Difference between revisions of "Removing the Depth Dependence"
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</math> | </math> | ||
| − | where <math>k</math> is the positive root of the [[Dispersion | + | where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]] |
then the problem reduces to [[Helmholtz's Equation]] | then the problem reduces to [[Helmholtz's Equation]] | ||
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in the region not occupied by the scatterers. | in the region not occupied by the scatterers. | ||
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| + | [[Category:Linear Water-Wave Theory]] | ||
Revision as of 01:28, 2 June 2006
We are considering the Frequency Domain Problem for linear wave waves. If we have a problem in which the water depth is of constant depth [math]\displaystyle{ z=-d }[/math] (we are assuming the free surface is at [math]\displaystyle{ z=0 }[/math]) and all the scatters are also constant with respect to the depth then we can remove the depth dependence by assuming that the dependence on depth is given by
[math]\displaystyle{ \Phi(x,y,z) = \cosh \big( k (z+d) \big) \phi(x,y) }[/math]
where [math]\displaystyle{ k }[/math] is the positive root of the Dispersion Relation for a Free Surface then the problem reduces to Helmholtz's Equation
[math]\displaystyle{ \nabla^2 \phi - k^2 = 0 }[/math]
in the region not occupied by the scatterers.
