A common way by which we experience the change of frequency (thus also the period) of a wave, is, for example, the sound of an ambulance’s siren, as it passes by. As the sound wave is deformed, the pitch goes higher as the ambulance approaches, and it goes lower as the ambulance recedes.
This is the Doppler shift for sound, and the frequency is given as follows. If fE is the frequency of the emitter (the moving ambulance), fR is the frequency of the receiver (an observer close by), vE is the speed of the ambulance, vR is the speed of the observer (assuming that the observer is also in motion), c is the speed of the sound wave (here it is not the speed of light), and λ is the wavelength of the sound wave, then we have
where the plus or minus sign depends on whether the source and the receiver approach or recede one from the other.
This formula implies that the two speeds, vE and vR, of the emitter and of the receiver, respectively, are calculated with respect to a fixed point in the medium (the air in this case), so that the medium can be considered as the absolute frame of reference to which both observers refer. However if we treat the source (the emitter) as stationary, and suppose that the observer (the receiver) moves with respect to the source, so that we set vE=0 and vR=v, the previous formula takes the form
This equivalent expression contains the relative speed v of the moving observer (the receiver) with respect to the emitter (the source), so that the medium is ignored.
From this formula we can reach the expression for the (relativistic) Doppler shift for light, if we additionally suppose that the wavelength λR of the light wave, as perceived by the receiver (the observer moving at the speed v), is contracted by the Lorentz factor γL, so that the calculation is as follows,
where here c is the speed of light, while the expression
is based on Lorentz transformation for length contraction in relativity.
We may notice that although the previous relationship for the change of frequency was derived by assuming that the speed v of the moving observer is relative to a source (the emitter) standing still, the same speed may also be related to a fixed point in the medium (the wave itself), independently of any external source or observer.
This is the Doppler shift for sound, and the frequency is given as follows. If fE is the frequency of the emitter (the moving ambulance), fR is the frequency of the receiver (an observer close by), vE is the speed of the ambulance, vR is the speed of the observer (assuming that the observer is also in motion), c is the speed of the sound wave (here it is not the speed of light), and λ is the wavelength of the sound wave, then we have
This formula implies that the two speeds, vE and vR, of the emitter and of the receiver, respectively, are calculated with respect to a fixed point in the medium (the air in this case), so that the medium can be considered as the absolute frame of reference to which both observers refer. However if we treat the source (the emitter) as stationary, and suppose that the observer (the receiver) moves with respect to the source, so that we set vE=0 and vR=v, the previous formula takes the form
From this formula we can reach the expression for the (relativistic) Doppler shift for light, if we additionally suppose that the wavelength λR of the light wave, as perceived by the receiver (the observer moving at the speed v), is contracted by the Lorentz factor γL, so that the calculation is as follows,
We may notice that although the previous relationship for the change of frequency was derived by assuming that the speed v of the moving observer is relative to a source (the emitter) standing still, the same speed may also be related to a fixed point in the medium (the wave itself), independently of any external source or observer.
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