j
jv1j
=
1
42
1
��3
v2b
c
cs
2
vbc
��
1 )
vbc
cs 8��2 (2.54)