| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
53 |
|
Paper 4, Section II, B |
2010 |
Defining the function $G_{f_{3}}\left(\mathbf{r} ; \mathbf{r}{0}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}{0}\right|\right)$, prove Green's third identity for functions $u(\mathbf{r})$ satisfying Laplace's equation in a volume $V$ with surface $S$, namely
$$u\left(\mathbf{r}{0}\right)=\int{S}\left(u \frac{\partial G_{f_{3}}}{\partial n}-\frac{\partial u}{\partial n} G_{f_{3}}\right) d S$$
A solution is sought to the Neumann problem for $\nabla^{2} u=0$ in the half plane $z>0$ :
$$u=O\left(|\mathbf{x}|^{-a}\right), \quad \frac{\partial u}{\partial r}=O\left(|\mathbf{x}|^{-a-1}\right) \text { as }|\mathbf{x}| \rightarrow \infty, \quad \frac{\partial u}{\partial z}=p(x, y) \text { on } z=0$$
where $a>0$. It is assumed that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) d x d y=0$. Explain why this condition is necessary.
Construct an appropriate Green's function $G\left(\mathbf{r} ; \mathbf{r}_{0}\right)$ satisfying $\partial G / \partial z=0$ at $z=0$, using the method of images or otherwise. Hence find the solution in the form
$$u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) f\left(x-x_{0}, y-y_{0}, z_{0}\right) d x d y$$
where $f$ is to be determined.
Now let
$$p(x, y)= \begin{cases}x & |x|,|y|<a \ 0 & \text { otherwise }\end{cases}$$
By expanding $f$ in inverse powers of $z_{0}$, show that
$$u \rightarrow \frac{-2 a^{4} x_{0}}{3 \pi z_{0}^{3}} \quad \text { as } \quad z_{0} \rightarrow \infty .$$