Removing the Depth Dependence

We are considering the Frequency Domain Problem for linear wave waves. The Standard Linear Wave Scattering Problem in Finite Depth for a fixed body is

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math]) The body boundary condition for a rigid body is just

[math]\displaystyle{ \partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}}, }[/math]

The equation is subject to some radiation conditions at infinity. We assume the following. [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,

[math]\displaystyle{ \phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \, }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math]) and

[math]\displaystyle{ \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h} }[/math]

In two-dimensions the Sommerfeld Radiation Condition is

[math]\displaystyle{ \left( \frac{\partial}{\partial|x|} - \mathrm{i} k \right) (\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]

where [math]\displaystyle{ \phi^{\mathrm{{I}}} }[/math] is the incident potential.

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 why 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]