# Eigenfunctions of the Helmholtz equation in Toroidal geometry

the Helmholtz equation

has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an assumed harmonic time dependence $e^{\pm\mathrm{i}\omega t}$.

$k$ can be seen as some kind of potential – the equation is analogue to the stationary Schrödinger equation.

The existance of solutions is to my knowledge linked to the separability of the Laplacian $\Delta$ in certain coordinate systems. Examples are cartesian, elliptical and cylindrical ones.

For now I am interested in a toroidal geometry,

where $T^2 = \left\{ (x,y,z):\, r^2 \geq \left( \sqrt{x^2 + y^2} - R\right)^2 + z^2 \right\}$

Hence the question:

Are there known solutions (in terms of eigenfunctions) of the Helmholtz equation for the given geometry?

Thank you in advance
Sincerely

Robert

Edit: As Hans pointed out, there might not be any solution according to a corresponding Wikipedia article. Unfortunately, there is no reference given – does anyone know where I could find the proof?

## Answer

• Normally $T^2$ means the Torus, which is a 2-manifold: $T^2 \cong [0,2\pi r]\times[0,2\pi R]$, the solution to

bears the form: for $m\in \mathbb{Z}^2$, $\psi_k = e^{ i m\cdot x}$, with $|m| = \sqrt{m_1^2 +m_2^2} = k.$
The reason behind this is that $\mathbb{T}^2 \cong \mathbb{S}^1(r)\times \mathbb{S}^1(R)$, and for (1) on $\mathbb{S}^1$ has eigenvectors $e^{imx}$ where $|m| = k$, then the Fourier expansion on product spaces use basis $\prod e^{i m_i x_i}$.

• In your case it is actually a Toroid, according to the Field Theory Handbook the chapter about rotational system, the Helmholtz equation is not separable in toroidal geometry. Only Laplace equation is separable, please see section 6 in here.

• By that wikipedia article about Toroidal coordinates: we make the substitution for (1) as well:

then by the Laplacian in the toroidal geometry in that wiki entry:

(one extra thing to mention, the wiki entry failed to mention that $a^2 = R^2-r^2$) Equation (1) can be reduced as follows:

For above equation, though we separate it in three variables in toroidal coordinates, we can separate the $\phi$ variable:

The equation becomes:

and

Hence $u_m = K(\tau,\sigma)e^{im\theta}$, and $K$ satisfies (2). If someone knows how to proceed using analytical method for (2), I am interested in it as well.

Attribution
Source : Link , Question Author : Robert Filter , Answer Author : Shuhao Cao