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Gyroscopic Coefficients
Let
where use has been made of Equations (12.235) and (12.236). It follows from Equations (12.262) and (12.263) that
However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.261). Hence, we are left with
![$\displaystyle \beta_{r,\,s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\le...
...\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\theta}\right)d{\mit\Omega}.$](img4869.png) |
(12.271) |
Let
where use has been made of Equations (12.235) and (12.236). It follows from Equations (12.272) and (12.273) that
However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.271). Hence, we are left with
![$\displaystyle \beta_{-r,\,s}= -\int_{\mit\Omega}\cos\theta\left(\frac{\partial{...
...}{\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$](img4874.png) |
(12.274) |
Let
where use has been made of Equations (12.248) and (12.261). It follows from Equations (12.262) and (12.263) that
However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.249). Hence, we are left with
![$\displaystyle \beta_{r,\,-s}= \int_{\mit\Omega}\cos\theta\left(\frac{\partial{\...
...}{\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$](img4880.png) |
(12.277) |
Finally, let
where use has been made of Equations (12.248) and (12.271). It follows from Equations (12.272) and (12.273) that
However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.249). Hence, we are left with
![$\displaystyle \beta_{-r,\,-s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\...
...\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\theta}\right)d{\mit\Omega}.$](img4885.png) |
(12.280) |
Incidentally, the
,
,
, and
are known collectively as gyroscopic coefficients (Proudman 1916).
Next: Proudman Equations
Up: Terrestrial Ocean Tides
Previous: Auxilliary Eigenfunctions
Richard Fitzpatrick
2016-03-31