Doppler effect

The Doppler effect (also known as Doppler shift) is the change in the apparent frequency of a wave due to the relative motion between the source and the observer (or receiver).

The same phenomenon is experienced by both electromagnetic waves (light) and mechanical waves (sound). In this page we will focus on the latter ones.

A real-life example of the Doppler effect is the change in the pitch of the siren of a police car or an ambulance as they approach us and then speed away from us. Another example is the change in the pitch of a train whistle as it approaches or moves away. In both cases, as the source approaches the sound seems to get higher in pitch, while it gets lower as the source speeds away.

In order to explain this phenomenon, let us consider an ambulance as the one represented in the figure below. Its siren is emitting a whine. Two observers (or listeners), are located behind the ambulance (A) and ahead of it (B). We will suppose that the air (the medium) is homogeneous, so the sound waves emitted by the source will travel at the same speed in all directions.

When the ambulance is at rest, the wavefronts of the sound wave emitted by its siren are evenly spaced. The distance between two consecutive wavefronts is the wavelength λ. In this situation both observers A and B perceive the same frequency, υ₀, which is the frequency of the sound waves emitted by the source.

The period T₀ of the sund wave emitted by the siren is given by:

When the ambulance is in motion, the wavefronts reaching observer B will be closer together, whereas for observer A (behind the moving source) they will be farther apart. Thus, the frequency perceived by both observers will be different. This frequency will also depend on the speed of both the medium and the observer.

We are going to derive an expression for the frequency υ of the sound perceived by the observer when both he and the source are in motion. We will also take into account the speed of the medium (think of a windy day). From now on, we will use the following magnitudes (which we will assume to be constant):

• • |v|: is the speed of the waves with respect to the medium.
  • v_m: is the speed of the medium with respect to Earth.
  • v_S: is the speed of the source emitting the waves.
  • v_R: is the speed of the observer (or listener).
  • υ₀: is the frequency of the waves emitted by the source.

First we are going to find the distance between two consecutive wavefronts λ’ when the source is moving at speed v_S.

In the figure below both wavefronts when they are emitted by the source are shown.

At the initial instant the source emits the first wavefront (1). The second wavefront will be emitted after a time T (which is the period of the wave). During this time T, the first wavefront will have traveled a distance d₁ given by its speed (measured with respect to Earth) times the time T:

The speed of the sound wave with respect to Earth is obtained using the Galilean transformation.

When the source emits the second wavefront (2 in the picture above) it (the source) will have traveled a distance d_S given by:

Therefore, the separation λ’ between wavefronts 1 and 2 will be:

The time T’ it takes for the second wavefront to reach the observer once the first one has reached it will be λ’ divided by the speed of the wave with respect to him:

And the frequency perceived by the observer will be the inverse of T’:

Check out Doppler effect – problems and solutions to see how this expression is used in different situations.

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