| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
51 |
|
Paper 3, Section II, D |
2019 |
By differentiating the expression $\psi(t)=H(t) \sin (\alpha t) / \alpha$, where $\alpha$ is a constant and $H(t)$ is the Heaviside step function, show that
$$\frac{d^{2} \psi}{d t^{2}}+\alpha^{2} \psi=\delta(t)$$
where $\delta(t)$ is the Dirac $\delta$-function.
Hence, by taking a Fourier transform with respect to the spatial variables only, derive the retarded Green's function for the wave operator $\partial_{t}^{2}-c^{2} \nabla^{2}$ in three spatial dimensions.
[You may use that
$$\frac{1}{2 \pi} \int_{\mathbb{R}^{3}} e^{i \mathbf{k} \cdot(\mathbf{x}-\mathbf{y})} \frac{\sin (k c t)}{k c} d^{3} k=-\frac{i}{c|\mathbf{x}-\mathbf{y}|} \int_{-\infty}^{\infty} e^{i k|\mathbf{x}-\mathbf{y}|} \sin (k c t) d k$$
without proof.]
Thus show that the solution to the homogeneous wave equation $\partial_{t}^{2} u-c^{2} \nabla^{2} u=0$, subject to the initial conditions $u(\mathbf{x}, 0)=0$ and $\partial_{t} u(\mathbf{x}, 0)=f(\mathbf{x})$, may be expressed as
$$u(\mathbf{x}, t)=\langle f\rangle t$$
where $\langle f\rangle$ is the average value of $f$ on a sphere of radius $c t$ centred on $\mathbf{x}$. Interpret this result.