Difference between revisions of "Removing the Depth Dependence"
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extend throughout the depth then we can remove the depth dependence by assuming | extend throughout the depth then we can remove the depth dependence by assuming | ||
that the dependence on depth is given by | that the dependence on depth is given by | ||
| − | + | <center> | |
<math> | <math> | ||
\Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y) | \Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y) | ||
</math> | </math> | ||
| + | </center> | ||
where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]]. | where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]]. | ||
Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]] | Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]] | ||
| − | + | <center> | |
<math>\nabla^2 \phi - k^2 \phi = 0 </math> | <math>\nabla^2 \phi - k^2 \phi = 0 </math> | ||
| − | + | </center> | |
in the region not occupied by the scatterers. | in the region not occupied by the scatterers. | ||
[[Category:Linear Water-Wave Theory]] | [[Category:Linear Water-Wave Theory]] | ||
Revision as of 14:17, 22 August 2008
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=-h }[/math] (we are assuming the free surface is at [math]\displaystyle{ z=0 }[/math]) and all the scatterers are of constant cross sections and extend throughout 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+h) \big) \phi(x,y) }[/math]
where [math]\displaystyle{ k }[/math] is the positive root of the Dispersion Relation for a Free Surface. Since [math]\displaystyle{ \Phi }[/math] satisfies Laplace's Equation, then [math]\displaystyle{ \phi }[/math] satisfies Helmholtz's Equation
[math]\displaystyle{ \nabla^2 \phi - k^2 \phi = 0 }[/math]
in the region not occupied by the scatterers.
