- #1

Dopplershift

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## Homework Statement

Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions:

\begin{equation}

\frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0

\end{equation}

\begin{equation}

u(x,0) = cos(x) \text{for}\ 0<x<\pi

\end{equation}

\begin{equation}

u(x,y) = \to 0 \ \text{as} \ y \to 0 \ \text{for}\ 0<x<\pi

\end{equation}

## Homework Equations

General Solution:

\begin{equation}

\begin{split}

u(x,y) = \frac{a_0}{2L}(L-y) + \sum_{n = 1}^\infty \alpha_n sinh(\frac{n\pi}{L})(L-y)cos(\frac{n\pi x}{L}) \\

\text{where} \ \alpha_n = \frac{a_n}{sinh(n\pi} \\

\text{where} \ a_n = \frac{2}{L} \int_0^L cos(\frac{n\pi x}{L}) f(x) dx

\end{split}

\end{equation}

## The Attempt at a Solution

I am asking if I setup the problem right so far.

The visualization (i.e my picture of how I visualize the problem) The boundary conditions.

My attempt so far:

\begin{equation}

u_{xx} + u_{yy} = 0

\end{equation}

Assume:

\begin{equation}

u = F(x)G(y)

\end{equation}

Thus:

\begin{equation}

\begin{split}

u_{xx} = F''(x)G(y) \\

u_{yy} = F(x)G''(y) \\

F''(x)G(y) + F(x)G''(y) = 0 \\

\end{split}

\end{equation}

From this, we can define the following:

\begin{equation}

\frac{F''(x)}{F(x)} = \frac{G''(y)}{G(y)} = K \ \text{some constant}

\end{equation}

If u(x,0) = cos(x)

\begin{equation}

F(x)G(0) = cos(x) \\

\end{equation}

Can I assume that G(y) is equal to sin(y)?

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