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When two waves of slightly different frequency overlap a phenomenon known as beats results. The effect is a kind of throbbing sensation, which can sometimes be heard when two musicians such as oboeists are playing together.

If they are attempting to play the same note but are fractionally out of tune with each other beats will be heard: the frequency will vary about a mean value. The closer the two frequencies the lower will be the beat frequency, and this will become zero when they are perfectly in tune. The human ear is normally very sensitive to pitch and the two notes have to be well within one semitone for true beats to be heard.

Even if we take C and B on the musicians' scale, one semitone apart, the beat frequency would be 261.6 - 247 = 14.6 Hz, that is, nearly fifteen beats per second! This would not give audible beats, only an unpleasant discord.

Consider two waves of slightly different frequencies f1 and f2 (f1 > f2) but of the same amplitude. Figure 1 is a diagram of the two waves and their resultant.

Proof of the formula for beat frequency

Let the two displacements at a point be y1 and y2.

y1 = a sin 2pf1t and y2 = a sin 2pf2t

The final displacement (y) is given by:

y = y1 + y2 = a (sin 2pf1t + sin 2pf2t) = 2a cos 2p([f1 f2]t/2 x sin 2p([f1 + f2]t/2)

The first term shows a slow amplitude variation and the second a rapid displacement variation.

(a) The amplitude varies with time with a frequency [f1 f2]/2

(b) Since the ear is sensitive to the intensity and not the amplitude of a vibration, the beat frequency f is the number of times that the magnitude of the amplitude reaches a maximum each second (positive or negative):

Beat frequency (f) = 2 x amplitude frequency = [f1 f2]

Principle of superposition

The addition of two or more waves at a point to give a resultant disturbance uses the principle of superposition. This states that the final disturbance is simply the vector sum of each disturbance at that point. The principle is used in the equations for standing waves, beats, diffraction and interference.

The effect can easily be observed in the laboratory with two signal generators and loudspeakers.

Example of beats between two waves

The diagrams in Figure 2 show the result of adding two waves, one of frequency f and the other of frequency 1.3f.

Beats are used in police radar speed traps. The outgoing and reflected signals are fed to the detector and the speed of the car is determined from the beat frequency using the Doppler shift.

Student investigation
A striking example of standing waves and resonance may be observed using a Slinky spring hanging vertically with its lower end fixed to a vibration generator. Set up suitable apparatus and investigate the relation between frequency and wavelength for the standing waves on the spring.

What happens if the tension is altered, either by raising or lowering the support or by using a shorter length of spring?

schoolphysics beats animation

To see an animation of beats formed between two waves please click on the animation link.

schoolphysics beats animation

To see an animation of beats formed between two waves of different frequency from the above example please click on the animation link.

© Keith Gibbs