Eigenfunctions of the Helmholtz equation in Toroidal geometry

the Helmholtz equation
Δψ+k2ψ=0
has a lot of fundamental applications in physics since it is a form of the wave equation Δϕc2ttϕ=0 with an assumed harmonic time dependence e±iωt.

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 Δ in certain coordinate systems. Examples are cartesian, elliptical and cylindrical ones.

For now I am interested in a toroidal geometry,
k(r)={ktorT2koutelse

where T2={(x,y,z):r2(x2+y2R)2+z2}

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 T2 means the Torus, which is a 2-manifold: T2[0,2πr]×[0,2πR], the solution to
    Δψ+k2ψ=0
    bears the form: for mZ2, ψk=eimx, with |m|=m21+m22=k.
    The reason behind this is that T2S1(r)×S1(R), and for (1) on S1 has eigenvectors eimx where |m|=k, then the Fourier expansion on product spaces use basis eimixi.

  • 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:
    ψ=ucoshτcosσ,
    then by the Laplacian in the toroidal geometry in that wiki entry:
    Δψ=(coshτcosσ)3a2sinhτ[sinhτσ(1coshτcosσΦσ)+τ(sinhτcoshτcosσΦτ)+1sinhτ(coshτcosσ)2Φϕ2].
    (one extra thing to mention, the wiki entry failed to mention that a2=R2r2) Equation (1) can be reduced as follows:
    2uτ2+coshτsinhτuτ+1sinh2τ2uϕ2+2uσ2+((R2r2)k2(coshτcosσ)2+14)u=0.
    For above equation, though we separate it in three variables in toroidal coordinates, we can separate the ϕ variable:
    u=K(τ,σ)Φ(ϕ).
    The equation becomes:
    Δτ,σK+coshτsinhτKτ+((R2r2)k2(coshτcosσ)2+14m2sinh2τ)K=0,
    and
    Φ+m2Φ=0.
    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

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