Doppler effect

The change in the received frequency of vibration with the speed of motion of the source of the vibration with respect to the observer or vice versa. If the distance between the source and the observer decreases, the frequency increases; if it increases, the frequency decreases. All types of waves give the Doppler effect: sound waves, elastic waves and electromagnetic waves.

Let $ x ^ {j} $ and $ {\overline{x}\; } {} ^ {j} $, where $ j = 0 , 1 , 2 , 3 , $ and $ x ^ {0} = t $, $ {\overline{x}\; } {} ^ {0} = \overline{t}\; $ are the time, be the inertial reference systems of the source of an electromagnetic wave and of the observer. Further, let the phase multiplier of the plane wave have the form $ \mathop{\rm exp} ( i \omega _ {j} x ^ {j} ) $ and $ \mathop{\rm exp} ( i {\overline \omega \; } _ {j} {\overline{x}\; } {} ^ {j} ) $ in the first and second reference frame, respectively. The Doppler effect is then expressed by a formula relating the time components (frequencies) $ \omega _ {0} $ and $ {\overline \omega \; } _ {0} $ of the four-dimensional wave vectors $ \{ \omega _ {j} \} $ and $ \{ {\overline \omega \; } _ {j} \} $:

$$ {\overline \omega \; } _ {0} = \omega _ {0} \frac{\sqrt {1 - v ^ {2} / c ^ {2} } }{1 - ( v / c) \cos \phi } . $$

Here $ v $ is the relative velocity between the source and the observer, $ \phi $ is the angle between the velocity $ v $ and the line of observation measured in the system of the observer, and $ c $ is the velocity of light in vacuum.

The effect was named after Ch. Doppler who was the first to predict it theoretically (1842).

References

[1]	L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1951) pp. Chapt. 6 (Translated from Russian)
[2]	S.E. Frish, A.V. Timoreva, "A course in general physics" , 1 , Moscow (1962) pp. Chapt. 12 (In Russian)

References

[a1]	E.T. Whittaker, "A history of the theories of aether and electricity" , I Chapt. XII, p. 368; Vol. II Chapt. II, p. 40–42 & Chapt. III, p. 92 , Harper Torchbooks (1960)
[a2]	J.W.S. Rayleigh, "The theory of sound" , II , Dover, reprint (1945) pp. 154–156