# Helmholtz equation on unit disk with angular Robin boundary condition

I'm trying to solve the Helmholtz equation on the unit disk with a prescribed boundary condition of the third kind (Robin) on the circle.

The Helmholtz equation in polar coordinates is given by

$\Delta p(r,\varphi)+k^2\cdot p(r,\varphi)=0$

and the particular boundary condition shall be

$p_r(1,\varphi)=f(\varphi)\cdot p(1,\varphi)$.

In particular, a solution for $f(\varphi)=-\sin(\varphi)$ is desired. Parameter $k$ has to be found such that the boundary condition is satisfied.

Solution attempt: Separation of variables fails for this example because the boundary condition is not separable with $p(r,\varphi)= R(r)\cdot \Phi(\varphi)$ because then

$\Phi(\varphi)\cdot R'(1) = f(\varphi)\cdot \Phi(\varphi)\cdot R(1) \Leftrightarrow R'(1)/R(1) = f(\varphi)\cdot \Phi(\varphi)$

which indeed would require $f(\varphi)\cdot \Phi(\varphi)=\mathrm{const.}$ and thus is not possible.

Questions:

1. Do you know of other approaches that are suitable for solving the above problem?
2. Can you name a specific term for referring to the type of above boundary condition (for me it is a spatially dependent Robin boundary condition)?