perturbation technique to compute surface wave stokes solutions

On Thu, 19 Apr 2012, c?dric ody wrote:

> I am working on oceanic surface waves and I would like to compute the
> well-known Stokes solution to this problem at various orders. The
> first order solution, which is also know as Airy's wave, is obtained
> from irrotational potential flow theory providing that the surface
> undulation is given (as a harmonic function). Solutions at higher
> order are obtained with a asymptotic/expansion technique.

If I've understood correctly, what happens in the leading order is
that your "given surface undulation" imposes separability in
co-ordinates of horizontal position, vertical position, and time.
Given this separability, Laplace's equation (implementing conservation
of volume) reduces to a second-order, linear, inhomogeneous ODE for
the potential as a function of the verical co-ordinate.  The Maxima
function "ode2" should be able to solve this equation, with or without
a no-penetration boundary condition at the ocean floor.  Substituting
(which can be done with the Maxima function "subst") this solution
into the combination of the linearized, generalized Bernoulli theorem
(implementing conservation of momentum) and the free surface boundary
condition (implementing the principle that the free surface moves at
the same vertical velocity as the fluid immediately below it), gives
you a second-order, linear, homogeneous ODE for the potential at mean
sea level as a function of time.  Again, the Maxima function "ode2"
should be able to solve this equation.

The process is pretty similar at each successive order of the
perturbation series, except that, on each occasion, the solutions from
the previous orders introduce a known (both to the user and to the CAS
that solved the previous orders) non-linear term in the generalized
Bernoulli theorem, and a known (both to the user and to the CAS that
solved the previous orders) difference in potential between mean sea
level and the free surface.  Mathematically, the effect of these is to
alter the inhomogeneity in the vertical ODE, and introduce an
inhomogenity in the temporal ODE; the Maxima functions "ode2" and
"subst" should still do the business.

So, I guess a good first step might be to search for "ode2" and
"subst" in the Maxima manual.

-- 

HTH,

Dan