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Unformatted text preview: 21 Doppler — cont’d Example 1: A space ship traveling between Earth and Moon is emitting a TEM wave at a radian frequency ω . The TEM wave reaching Earth is found to be oscillating with a radian frequency of ω E = 2 . 999 π 10 9 rad/sec while on the moon the wave frequency is measured as ω M = 3 . 001 π 10 9 rad/sec. (a) Determine ω , k and λ , where k and λ are the TEM wavenumber and wavelength, respectively, in the reference frame of the space ship. (b) Determine the velocity of the space ship in the Earth reference frame. Assume freespace propagation and that the distance between Earth and Moon is constant during the measurements. Solution: (a) Clearly ω E and ω M can differ from ω by ± kv (in nonrelativistic approx imation) where v is the relative speed of the space shift with respect to Earth and Moon and k is the wavenumber in the space ship frame. Since ω M > ω E , we must have ω M = ω + kv, ω E = ω kv. Hence, ω M + ω E = 2 ω ⇒ ω = ω M + ω E 2 = 3 π 10 9 rad/s . It follows that k = ω c = 10 π and λ = 2 π k = 0 . 2 m. (b) Taking the difference of the above equations we also find that ω M ω E = 2 kv ⇒ v = ω M ω E 2 k = 2 π 10 6 20 π = 10 5 m/s . Since ω E is redshifted with respect to ω , the space ship must be moving away from the Earth with the speed v . 1 Note that identifying the speed of the space shift in the Earth frame and its direction of motion is equivalent to identifying its velocity ....
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Electromagnet, Frequency

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