View source for Template:Removing the depth dependence

If we have a problem in which all the scatterers are of constant cross sections so
that 
<center>
<math>\partial\Omega = \partial\hat{\Omega} \times z\in[-h,0]
</math>
</center>
where <math>\partial\hat{\Omega} </math> is a function only of <math>x,y</math>
i.e. the boundary of the scattering bodies is uniform with respect to depth.
We can remove the depth dependence [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
and obtain that the dependence on depth is given by 
<center>
<math>
\phi(x,y,z) = \frac{\cos \big( k_0 (z+h) \big)}{\cos(k_0 h)} \Phi(x,y)
</math>
</center>
Since <math>\phi</math> satisfies [[Laplace's Equation]], then <math>\Phi</math> satisfies [[Helmholtz's Equation]]
<center>
<math>\nabla^2 \Phi + k_0^2 \Phi = 0 </math>
</center>
in the region not occupied by the scatterers. Not that this is not the standard way to write [[Helmholtz's Equation]]
because <math>k_0</math> is the pure imaginary, and it is more normal to write 
<center>
<math>\nabla^2 \Phi - k^2 \Phi = 0 </math>
</center>
where <math>k=-ik_0.</math>