2.3. EXAMPLE: ORTHOGONALLY OSCILLATING PLATES 15
We then dene the damping factor �� and the complex wavenumber k as
�� =
!
20c2s
4
3 + b
10��6; (2.26a)
k = (1 + i��)
!
cs
= (1 + i��)k0; (2.26b)
where the value of �� is for a typical water-lled channel exited at 2 MHz. Utilizing the
smallness of ��, Eq. (2.25) can to rst order in �� be expressed as
@2
yv1 + k2v1 = 0: (2.27)
The solution to the governing equation (2.27) and the boundary conditions Eq. (2.20) is
v1(y) = vbc
cos(ky)
cos(k w
2 )
; (2.28a)
1(y) = i0vbc
k
!
sin(ky)
cos(k w
2 )
; (2.28b)
where the solution for 1 is given by Eq. (2.24b). Expanding the trigonometric functions
to rst order in ��, the velocity and density becomes
v1(y) = vbc
cos(k0y) �� i��k0y sin(k0y)
cos(k0
w
2 ) �� i��k0
w
2 sin(k0
w
2 )
+ O
��
��2
; (2.29a)
1(y) = i0
vbc
cs
(1 + i��)
sin(k0y) + i��k0y cos(k0y)
cos(k0
w
2 ) �� i��k0
w
2 sin(k0
w
2 )
+ O
��
��2
; (2.29b)
where O
��
��2
indicates that the terms left out are smaller by a factor of ��2 with respect
to the leading order term in the expression. The system is at resonance when v1 and 1
have their maximum amplitudes. This occurs when the denominators in Eq. (2.29) are
minimized, which is approximately determined by
k0
w
2
= (n �� 1
2 ); n = 1; 2; 3; ::: (2.30)
The various values of n describe the higher order resonances, but only those for which
the main component of v1 is even in y, due to the choice of in-phase boundary conditions
Eq. (2.20). To obtain the odd v1-resonances, we would need to apply an anti-phase boundary
condition v1(w
2 ) = vbc, as done in Ref. 41. At the fundamental resonance n = 1
the velocity and density elds become
v1(y) = i
2
vbc
��
h
cos(k0y) �� i��k0y sin(k0y)
i
+ O
��
��2
; (2.31a)
1(y) = ��
2
vbc
��cs
0
h
sin(k0y) + i��
i
��
sin(k0y) + k0y cos(k0y)
+ O
��
��2
; (2.31b)
where k0 =
w. The spatial dependences of the terms in the solution Eq. (2.31) are sketched
in Fig. 2.1(b). The perturbation expansion of the governing equations is valid only if