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Surface Harmonics and Solid Harmonics

A surface harmonic of degree $ n$ (where $ n$ is a non-negative integer), denoted $ {\cal S}_n(\theta,\varphi)$ , is defined as a well-behaved solution to

$\displaystyle r^{\,2}\,\nabla^{\,2} {\cal S}_n + n\,(n+1)\,{\cal S}_n = 0$ (12.36)

on the surface of a sphere (i.e., $ r=$ constant). Here, $ r$ , $ \theta $ , $ \varphi$ are standard spherical coordinates, and $ \nabla^{\,2}$ is the Laplacian operator. It follows that (Love 1927)

$\displaystyle {\cal S}_n(\theta,\varphi) = \sum_{m=0,n} \left[c_n^{\,m}\,P_n^{\...
...\,\cos(m\,\varphi) + d_n^{\,m}\,P_n^{\,m}(\cos\theta)\,\sin(m\,\varphi)\right],$ (12.37)

where the $ c_n^{\,m}$ and $ d_n^{\,m}$ are arbitrary coefficients, and the $ P_n^{\,m}(x)$ associated Legendre functions.

A solid harmonic of degree $ n$ (where $ n$ is a non-negative integer), denoted $ {\cal R}_n(r,\theta,\varphi)$ , is defined as a well-behaved solution to

$\displaystyle \nabla^{\,2} {\cal R}_n = 0$ (12.38)

in the interior of a sphere (i.e., the region $ r<$ constant). It follows that (Love 1927)

$\displaystyle {\cal R}_n(r,\theta,\varphi) \propto r^{\,n}\,{\cal S}_n(\theta,\varphi).$ (12.39)

In particular, the functions $ {\cal R}_n^{\,(\pm m)}(r,\theta,\varphi)$ and $ {\cal R}_n^{\,(0)}(r,\theta)$ , introduced in Section 12.3, are solid harmonics of degree $ n$ . Note that the Cartesian coordinates $ x_i$ (where $ i$ runs from $ 1$ to $ 3$ ) are solid harmonics of degree $ 1$ . Moreover, $ \partial {\cal R}_n/\partial x_i$ is a solid harmonic of degree $ n-1$ . Finally, $ {\mit\Phi}_{\rm tide}({\bf r})$ , specified in Equation (12.35), is a solid harmonic of degree 2.

The following results regarding solid harmonics are useful (Love 1927):

$\displaystyle x_i\,\frac{\partial {\cal R}_n}{\partial x_i}$ $\displaystyle = r\,\frac{\partial {\cal R}_n}{\partial r} = n\,{\cal R}_n,$ (12.40)
$\displaystyle \nabla^{\,2}(x_i\,{\cal R}_n)$ $\displaystyle = \nabla\cdot({\cal R}_n\,\nabla x_i + x_i\,\nabla{\cal R}_n) = 2\,\nabla x_i\cdot\nabla{\cal R}_n = 2\,\frac{\partial {\cal R}_n}{\partial x_i},$ (12.41)
$\displaystyle \nabla^{\,2}(r^{\,m}\,{\cal R}_n)$ $\displaystyle = \nabla^{\,2}(r^{\,m+n}\,{\cal S}_n) = \frac{1}{r^{\,2}}\,\frac{...
...,\frac{d}{dr}(r^{\,m+n}\,{\cal S}_n)\right] - n\,(n+1)\,r^{\,m+n-2}\,{\cal S}_n$    
  $\displaystyle = m\,(m+2\,n+1)\,r^{\,m-2}\,{\cal R}_n.$ (12.42)

Here, use has been made of the Einstein summation convention (Riley 1974).


next up previous
Next: Planetary Rotation Up: Terrestrial Ocean Tides Previous: Expansion of Tide Generating
Richard Fitzpatrick 2016-01-22