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 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+d) \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 = 0 }[/math]

in the region not occupied by the scatterers.