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(x,y,t) = (y) exp ik(x-ct) (3.101)
Substitution in Equation 3-100 leads to the so-called “Rayleigh equation”
(U-c){ "(y) -k 2 } - U" (y) = 0 (3.102)
where k is a specified wavenumber, and (y) is the modal eigenfunction; the
factor exp ik(x-ct) allows wave propagation in the x direction, and c = c r + i c i is
the complex eigenvalue .
3.4.2 Boundary conditions for (y).
We will assume a rigid wall placed along y = - H < 0, at which the normal
velocity v vanishes. Thus,
(-H) = 0 (3.103)
Now consider a flexible membrane whose deviation from equilibrium y = 0
satisfies y = (x,t), where (x,t) is an unknown in addition to (y). The
disturbance pressure acting on the membrane satisfies the force balance
p'(y=0) = g - T xx + s + m tt (3.104)
Here we assume a constant fluid density , a gravitational acceleration g, a
membrane tension T, a spring constant s, and a lineal mass density m. If we
now take
p'(x,y,t) = p (y) exp ik(x-ct)
(3.105)
(x,t) = a exp ik(x-ct)
(3.106)
we obtain
p (0) = ( g + Tk 2 + s - mk 2 c 2 ) a
(3.107)
The kinematic condition
v =
t + U
x
(3.108)
leads to
(0) = -{U(0) - c} a
(3.109)
He nc e
'(0)/ (0) = U'(0)/{U(0) - c} +
{ g + Tk 2 + s - mk 2 c 2 }/{ (U(0)-c) 2 }
(3.110)
Equations 3.102, 3.103, and 3.110 specify the entire “inviscid stability
problem.” The reader should examine the similarities and differences between
the present formulation and our previous one for water waves.
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