# Helmholtz equation with moving boundary in the plane

Let us assume we have a unit disk $$D\subset\mathbb{R}^2$$ s.t. $$\vec{0}\in D$$. To obtain the eigenfrequencies and eigenmodes (or eigenvalues and -functions if you like) we must solve $$\Delta \psi + \lambda^2 \psi = 0$$ with appropriate boundary conditions.

This can be done analytically using the ansatz $$\psi(r,\phi)=R(r)\Phi(\phi)$$ to obtain Bessel's equation. After a long but simple derivation one obtains the roots of Bessel's functions of the first kind as eigenvalues.

Here's my problem then:

Assuming we start to rotate the disk in the plane around the origin will result in a moving boundary.

Can we still determine the eigenvalues of the unit disc analytically?

I know that one can obtain the eigenvalues employing coordinate transformations which will modify the equation, transforming it into a coordinate frame where the boundary is at rest and hence Dirichlet conditions may be used. But I'm interested specifically in an analytic solution where the differential equation is not modified.

Does such a solution exist?

• If the equation itself doesn't depend on time, then the eigenvalues remain the same. The final solution just has time as a parameter – Dylan Jan 11 at 4:43
• I can argue, that the eigenvalues should remain the same from a physical point of view (changing how I look at the system, not the system itself). I need either a math. proof of identity of EV or a way to derive an explicit expression of the EV for moving boundary. – Nox Jan 16 at 9:35
• Apply separation of variables: $\psi = R(r)\Phi(\phi)T(t)$. If you plug this in the original equation, the time parts would all cancel out, and you get same equations for $R(r)$ and $\Phi(\phi)$ as before. The only thing that changes is the coefficients in the series solution would be dependent on $t$ – Dylan Jan 16 at 11:16