The two dimensional helmholtz equation is $$\frac{\partial ^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}+k^2 \phi=0$$ and I have that $$\nabla^2 u(r,\theta)=\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r} \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}$$
How can I derive the helmholstz equation in polar coordinates from this information?
Helmholtz in cartesian coordinates being
$\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} + k^2 \phi = 0, \tag{1}$
we recall that
$\nabla^2 \phi = \dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2}; \tag{2}$
since
$\nabla^2 u(r,\theta)=\dfrac{\partial^2 u}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r} \dfrac{1}{r^2} \dfrac{\partial^2 u}{\partial \theta^2} \tag{3}$
is a general expression for $\nabla^2$ in polars, valid for any function $u$, we have the polar expression for $\nabla^2 \phi$:
$\nabla^2 \phi(r,\theta)=\dfrac{\partial^2 \phi}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial \phi}{\partial r} \dfrac{1}{r^2} \dfrac{\partial^2 \phi}{\partial \theta^2}; \tag{4}$
(1) then becomes
$\dfrac{\partial^2 \phi}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial \phi}{\partial r} \dfrac{1}{r^2} \dfrac{\partial^2 \phi}{\partial \theta^2} + k^2 \phi = 0. \tag{5}$
Pretty simple, no?
