Does it make sense to talk about "frequency" when expanding a function using spherical harmonics? If I had a function defined in the sphere and I expand it using spherical harmonics, such as
$$
f(\theta,\phi) = \sum_{l=0}^{+\infty} \sum_{m=-l}^{l} c_{lm}Y_l^m(\theta,\phi)
$$
does it make any sense to talk about "low frequency" and "high frequency"? If yes, what is usually meant by that?
My analogy is with the Fourier analysis where low frequency is usually associated wit the first terms of a series. I think something similar can be said here, is it correct?
 A: Start with a uniform string stretched between $-L$ and $L$. The wave equation for this is
$$
                           u_{tt}=cu_{xx},\;\;\; u(t,\pm L)=0, c >0.
$$
The resonant, or characteristic, frequencies of the string are the eigenvalues of the operator $Df = cf''$ subject to $f(\pm L)=0$. For any such $f$ with eigenvalue $\lambda$, there is a stationary (standing wave) solution of the wave equation
$$
                     f(x)\sin(\sqrt{\lambda}t+\phi).
$$
The string has evenly spaced $\sqrt{\lambda}$.
If you have a spherical balloon of radius $1$, then the wave equation is
$$
                u_{tt}=c\Delta_S u,
$$
where $\Delta_S$ is the Laplacian on the sphere. The stationary solutions correspond to the eigenfunctions of the Laplacian operator on the sphere, and the frequency of this solution is proportional to $\sqrt{\lambda}$, where $\lambda$ is the eigenvalue of the spherical Laplacian. There are the natural resonant frequencies of the balloon. The spherical harmonics are the eigenfunctions of the Laplacian on the unit spherical shell, and these are the displacement distributions of the natural resonant modes.
The spherical harmonics are the eigenfunctions of the spherical Laplacian:
$$
          \Delta_S Y_{l}^{m}(\theta,\varphi)= -l(l+1)Y_{l}^{m}.
$$
The corresponding frequencies in $t$ are $\sqrt{l(l+1)}$, which are not evenly spaced, and one of the reasons an excited balloon does not sound very musical. And there are multiple shapes for the given $l$, and these all have the same natural frequency. The displacements of the natural shapes are described by the spherical harmonics, and the frequency of their oscillations when excited is given as a constant times the square root of the eigenvalue.
http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/ClassicalWaveEquations1.htm
https://en.wikipedia.org/wiki/Spherical_harmonics
