Removing the Depth Dependence: Difference between revisions
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We are considering the [[Frequency Domain Problem]] for linear wave waves. | 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>z=-d </math> (we are assuming | If we have a problem in which the water depth is of constant depth <math>z=-d </math> (we are assuming | ||
the free surface is at <math>z=0</math>) and all the scatterers | the free surface is at <math>z=0</math>) and all the scatterers are of constant cross section and | ||
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 | ||
Revision as of 03:04, 21 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=-d }[/math] (we are assuming the free surface is at [math]\displaystyle{ z=0 }[/math]) and all the scatterers are of constant cross section 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+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.
