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 | the free surface is at <math>z=0</math>) and all the scatterers | ||
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 03:02, 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 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.
