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Some waves take time to get places, so I was wondering how to describe a wave source, such as a disturbance in a pond, or light going in every direction. I have seen the following description, $$\psi(r,t) = \frac{\mathcal{A}}{r} e^{i(kr-\omega t)}$$But doesn't this assume that the wave already permeates all of space?

For example, could one use a piecewise function like this? $$\psi(r,t) = \begin{cases} 0 & t-t_0 < \frac{r}{v} \\\frac{\mathcal{A}}{r} e^{i(kr-\omega t)} & \text{otherwise}\end{cases}$$

Where $t_0$ is the time the source started emanating the wave, $v$ is the velocity of the wave.

How could one describe such a wave in a different way?

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    $\begingroup$ Yes, of course there are waves that don't exist everywhere. The second wave written in the question is a perfectly acceptable one. You may want to reword the question to invite more than just a yes/no answer. $\endgroup$ – DanielSank Jul 24 '18 at 1:48
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A time-space domain description works better than a frequency domain description. Your complex exponentials are frequency domain descriptions. Time-space domain descriptions include pulse like waves which do not 'permeate all of space'.

For example in three dimensions $\frac{1}{4\pi r}\delta (r-ct) $ is a spherical impulsive wave. (r is the distance from the center of the sphere and c is the propagation speed) This is often considered to be the elementary solution of the wave equation. Notice that the wave exists only on the surface of the sphere where r=ct, so it does not permeate all of space. It is an expanding spherical shell with an amplitude inversely proportional to its radius.

Other waves can be derived from it by convolution.

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  • $\begingroup$ Specify "works better". $\endgroup$ – DanielSank Jul 24 '18 at 18:56
  • $\begingroup$ Easier to analyze and understand. $\endgroup$ – user45664 Jul 24 '18 at 19:00
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A common approach is to apply an envelope to the function. Typically this is done in the time domain rather than the frequency domain because the intuitive bounds you intended to apply are typically phrased in time and space. For example, it is quite typical to want to describe a wave source which is "turned on" at a given moment and propagates outwards. If you multiply your wave function by $u(t)$ where $u$ is the step function, you get this behavior.

In some senses, this is really just a different way of phrasing the piecewise notation you used. However, it has a distinct advantage because it can be converted to frequency space. Multiplication in the time domain is the same as convolution in the frequency domain. Knowing this, we can actually calculate the spectral effects of that step function.

Indeed this shows why you had trouble coming up with a closed form solution in the frequency domain. The result of a sharp cutoff is an infinite series of waves at different harmonics, where the low frequencies have higher amplitudes than the high frequencies. You would not be able to come up with a closed form solution in the frequency domain without an infinite summation.

This approach also permits you to use more "gentle" envelope functions. It's common to talk about signals that are bounded by a sinc envelope ($\frac{\sin x}{x}$), and they behave rather well when you convolve them.

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