Template:Removing the depth dependence

If we have a problem in which all the scatterers are of constant cross sections so that

[math]\displaystyle{ \partial\Omega = \partial\hat{\Omega} \times z\in[-h,0] }[/math]

where [math]\displaystyle{ \partial\hat{\Omega} }[/math] is a function only of [math]\displaystyle{ x,y }[/math] i.e. the boundary of the scattering bodies is uniform with respect to depth. We can remove the depth dependence separation of variables and obtain that the dependence on depth is given by

[math]\displaystyle{ \phi(x,y,z) = \frac{\cos \big( k_0 (z+h) \big)}{\cos(k_0 h)} \Phi(x,y) }[/math]

Since [math]\displaystyle{ \phi }[/math] satisfies Laplace's Equation, then [math]\displaystyle{ \Phi }[/math] satisfies Helmholtz's Equation

[math]\displaystyle{ \nabla^2 \Phi + k_0^2 \Phi = 0 }[/math]

in the region not occupied by the scatterers. Not that this is not the standard way to write Helmholtz's Equation because [math]\displaystyle{ k_0 }[/math] is the pure imaginary, and it is more normal to write

[math]\displaystyle{ \nabla^2 \Phi - k^2 \Phi = 0 }[/math]

where [math]\displaystyle{ k=-ik_0. }[/math]