Author 
Comment/Response 
Michael

07/26/12 07:56am
I'm not sure what you're asking. It seems, but I may be wrong, that you know that
the partial dh/dt in Mathematica is D[h[theta,phi,t], t]
the 2nd partial is D[h[theta,phi,t], t, t]
and so on for partials with resp. to theta, phi. So translating the derivatives doesn't seem a problem.
You have probably found out that sin(phi) in Mathematica is written Sin[phi]
Write == instead of = for equations.
The form of NDSolve is NDSolve[{yourSphericalDE, listOfInitialConditions}, h, variables]. You'll need to figure out the initial conditions. If theta is the longitudinal variable, then one condition is h[0,phi,t]==h[2 Pi, phi,t] for h to be continuous. Specifying the derivative at the boundary is trickier: Derivative[1,0,0][h][0,phi,t]==Sin[phi], or whatever.
Derivative[1,0,0][h] represents the 1st partial derivative of h with respect the first argument.
Derivative[2,0,0][h] represents the 2nd partial derivative of h with respect the first argument.
Derivative[0,1,0][h] represents the 1st partial derivative of h with respect the second argument.
And so on.
Those are my first thoughts...
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