I need to show that a mode shape $\psi(x,y,z)$ given by:
$p(x,y,z) = \psi(x,y,z)e^{-jk_0\lambda z}$
where $k_0 = \omega_0/c_0$, $c_0$ is the sound speed, and $\lambda$ is an unknown, non-dimensional axial wavenumber.
Satisfies the 2D Helmholtz equation:
$\frac{\delta^2 \psi}{\delta x^2} +\frac{\delta^2 \psi}{\delta y^2} + k_0^2(1-\lambda^2)\psi$ = 0
With the following boundary condition:
$\frac{\delta \psi}{\delta n} = -k_0A\psi$
On $S$, where $A = \frac{\rho_0 c_0}{Z_n(w)}$
I know that specific version of the Helmholtz equation is found by substituting the mode shape into the "normal" 2D Helmholtz equation, but I'm not sure how I use such a general boundary condition on an arbitrary surface $S$ to show that the modeshape given is a solution.
Thanks for any help!