Main: The Wave equation

Exercises

1.

Consider the wave equation problem

$\begin{displaymath} {\partial^2 \phi\over \partial t^2} = {\hbox{\boldmath $\De... ... $x$}},0) = H(1-\left\vert{\hbox{\boldmath $x$}}\right\vert), \end{displaymath}$

where $H(\cdot)$ is the unit Heaviside function. Use the transformation $\psi({\hbox{\boldmath$x$}},t)=H(t)\phi({\hbox{\boldmath$x$}},t)$ to convert the problem into the form

$\begin{displaymath} {{\partial^2 \psi}\over{\partial {t}^2}}-{\hbox{\boldmath $\Delta$}}\psi = f({\hbox{\boldmath $x$}},t). \end{displaymath}$

Use the three dimensional Green's function to find the solution of this problem in integral form and hence show

$\begin{displaymath} \phi({\hbox{\boldmath $0$}},t) = tH(1-t), \quad t>0. \end{displaymath}$

[Note: the delta function in the Green's function for the three-spatial dimensional wave equation is a ONE dimensional Green's function, NOT a vector Green's function. You will have to use spherical polar co-ordinates to get the correct answer.]

2.

Find the solution of the following problem

$\begin{displaymath} {\partial^2 \phi\over \partial t^2} = {\partial^2 \phi\ov... ...x),\quad {\partial \phi\over \partial t}({x},0) = 2xe^{-x^2} \end{displaymath}$

for t>0. Show that this can be written as the sum of two waves, one moving to the left and one moving to the right, both at unit speed.

3.

By writing $\psi=H(t)\phi$ , show that the solution of

$\begin{displaymath} {\partial^2 \phi\over \partial t^2} - {\partial^2 \phi\over \partial x^2} = \delta(x)\sin(\omega t),\quad t>0, \end{displaymath}$

with

$\begin{displaymath} \phi(x,0) = \cos(x),\quad {\partial \phi\over \partial t}({x},0) = 0 \end{displaymath}$

$\begin{displaymath} \phi(x,t) = \hbox{$1\over 2$}\left[ \cos(x+t)+\cos(x-t) + ... ...left(1-\cos(\omega (t-\left\vert x\right\vert))\right)\right]. \end{displaymath}$

Main: The Wave equation

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