Main: The Wave equation

Retarded Potentials

The solution of the problem

$\begin{displaymath} {1\over c^{2}}{{\partial^2 \phi}\over{\partial {t}^2}} - {\hbox{\boldmath $\Delta$}}_{3}\phi = f({\hbox{\boldmath $x$}},t) \end{displaymath}$

$\begin{displaymath} \phi({\hbox{\boldmath $x$}},t) = \int^{\infty}_{-\infty}\ke... ...\right\vert\right) d^{3}{\hbox{\boldmath $y$}}\kern 4pt d\tau \end{displaymath}$

and the $\tau$ integral picks out the value of the integrand at $\tau = t-\left\vert{\hbox{\boldmath$x$}}-{\hbox{\boldmath$y$}}\right\vert/c$ , hence

$\begin{displaymath} \phi({\hbox{\boldmath$x$}},t) = {1\over 4\pi} \int\limits... ...-{\hbox{\boldmath$y$}}\right\vert} d^{3}{\hbox{\boldmath$y$}}. \end{displaymath}$

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This is referred to as the Retarded Potential Integral. Essentially it represents the superposition (ie, the integral) of disturbances at points ${\hbox{\boldmath$y$}}$ . The disturbance which eminates from point ${\hbox{\boldmath$y$}}$ at time t reaches the point ${\hbox{\boldmath$x$}}$ at time $t+\left\vert{\hbox{\boldmath$x$}}-{\hbox{\boldmath$y$}}\right\vert/c$ (ie, at t plus the time it takes the wave to travel from ${\hbox{\boldmath$y$}}$ to ${\hbox{\boldmath$x$}}$ ). The disturbance is attenuated with the distance between ${\hbox{\boldmath$y$}}$ and ${\hbox{\boldmath$x$}}$ . The following example makes this idea somewhat clearer.

Example:

Find the field generated by a time harmonic point source at the origin which is switched on at time t=0.

In terms of the wave equation the problem is to solve

$\begin{displaymath} {1\over c^{2}}{{\partial^2 \phi}\over{\partial {t}^2}} - ... ...a$}}\phi = H(t)\delta({\hbox{\boldmath $x$}}) e^{{i\omega t}}. \end{displaymath}$

Using the retarded potential integral (8.4) we have

$\begin{displaymath} \phi({\hbox{\boldmath $x$}},t) = {1\over 4\pi} \int\limit... ...{\hbox{\boldmath $y$}}\right\vert} d^{3}{\hbox{\boldmath $y$}} \end{displaymath}$

where $f({\hbox{\boldmath$y$}},t) = \delta({\hbox{\boldmath$y$}})H(t)e^{i\omega t}$ , so

$\begin{displaymath} \phi({\hbox{\boldmath $x$}},t) = {1\over 4\pi} \int\limits... ...$y$}}\right\vert/c\right) \right) d^{3}{\hbox{\boldmath $y$}} \end{displaymath}$

and, as the $\delta({\hbox{\boldmath$y$}})$ term simply selects the value of the integrand at ${\hbox{\boldmath$y$}}={\hbox{\boldmath$0$}}$ ,

$\begin{displaymath} \phi({\hbox{\boldmath $x$}},t) = {H(t-\left\vert{\hbox{\bol... ...t-\left\vert{\hbox{\boldmath $x$}}\right\vert/c\right)\right). \end{displaymath}$

The term

$\begin{displaymath} H(t-\left\vert{\hbox{\boldmath $x$}}-{\hbox{\boldmath $y$}}\right\vert/c) \end{displaymath}$

means that the disturbance is confined to the expanding sphere

$\begin{displaymath} \left\vert{\hbox{\boldmath $x$}}\right\vert \le ct \end{displaymath}$

whose radius expands at the wave speed c. The surface of the sphere

$\begin{displaymath} \left\vert{\hbox{\boldmath $x$}}\right\vert = ct \end{displaymath}$

represents the wave front of the expanding sphere of disturbance. Once the wave front has passed a point ${\hbox{\boldmath$x$}}$ , (so ${\hbox{\boldmath$x$}}$ is inside the sphere, ie, $\left\vert{\hbox{\boldmath$x$}}\right\vert<ct$ ) the disturbance is simply

$\begin{displaymath} \phi = {1\over 4\pi \left\vert{\hbox{\boldmath$x$}}\right\v... ...i\omega(t-\left\vert{\hbox{\boldmath$x$}}\right\vert/c)\right) \end{displaymath}$

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and the phase of the oscillation at ${\hbox{\boldmath$x$}}$ differs from the phase of the oscillation at the source of the disturbance, ${\hbox{\boldmath$0$}}$ , by a factor of $\left\vert{\hbox{\boldmath$x$}}\right\vert/c$ . This phase difference is simply the time it takes the disturbance to propagate from ${\hbox{\boldmath$0$}}$ to ${\hbox{\boldmath$x$}}$ , travelling at speed c.

In other words, when the disturbance reaches the point $\left\vert{\hbox{\boldmath$x$}}\right\vert$ it has the same phase that it had when it left the origin. Put another way, it will take a time $\left\vert{\hbox{\boldmath$x$}}\right\vert/c$ before a disturbance at the origin will propagate out to an observer at position ${\hbox{\boldmath$x$}}$ . When the disturbance does reach the observer at ${\hbox{\boldmath$x$}}$ it will have the same phase that it had when it left the origin.

The factor of $1/( 4\pi\left\vert{\hbox{\boldmath$x$}}\right\vert)$ says that the amplitude of the disturbance decays inversely with the distance from the source of the disturbance (at least it does so inside the sphere of disturbance).

The retarded potential integral (8.4) simply says that we can regard a continuous source term $f({\hbox{\boldmath$y$}},\tau)$ as an integral of a lot of point source terms. The effect of the continuous source at position ${\hbox{\boldmath$x$}}$ time t, ie, $\phi({\hbox{\boldmath$x$}},t)$ , is simply the sum of all disturbances from point sources ${\hbox{\boldmath$y$}}$ , time $\tau$ divided by the distance between ${\hbox{\boldmath$x$}}$ and ${\hbox{\boldmath$y$}}$ (ie, the $1/\left\vert{\hbox{\boldmath$x$}}-{\hbox{\boldmath$y$}}\right\vert$ term) with the phase adjustment for the time it takes the disturbance at point ${\hbox{\boldmath$y$}}$ to reach the point ${\hbox{\boldmath$x$}}$ (ie, the $t-\left\vert{\hbox{\boldmath$x$}}-{\hbox{\boldmath$y$}}\right\vert/c$ term).

Discarding the time factor in (8.5), this has the same form as the Green's function for the Helmholtz equation (but with the opposite sign). This is not surprising since, for $\left\vert{\hbox{\boldmath$x$}}\right\vert<ct$ , we have

$\begin{displaymath} {1\over c^{2}}{{\partial^2 \phi}\over{\partial {t}^2}} = -{\omega^{2}\over c^{2}}\phi = -k^{2}\phi \end{displaymath}$

and hence

$\begin{displaymath} {1\over c^{2}}{{\partial^2 \phi}\over{\partial {t}^2}} - {\... ...ta$}}\phi = H(t)\delta({\hbox{\boldmath $x$}}) e^{i\omega t} \end{displaymath}$

reduces to

$\begin{displaymath} ({\hbox{\boldmath $\Delta$}}+k^{2})\phi = -H(t)\delta({\hbox{\boldmath $x$}})e^{i\omega t} \end{displaymath}$

and writing $\phi=H(t)e^{i\omega t}\psi$ this becomes the problem for (minus) the Helmholtz Green's function;

$\begin{displaymath} ({\hbox{\boldmath $\Delta$}}+ k^{2})\psi = -\delta({\hbox{\boldmath $x$}}) \end{displaymath}$

Main: The Wave equation

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