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Next: Gyroscopic Coefficients Up: Terrestrial Ocean Tides Previous: Basis Eigenfunctions

Auxilliary Eigenfunctions

Let

$\displaystyle X$ $\displaystyle = -\frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Phi}_r}{\partial \phi},$ (12.249)
$\displaystyle Y$ $\displaystyle = \cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta},$ (12.250)
$\displaystyle P$ $\displaystyle ={\mit\Phi}_r',$ (12.251)
$\displaystyle Q$ $\displaystyle = {\mit\Psi}_r''$ (12.252)

in Equations (12.191)-(12.196). It follows that

$\displaystyle D\,{\mit\Phi}_r'$ $\displaystyle = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left...
...hi}\left(\cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta}\right)\right],$ (12.253)
$\displaystyle D\,{\mit\Psi}_r''&= \frac{1}{\sin\theta}\left[ \frac{\partial}{\p...
...hi}\left( \cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial \phi}\right)\right],$ (12.254)

and

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial {\mit\Phi}_r'(\theta,\phi_\pm)}{\partial\phi}$ $\displaystyle = -\cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta},$ (12.255)
$\displaystyle {\mit\Psi}_r''(\theta,\phi_\pm)$ $\displaystyle = 0,$ (12.256)

as well as

$\displaystyle -\frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Phi}_r}{\partial\phi}$ $\displaystyle = - \frac{\partial{\mit\Phi}_r'}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}_r''}{\partial\phi},$ (12.257)
$\displaystyle \cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta}$ $\displaystyle = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}_r'}{\partial\phi} + \frac{\partial{\mit\Psi}_r''}{\partial\theta}.$ (12.258)

Let

$\displaystyle X$ $\displaystyle = \cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial \theta},$ (12.259)
$\displaystyle Y$ $\displaystyle = \frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Psi}_r}{\partial\phi},$ (12.260)
$\displaystyle P$ $\displaystyle ={\mit\Phi}_r'',$ (12.261)
$\displaystyle Q$ $\displaystyle = {\mit\Psi}_r'$ (12.262)

in Equations (12.191)-(12.196). It follows that

$\displaystyle D\,{\mit\Psi}_r''&= - \frac{1}{\sin\theta}\left[\frac{\partial}{\...
...hi}\left( \cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial \phi}\right)\right],$ (12.263)
$\displaystyle D\,{\mit\Psi}_r'$ $\displaystyle = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left...
...hi}\left(\cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial\theta}\right)\right],$ (12.264)

and

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial {\mit\Phi}_r''(\theta,\phi_\pm)}{\partial\phi}$ $\displaystyle =- \frac{\cos\theta}{\sin\theta}\,\frac{\partial {\mit\Psi}_r(\theta,\phi_\pm)}{\partial\phi},$ (12.265)
$\displaystyle {\mit\Psi}_r'(\theta,\phi_\pm)$ $\displaystyle = 0,$ (12.266)

as well as

$\displaystyle \cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial\theta}$ $\displaystyle = - \frac{\partial{\mit\Phi}_r''}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}_r'}{\partial\phi},$ (12.267)
$\displaystyle \frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Psi}_r}{\partial\phi}$ $\displaystyle = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}_r''}{\partial\phi} + \frac{\partial{\mit\Psi}_r'}{\partial\theta}.$ (12.268)


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Next: Gyroscopic Coefficients Up: Terrestrial Ocean Tides Previous: Basis Eigenfunctions
Richard Fitzpatrick 2016-03-31