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Doppler Effect |
The Doppler effect, named after Christian Johann Doppler, is a phenomenon that occurs in both classical and relativistic physics. It causes the frequency of a wave measured by an observer to differ from that emitted by a source. In both classical and relativistic physics, the frequency is lowered when the source is moving away from the observer and raised when it is moving toward the observer.
The
Classical Doppler Equation
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· fL = the frequency observed by the listener · fS = the frequency emitted by the source · c = the speed of the wave through the medium · vL = the velocity of the listener through the medium · vS = the velocity of the source through the medium · The positive direction is from listener to source |
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The classical Doppler equation pertains to sound and other waves that are transmitted at non-relativistic speeds through a medium when both the listener and the source are moving slower than the wave along a straight line. As you can see, the classical Doppler effect depends upon the motion of both the source and the listener. The derivation of this equation can be found in most university physics books.
The
Relativistic Doppler Equation
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· f = the relative frequency observed in the rest frame · fo = the proper frequency emitted in the moving frame · c = speed of light in a vacuum · v = the velocity of the moving frame · The positive direction is from observer to source |
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The relativistic Doppler equation pertains to light and other electromagnetic radiation traveling through a vacuum when the source is moving directly away from the observer. If the source is moving toward the observer, the sign of v needs to be reversed. Because there is no detectable medium through which light travels, only the relative velocity of the source occurs in the equation. The velocity of the observer is always zero. The derivation of this equation can be found in most modern physics books, including Serway p.22-24 and Tipler p.45-49.
The
Transverse Doppler Equation
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· f = the frequency observed in the rest frame · fo = the frequency emitted in the moving frame · v = the velocity of the moving frame · g = the gamma factor for velocity v · q = the angle between the velocity of and the direction to the source |
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The transverse Doppler equation pertains to light and other electromagnetic radiation when the source is moving in some direction other than directly away from the observer. (See Tipler p.49.)
Questions:
Q1. One
of the bright spectral lines of hydrogen has a wavelength of 656.3 nm. If this
line is observed for a distant galaxy to have a wavelength of twice that value,
how fast is that galaxy traveling? (A) 0.5 c (B) 0.6 c (C) 0.75 c (D) 0.9 c (E)
None of these.
Problems:
P1. Sound at a Railroad Crossing
An automobile traveling northward at 30 m/s into a head wind of 10 m/s approaches a railroad crossing while the signals are sounding. If the driver hears a frequency of 1000 hertz before crossing the railroad tracks, (a) what frequency are the signals emitting? (b) What frequency does the driver hear after crossing the railroad tracks? Assume that the speed of sound in air is 345 m/s.
P2. Age of a Quasar
Astronomers use a factor z = (fo – f)/f
to describe the red shift of a receding source of light. (a) Combine this
equation for z with the Doppler equation and derive the equation 
, which determines the recession velocity of a source as a
function of z. (b) From this equation determine the recession velocity
of a distant quasar for which z = 3.50. (Note: z-factors greater
than this value have actually been measured.) (c) Assume the quasar has been
traveling away from us for 10 billion years, how far away was the quasar when
the light we see today was emitted? (d) How old was the quasar when the light was
emitted?
P2. Transverse Doppler Effect
(a) Show that the transverse Doppler equation reduces to the relativistic Doppler equation when the source is moving directly away from the observer. (b) Show that when q = p = 180o, the transverse Doppler equation reduces to the relativistic Doppler equation for a source moving directly toward the observer. (c) Show that when q = p/2 = 90o, the transverse Doppler equation reduces to the time dilation equation relating the periods of oscillation t = gto.
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