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rigid-free boundary

The solution for the case when the top surface is free and the bottom surface is rigid can be deduced from the odd solution of the rigid-rigid case. The problem is defined by

$\displaystyle (D^2-a^2)^3W=-a^2 R_a W$ (67)

subject to boundary condition
$\displaystyle W=DW = (D^2-a^2)^2 W = 0$   at$\displaystyle \quad z=0$     (68)
$\displaystyle W=D^2 W = D^4 W = 0$   at$\displaystyle \quad z=1$     (69)

The boundary conditions at the mid height for the odd solution is the (69). Accordingly an odd solution for the rigid-rigid boundary at depth $ d$ provides solution for the rigid-free boundary for a depth $ d/2$. Thus using the stability results from the rigid-rigid case, we have

$\displaystyle a_c=5.365/2\approx 2.682$   and$\displaystyle \quad R_c=17610.39/2^4\approx 1100.65$ (70)

[Note: From $ \Delta_1^2 f + k^2 f$ in dimensional form, we have $ a=kL$ ($ L$ is the scaling length) as the nodimensional wave number.]
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Next: About this document ... Up: Exact solutions of characteristics Previous: rigid-rigid boundaries
A/C for Homepage of Dr. S Ghorai 2003-01-16