Beats

 

Number of beats heard per second

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Qualitative treatment

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Suppose two tuning forks having frequencies 256 and 257 per second respectively, are sounded together. If at the beginning of a given second, they vibrate in the same phase so that the compressions (or rarefactions) of the corresponding waves reach the ear together, the sound will be reinforced (strengthened). Half a second later, when one makes 128 and the other vibrations, they are in opposite phase, i.e., the compression of one wave combines with the rarefaction of the other and tends to produce silence. At the end of one second, they are again be in the same phase and the sound is reinforced. By this time, one fork is ahead of the other by one vibration.

Thus, in the resultant sound, the observer hears maximum sound at the interval of one second. Similarly, a minimum loudness is heard at an interval of one second. As we may consider a single beat to occupy the interval between two consecutive maxima or minima, the beat produced in one second in this case, is one in each second. If the two tuning forks had frequencies 256 and 258, a similar analysis would show that the number of beats will be two per second. Thus, in general, the number of beats heard per second will be equal to the difference in the frequencies of the two sound waves.

Analytical treatment

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Consider two simple harmonic sound waves each of amplitude A, frequencies f₁ and f₂ respectively, travelling in the same direction. Let y₁ and y₂ represent the individual displacements of a particle in the medium, that these waves can produce. Then the resultant displacement of the particle, according to the principle of superposition will be given by

This equation represents a periodic vibration of amplitude R and  frequency . The amplitude and hence the intensity of the resultant wave, is a function of the time. The amplitude varies with a  frequency

Since intensity (amplitude)², the intensity of the sound is maximum in all these cases. For  to assume the above values like 0, p, 2p, 3p, 4p,....

Thus, the time interval between two maxima or the period of beats =

When the difference in the frequency of the two waves is small, the variation in intensity is readily detected on listening to it. As the difference increases beyond 10 per second, it becomes increasingly difficult to distinguish them. If the difference in the frequencies reaches the audible range, an unpleasant note of low pitch called the beat note is produced. The ability to hear this beat note is largely due to the lack of linearity in the response of the ear.

Demonstration of beats

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Let two tuning forks of the same frequency be fitted on suitable resonance boxes on a table, with the open ends of the boxes facing each other. Let the two tuning forks be struck with a wooden hammer. A continuous loud sound is heard. It does not rise or fall. Let a small quantity of wax be attached to a prong of one of the tuning forks.. This reduces the frequency of that tuning fork. When the two forks are sounded again beats will be heard.

Uses of beats

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• The phenomenon of beats is used for tuning a note to any particular frequency. The note of the desired frequency is sounded together with the note to be tuned. If there is a slight difference in frequencies, then beats are produced. When they are exactly in unison, i.e., have the same frequency, they do not produce any beats when sounded together, but produce the same number of beats with a third note of slightly different frequency. Stringed musical instruments are tuned this way. The central note of a piano is tuned to a standard value using this method.

• The phenomenon of beats can be used to determine the frequency of a tuning fork. Let A and B be two tuning forks of frequencies f_A (known) and f_B (unknown). On sounding A and B, let the number of beats produced be n. Then one of the following equations must be true.

f_A - f_B = n ……………. (i)

or f_B - f_A = n ……………. (ii)

To find the correct equation, B is loaded with a little wax so that its frequency decreases. If the number of beats increases, then equation (i) is to be used. If the number of beats decreases, then equation (ii) is to be used. Thus, knowing the value of f_A and the number of beats, f_B can be calculated.

• Sometimes, beats are deliberately caused in musical instruments in a section of the orchestra to create sound of a special tonal quality.

• The phenomenon of beats is used in detecting dangerous gases in mines. The apparatus used for this purpose consists of two small and exactly similar pipes blown together, one by pure air from a reservoir and the other by the air in the mine. If the air in the mine contains methane, its density will be less than that of pure air. The two notes produced by the pipes will then differ in the pitch and produce beats. Thus, the presence of the dangerous gas can be detected.

• The super heterodyne type of radio receiver makes use of the principle of beats. The incoming radio frequency signal is mixed with an internally generated signal from a local oscillator in the receiver. The output of the mixer has a carrier frequency equal to the difference between the transmitted carrier frequency and the locally generated frequency and is called the intermediate frequency. It is amplified and passed through a detector. This system enables the intermediate frequency signal to be amplified with less distortion, greater gain and easier elimination of noise.