Green's Functions with Applications, 2nd Edition

Appendix: Relationship between Solutions of Helmholtz’s and Laplace’s Equations in Cylindrical and Spherical Coordinates

In Section 6.1 we showed how solutions to the Helmholtz or scalar wave equation in one coordinate system can be re-expressed as a superposition (integral) of solutions in another coordinate system. We extend these thoughts as they apply to product solutions involving Laplace’s and Helmholtz’s equations.

Consider the three-dimensional Helmholtz equation in cylindrical and spherical coordinates

$\begin{array}{cc}\frac{{\partial }^{2}u}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial u}{\partial \rho }+\frac{1}{{\rho }^2}\frac{{\partial }^{2}u}{\partial {\phi }^2}+\frac{{\partial }^{2}u}{\partial z^2}+k_0^{2}u=0,& \left(\mathbf{\text{A}}\mathbf{.1}\right)\end{array}$

and

$\begin{array}{cc}\frac{{\partial }^{2}u}{\partial r^2}+\frac{2}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2\sin \left(\theta \right)}\frac{\partial }{\partial \theta }\left[\sin \left(\theta \right)\frac{\partial u}{\partial \theta }\right]+\frac{1}{r^2{\sin }^2\left(\theta \right)}\frac{{\partial }^{2}u}{\partial {\phi }^2}+k_0^{2}u=0,& \left(\mathbf{\text{A}}\mathbf{.2}\right)\end{array}$

Separation of variables1 leads to the product solutions

$\begin{array}{cc}{Z}_{m,{\theta }_0}\left(\rho ,\phi ,z\right)=e^{i{k}_{0}z\cos \left({\theta }_0\right)}{J}_m\left[k_0\rho \sin \left({\theta }_0\right)\right]e^{im\phi },& \left(\mathbf{\text{A}}\mathbf{\text{.3}}\right)\end{array}$

and

$\begin{array}{c}{\text{ψ}}_n^m(r,\end{array}$