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Basis Eigenfunctions

Suppose that the $ {\mit\Phi}_r(\theta,\phi)$ are well-behaved solutions of the eigenvalue equation

$\displaystyle D\,{\mit\Phi}_r +\lambda_r\,{\mit\Phi}_r=0,$ (12.230)

subject to the boundary conditions

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial {\mit\Phi}_r(\theta,\phi_\pm)}{\partial \phi} =0.$ (12.231)

It immediately follows from Equations (12.234) and (12.235) that

$\displaystyle \int_{\mit\Omega} ({\mit\Phi}_r^{\,\ast}\,D\,{\mit\Phi}_r - {\mit...
...a_r^{\,\ast})\int_{\mit\Omega} \vert{\mit\Phi}_r\vert^{\,2}\,d{\mit\Omega} = 0.$ (12.232)

Hence, we deduce that the $ \lambda_r$ are real. It follows that we can choose the $ {\mit\Phi}_r$ to be real functions. Equations (12.234) and (12.235) also yield

$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r\,D\,{\mit\Phi}_r\,d{\mit\Omega} = ...
...heta}\,\frac{\partial {\mit\Phi}_r}{\partial\phi}\right)^2\right]d{\mit\Omega},$ (12.233)

which implies that the $ \lambda_r$ are positive. Integration of Equation (12.235), subject to the boundary condition (12.236), gives

$\displaystyle \lambda_r\int_{\mit\Omega} {\mit\Phi}_r\,d{\mit\Omega} = 0.$ (12.234)

Because $ \lambda_r$ is positive, this implies that

$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r\,d{\mit\Omega} = 0.$ (12.235)

Finally, Equations (12.234) and (12.235) yield

$\displaystyle \int_{\mit\Omega} ({\mit\Phi}_s\,D\,{\mit\Phi}_r-{\mit\Phi}_r\,D\...
...da_s-\lambda_r)\int_{\mit\Omega} {\mit\Phi}_s\,{\mit\Phi}_r\,d{\mit\Omega} = 0.$ (12.236)

It follows that

$\displaystyle \int_{\mit\Omega} {\mit\Phi}_s\,{\mit\Phi}_r\,d{\mit\Omega} = 0$ (12.237)

if $ \lambda_s\neq \lambda_r$ . As is well known (Riley 1974), if $ {\mit\Phi}_r$ and $ {\mit\Phi}_{r'}$ are linearly independent solutions of (12.235) corresponding to the same eigenvalue, $ \lambda_r$ , then it is always possible to choose linear combinations of them that satisfy

$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r\,{\mit\Phi}_{r'}\,d{\mit\Omega} = 0.$ (12.238)

This argument can be extended to multiple linearly independent solutions corresponding to the same eigenvalue. Hence, we conclude that it is possible to choose the $ {\mit\Phi}_r$ such that they satisfy the orthonormality condition

$\displaystyle \lambda_r\int_{\mit\Omega} {\mit\Phi}_r\,{\mit\Phi}_s\,d{\mit\Ome...
...a_s\int_{\mit\Omega} {\mit\Phi}_r\,{\mit\Phi}_s\,d{\mit\Omega} = \delta_{r\,s}.$ (12.239)

Let $ F(\theta,\phi)$ be a well-behaved function. Suppose that

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial F(\theta,\phi_\pm)}{\partial \phi} =0.$ (12.240)

We can automatically satisfy the previous boundary condition by writing

$\displaystyle F= \sum_{r=1,\infty} a_r\,{\mit\Phi}_r.$ (12.241)

(Note that $ F$ is undetermined to an arbitrary additive constant which is chosen so as to ensure that $ \int_{\mit\Omega} F\,d{\mit\Omega}=0.$ ) Here, $ \lambda_1$ is the smallest eigenvalue of Equation (12.235), $ \lambda_2$ the next smallest eigenvalue, and so on. It follows from Equation (12.244) that

$\displaystyle a_r = \lambda_r\int_{\mit\Omega} F\,{\mit\Phi}_r\,d{\mit\Omega}.$ (12.242)

Suppose that the $ {\mit\Psi}_r(\theta,\phi)$ are well-behaved solutions of the eigenvalue equation

$\displaystyle D\,{\mit\Psi}_r +\mu_r\,{\mit\Psi}_r=0,$ (12.243)

subject to the boundary conditions

$\displaystyle {\mit\Psi}_r(\theta,\phi_\pm)=0.$ (12.244)

Using analogous arguments to those employed previously, we can show that the $ \mu_r$ are real and positive, and that the $ {\mit\Psi}_r$ can be chosen so as to satisfy the orthonormality constraint

$\displaystyle \mu_r\int_{\mit\Omega} {\mit\Psi}_r\,{\mit\Psi}_s\,d{\mit\Omega}=\mu_s\int_{\mit\Omega} {\mit\Psi}_r\,{\mit\Psi}_s\,d{\mit\Omega} = \delta_{r\,s}.$ (12.245)

Let $ F(\theta,\phi)$ be a well-behaved function. Suppose that

$\displaystyle F(\theta,\phi_\pm) =0.$ (12.246)

We can automatically satisfy the previous boundary condition by writing

$\displaystyle F= \sum_{r=1,\infty} a_r\,{\mit\Psi}_r.$ (12.247)

It follows from Equation (12.250) that

$\displaystyle a_r = \mu_r\int_{\mit\Omega} F\,{\mit\Psi}_r\,d{\mit\Omega}.$ (12.248)


next up previous
Next: Auxilliary Eigenfunctions Up: Terrestrial Ocean Tides Previous: Another Useful Lemma
Richard Fitzpatrick 2016-01-22