# Laplace's Equation in the semi infinite strip

The problem is $$\Delta u = 0$$ in the semi infinite strip $$S = \{(x,y)|00\}$$ subject to the boundary conditions $$\begin{cases} u(0,y) = 0,\ y\ge0\\ u(1,y) = 0,\ y\ge0\\ u(x,0) = f(x),\ 0\le x \le 1 \end{cases}$$ where $$f$$ is a given function with $$f(0) = f(1) = 0$$. Write $$f(x) = \sum_{n=1}^{\infty} a_nsin(n\pi x)$$ and expand the general solution in terms of the special solutions given by $$u_n(x,y) = e^{-n \pi y}sin(n \pi x)$$.

The above is the problem.

So by using separation of variables and analyzing the first two boundary conditions, I got $$u(x,y) = \sum_{n=1}^{\infty} sin(n\pi x)(C_n e^{-n \pi y}+D_n e^{n \pi y})$$, where $$C_n$$ and $$D_n$$ are constants. My question is if I could get rid of the $$e^{n \pi y}$$ part from the general solution. I've seen someone did so but usually the boundary conditions would include for example $$\lim_{y\to\infty} u(x,y)$$ is finite. But here the boundary conditions do not say something like this. Can I still get rid of that part based on the give information? I'm a bit confused since the later instructions of the problem seem to suggest only $$e^{-n \pi y}$$ part remains.

Thank you so much for any insights.

• Unfortunately you need two conditions to determine two constants. The solution may be implied to be bounded at $y \to \infty$ which would give $D_n=0$ – Dylan Feb 3 at 9:00

Yes, there is one condition missing, as your semi-infinite strip has, very simply put, four boundary segments, at $$x \in \{0,1\}$$ and at $$y \in \{0,\infty\}$$. But you have prescribed conditions on only three of the boundary segments so far, which leaves you with arbitrary constants in the solution (you obtain $$C_n + D_n = a_n$$ where only $$a_n$$ is known).
The boundedness of the solution as $$y \rightarrow \infty$$ usually follows from some physical considerations.