DOPPLER EFFECT

DOPPLER EFFECT IN ACOUSTICS

Doppler effect describes the frequency shift of the signal in relation to the relative motion of a source and an observer. The wave generated by a source that moves away from an observer/receiver appears to him to be of lower frequency than the wave generated by a stationary source, or generated by a source moving toward the observer. The frequency of the signal detected by the receiver moving toward the still source is higher, compared to the frequency detected by the still receiver, or a receiver moving away from the source. This theory was formulated by Austrian physicist Ch.Doppler in 1842.

1. Source moves, receiver is still.

Let us consider the source of wave moving toward the observer with velocity v and emitting the impulses with period T. At the moment t=0 the length between the source and the observer is L. The first impulse will reach the observer at time t=L/u, where u in the velocity of the waves. The second impulse will be sent to observer at time T. At this time the distance between the source and observer will be L₁=L-vT, so the second impulse will reach the observer at time t₁=T+(L-vT)/u. As a result the observer will detect the impulses with period

T_dop=t₁-t= T(1- v/u)

And the frequency f_dop registered by the observer equals:

f_dop=f / (1-v/u) (source is moving toward the stationary observer)

where f is the frequency emmited by the source. We see from this formula that when the source is moving toward the observer the frequency is increased by the value fv/u called Doppler shift. On the contrary, when source moves away from the receiver the frequency is diminished as follows:

f_dop=f / (1+v/u) (source is moving away from the stationary observer)

In case when the source is moving and the observer is still, the Doppler shift appears because the wavelength is changes (see the animation on CD).

2. Observer moves, source is still.

Next, we shall consider the case when observer moves and the source of the wave is still. In this case the wavelength is not changed and Doppler frequency shift appears because the velocity w of the wave relatively the observer is changed:

w = u + v (observer is moving toward the stationary source)

w = u - v (observer is moving away from the stationary source)

Because f_dop=w/l , initial f=u/l₀ and l =l₀ we find that

f_dop=f(1+v/u) (observer moves toward the stationary source)

f_dop=f(1-v/u) (observer is moving away stationary source)

We can see from these conclusions that for acoustic waves the frequency shift will be different depending on what is moving: source or observer. On the contrary, in the case of electro-magnetic wave the Doppler shift depends only on the relative motion of source and receiver.