|All Solutions of Type: Wave Phenomena|
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Wave Phenomena}Group Velocity
Recall the definition of index of refraction in terms of the speed of light in vacuum and the velocity of the light in medium , . Since the velocity of light in any medium is , the condition usually holds. However, even if rock salt has , the wave does not exceed the speed of light. The group velocity can travel faster than the speed of light, and apparently it is the group velocity at work in the equation .
One can also arrive at this conclusion via MOE (Method of Elimination):
(A) Relativity is a pretty general theory that supposedly applies everywhere. (Newtonian mechanics can be achieved using the proper approximation technique.) X-out this choice.
(B) An x-ray is a specific frequency in the electromagnetic spectrum. Nothing forbids an electromagnetic wave from transmitting signals, and thus this choice is out.
(C) Imaginary mass? If confused, save for last comparison.
(D) Historical precedence shouldn't change the correctness of a theory (at least not in the ideal world ETS lives in)... X-this out.
(E) One recalls that there's a difference between group and phase velocities. Could this difference allow one of them to exceed the speed of light? Probably. In either case, this is a much better choice than the other remaining candidate, choice (C). So, choose this.
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Wave Phenomena}Light Doppler Shift
One can derive the Doppler Shift for light as follows:
For source/observer moving towards each other, one has the wavelength emitted from the source decreasing, thus . Thus, .
For source/observer moving away from each other, one has the wavelength emitted from the source increasing, thus . Thus, .
Where in the last equality in the above, one applies time dilation from special relativity, and the fact that in general.
Now that one has the proper battle equipment, one can proceed with the problem.
This problem is essentially the difference in wavelengths seen from a red shift and blue shift, i.e., light moving towards and away from the observer.
, where the approximation is made since one assumes the particle is moving at a non-relativistic speed.
2 km is closest to choice (B).
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Since the wavelength of the wave does not change, as the pipe presumably stays approximately the same length, only the frequency varies. If the speed of sound changes, then the frequency changes. If the speed of sound is lower than usual, then the frequency is lower. Thus, choices (A), (B) and (C) remain. Calculate to get choice (B).
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