Removing the Depth Dependence

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.