6
$\begingroup$

I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line.

The main equation is $$\frac{d^2\eta}{dt^2} + \frac{d}{dx}\left(h\frac{d\eta}{dx}\right)=0,$$ where $\eta(x,t)$ is the wave equation, and $h$ is the depth. enter image description here

I divided the problem to two parts, one with the constant depth (zone 1) and the other with variable depth (zone 2). By assuming $h/h = 1$, for zone 1 the answer of main equation is $$\eta(x,t)=A_i e^{-ik(x+ct) }+A_r e^{ik(x-ct)}.$$

For zone 2 with variable depth I want to solve the main equation with Hermite polynomials. By assuming the answer like $\eta=\eta(x,t)=A(x)e^{-ikct}$, the goal is finding $A(x)$.

$$A(x)=\sum_{n=0}^\infty a_n H_n $$ and $$h=f(x)=\sum_{n=0}^\infty b_n H_n, $$ where $H_n$ is the $n$th Hermite Polynomial. Unfortunately I can’t achieve to an exact solution for the problem . Exact Solution Must be obtained by using the Hermite polynomials .

  • 0
    Please try re-formatting your post using [this website's built-in LaTeX-like support for equations](http://meta.math.stackexchange.com/q/1773/1543). Especially I'm not sure how to parse the expression in the last paragraph. Also, you seem to be missing a picture.2012-02-10
  • 0
    I changed your math to LaTeX format so it displays properly; please check that I preserved your meaning. You have a function $f(x)$ which is defined but never used. Also, there was a placeholder for an image which had no URL, so no image was displayed; I removed it.2012-02-10
  • 0
    What does "where $\eta(x,t)$ is the wave equation" mean? Maybe you mean "$\eta(x,t)$ is a solution of the wave equation"?2013-03-29

1 Answers 1