BELIEVE ME NOT!   A SKEPTIC's GUIDE
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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 wave^{14.5}at velocity . This doesn't require any elaborate derivation;
is just the position measured in such an observer's reference frame!
Footnotes
 . . . wave^{14.5}
 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