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## Speed of Propagation

Neither of the images in Fig. 14.1 captures the most important qualitative feature of the wave: namely, that it propagates -- i.e. moves steadily along in the direction of . If we were to let the snapshot in Fig. 14.1b become a movie, so that the time dependence could be seen vividly, what we would see would be the same wave pattern sliding along the graph to the right at a steady rate. What rate? Well, the answer is most easily given in simple qualitative terms:

The wave has a distance (one wavelength) between "crests." Every period , one full wavelength passes a fixed position. Therefore a given crest travels a distance in a time so the velocity of propagation of the wave is just

 (14.6)

where I have used as the symbol for the propagation velocity even though this is a completely general relationship between the frequency , the wave vector magnitude and the propagation velocity of any sort of wave, not just electromagnetic waves (for which has its most familiar meaning, namely the speed of light).

This result can be obtained more easily by noting that is a function only of the phase of the oscillation,

 (14.7)

and that the criterion for "seeing the same waveform" is   constant  or  . If we take the differential of Eq. (7) and set it equal to zero, we get

But  ,  the propagation velocity of the waveform. Thus we reproduce Eq. (6). This treatment also shows why we chose     for the time dependence so that Eq. (7) would describe the phase: if we used     then the phase would be     which gives  ,  - i.e. a waveform propagating in the negative direction (to the left as drawn).

If we use the relationship (6) to write   ,  so that Eq. (4) becomes

we can extend the above argument to waveforms that are not of the ideal sinusoidal shape shown in Fig. 14.1; in fact it is more vivid if one imagines some special shape like (for instance) a pulse propagating down a string at velocity . As long as is a function only of  , no matter what its shape, it will be static in time when viewed by an observer traveling along with the wave14.5at velocity . This doesn't require any elaborate derivation;    is just the position measured in such an observer's reference frame!

#### Footnotes

. . . wave14.5
Don't try this with an electromagnetic wave! The argument shown here is explicitly nonrelativistic, although a more mathematical proof reaches the same conclusion without such restrictions.

Next: The Wave Equation Up: Wave Phenomena Previous: Traveling Waves
Jess H. Brewer - Last modified: Sun Nov 15 17:58:14 PST 2015