# [Maxima] perturbation technique to compute surface wave stokes solutions

Richard Fateman fateman at eecs.berkeley.edu
Wed Apr 25 10:53:57 CDT 2012

```I don't know about the specifics of your problem, but years ago I worked
from a
book by Nayfeh on asymptotic methods, trying to use computer algebra.

Most methods seem to have a step in which some human judgment is used
to eliminate terms, so not so easy to totally automate.  In particular,
human
judgment is used in 2 ways.  (a) To figure out which simplification
functionality
to use to simplify an intermediate result, and (b) To figure out which
resulting
terms are "secular".
On the other hand, the rest of the algebra can be largely automated.
(Perhaps by using the ode2 program)

There are some methods that appear more mechanical than others, e.g.
method of multiple scales.

However, the bulk of the material I encountered looked like the presentation
of particular examples for which particular methods worked,  without any
hint of
what other problems could also be solved the same way or
how you could choose among various competing methods to try.

On 4/25/2012 8:45 AM, Dan wrote:
> On Thu, 19 Apr 2012, cédric 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.
>
>
>
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