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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 free-space 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 non-relativistic 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 red-shifted 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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