Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.
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]
On [math]\displaystyle{ x = 0 \, }[/math]:
[math]\displaystyle{ \partial_n\phi = \frac{\partial}{\partial x}
\left( \phi^{\mathrm{I}} + \phi^{mathrm{D}} \right) = 0 }[/math]
where [math]\displaystyle{ \phi^{mathrm{D}} }[/math] is the diffraction potential
or
[math]\displaystyle{ - \mathrm{i} k A_I + \mathrm{i} k A_R \Longrightarrow A_R = A_I \equiv A }[/math]
The refl
The corresponding velocity potentials are
[math]\displaystyle{ \Phi_I = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_I}{\omega} \frac{\cosh k k h (z+h)}{\cosh k h}
e^{-\mathrm{i} k x + \mathrm{i}\omega t} \right \} }[/math]
[math]\displaystyle{ \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} }[/math]
The resulting total velocity potential is
[math]\displaystyle{ \Phi = \Phi_I + \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i}gA_R}{\omega}
\frac{\cosh k (z+h}{\cosh kh} e^{\mathrm{i}\omega t} \left( e^{\mathrm{i}kx} + e^{-\mathrm{i}kx} \right) \right \} }[/math]
[math]\displaystyle{ = 2 A \mathrm{Re} \left \{ \frac{\mathrm{i}g}{\omega}
\frac{\cosh k (z+h)}{\cosh kh} \cos kx e^{\mathrm{i}\omega t} \right \} }[/math]
The resulting standing-wave elevation is:
[math]\displaystyle{ \zeta = \zeta_I + \zeta_R = 2 A \cos k x \cos \omega t \, }[/math]
