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Next: Another Useful Lemma Up: Terrestrial Ocean Tides Previous: Useful Lemma

Transformation of Laplace Tidal Equations

Let $ X=\xi$ , $ Y=\eta$ , $ P={\mit\Phi}$ , and $ Q={\mit\Psi}$ in Equations (12.191)-(12.196). It follows that [cf., Equation (12.161)]

$\displaystyle D\,{\mit\Phi}$ $\displaystyle =\zeta,$ (12.202)
$\displaystyle D\,{\mit\Psi}$ $\displaystyle = \frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,\eta) -\frac{\partial\xi}{\partial \phi}\right],$ (12.203)

where

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}(\theta,\phi_\pm,t)}{\partial \phi}$ $\displaystyle = 0,$ (12.204)
$\displaystyle {\mit\Psi}(\theta,\phi_\pm,t)$ $\displaystyle =0.$ (12.205)

Here, use has been made of Equations (12.186) and (12.190). Furthermore [cf., Equations (12.159) and (12.160)],

$\displaystyle \xi$ $\displaystyle = -\frac{\partial{\mit\Phi}}{\partial \theta} - \frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}}{\partial\phi},$ (12.206)
$\displaystyle \eta$ $\displaystyle = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}}{\partial\phi} +\frac{\partial{\mit\Psi}}{\partial\theta}.$ (12.207)

Let $ X=\cos\theta\,\eta$ , $ Y=-\cos\theta\,\xi$ , $ P={\mit\Phi}'$ , and $ Q={\mit\Psi}'$ in Equations (12.191)-(12.196). It follows that

$\displaystyle D\,{\mit\Phi}'$ $\displaystyle =-\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\si...
...eta\,\cos\theta\,\eta)-\frac{\partial}{\partial\phi}\,(\cos\theta\,\xi)\right],$ (12.208)
$\displaystyle D\,{\mit\Psi}'$ $\displaystyle = -\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\s...
...a\,\cos\theta\,\xi) +\frac{\partial}{\partial \phi}\,(\cos\theta\,\eta)\right],$ (12.209)

where

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}'(\theta,\phi_\pm,t)}{\partial \phi}$ $\displaystyle =\cos\theta\,\xi(\theta,\phi_\pm,t),$ (12.210)
$\displaystyle {\mit\Psi}'(\theta,\phi_\pm,t)$ $\displaystyle =0.$ (12.211)

Furthermore,

$\displaystyle \cos\theta\,\eta$ $\displaystyle = -\frac{\partial{\mit\Phi}'}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}'}{\partial\phi},$ (12.212)
$\displaystyle -\cos\theta\,\xi$ $\displaystyle = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}'}{\partial\phi} +\frac{\partial{\mit\Psi}'}{\partial\theta}.$ (12.213)

Substitution of Equations (12.211), (12.212), (12.217), and (12.218) into Equations (12.187) and (12.188) yields

$\displaystyle \frac{\partial A}{\partial\theta} + \frac{1}{\sin\theta}\,\frac{\partial B}{\partial\phi}$ $\displaystyle = 0,$ (12.214)
$\displaystyle \frac{\partial B}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial A}{\partial\phi}$ $\displaystyle = 0,$ (12.215)

where

$\displaystyle A$ $\displaystyle = \skew{5}\ddot{\mit\Phi} -2\,{\mit\Omega}\,\skew{5}\dot{\mit\Phi}' - \frac{g\,d}{a^{\,2}}\,(\zeta-F),$ (12.216)
$\displaystyle B$ $\displaystyle =\skew{5}\ddot{\mit\Psi} - 2\,{\mit\Omega}\,\skew{5}\dot{\mit\Psi}'.$ (12.217)

Equations (12.219) and (12.220) can be combined to give

$\displaystyle D\,A$ $\displaystyle = 0,$ (12.218)
$\displaystyle D\,B$ $\displaystyle =0.$ (12.219)

Moreover, it follows from Equations (12.188), (12.190), (12.209), (12.210), (12.215), and (12.216) that

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial A(\theta,\phi_\pm,t)}{\partial \phi}$ $\displaystyle = 0,$ (12.220)
$\displaystyle B(\theta,\phi_\pm,t)$ $\displaystyle =0.$ (12.221)

We have already seen that the solution of Equation (12.205), subject to the boundary condition (12.204), is $ P'={\rm constant}$ . It follows that the solution of Equation (12.223), subject to the boundary condition (12.225), is $ A={\rm constant}$ . Analogous arguments reveal that the solution of Equation (12.224), subject to the boundary condition (12.226), is $ B=0$ . Hence, we deduce that the Laplace tidal equations, (12.186)-(12.188), are equivalent to the following set of equations:

$\displaystyle D\,{\mit\Phi}$ $\displaystyle =\zeta,$ (12.222)
$\displaystyle \skew{5}\ddot{\mit\Phi} -2\,{\mit\Omega}\,\skew{5}\dot{\mit\Phi}' - \frac{g\,d}{a^{\,2}}\,(\zeta-{\cal F})$ $\displaystyle ={\rm constant},$ (12.223)
$\displaystyle \skew{5}\ddot{\mit\Psi} - 2\,{\mit\Omega}\,\skew{5}\dot{\mit\Psi}'$ $\displaystyle = 0,$ (12.224)

where $ {\mit\Phi}$ , $ {\mit\Psi}$ , $ {\mit\Phi}'$ , $ {\mit\Psi}'$ , and $ {\cal F}$ are defined in Equations (12.211), (12.212), (12.217), (12.218), and (12.189), respectively.


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