Beats: Sound is a form of energy that gives us the sensation of hearing. It is a longitudinal mechanical wave. The sound wave travels by the vibrations of the medium particles in which it is travelling. Our ears sense these vibrations and transform them into electric pulses, and sends them to the brain. When sound waves are coming from various sources, the sound wave mixes up, and there is a chance that interference will occur.

Interference is a phenomenon in which two or more waves overlap and superimpose on each other, changing the wave’s amplitude by either increasing it (constructive interference) or decreasing it (destructive interference). In this article, we will learn about beats that are the result of interference of the sound waves.

What are Beats?

Beats is a phenomenon in which interference between two or more sound waves occurs that changes with time. Beats occurs due to the interference of waves. A sound louder or fainter than any of the constituent sounds is heard. Depending on the frequency of the constituent sound waves, the beat can have some frequency.

Waves

We encounter waves daily around us. For example, light and sound are waves. Waves can transfer energy as well as momentum from one point to the other.
Based on the requirement of medium for propagation, waves can be classified as:

1. Electromagnetic waves: Waves that do not require a medium to propagate are classified as electromagnetic waves. Examples of electromagnetic waves are as follows: light waves, radio waves, infrared waves etc.
2. Mechanical waves: Waves that require a medium to propagate are classified as mechanical waves. Examples of mechanical waves are as follows: sound waves, waves formed in a string, waves formed on the surface of the water.

Mechanical waves can be further classified as transverse and longitudinal waves.

1. Transverse wave: Waves in which the medium particles oscillate in the direction perpendicular to the direction of propagation of the wave. Examples of transverse waves are waves formed in a string, waves formed on the surface of the water.
2. Longitudinal wave: Waves in which the medium particles oscillate in the direction of the propagation of the wave is known as Longitudinal wave. A sound wave is an example of a longitudinal wave.

Interference

Interference is a phenomenon in which two or more waves overlap with each other to superimpose on each other. The amplitude at a particular point at any instance is the sum of the amplitudes of the constituent waves.
The amplitude of the resultant wave will be a function of time and the position of the point. But if the sources are coherent or have the same frequency, then the resultant wave will only be the function of the position of the point.
If the positive maxima of the first wave coincide with the negative maxima of the second wave, then the amplitude of the resultant wave will be less. This is known as destructive interference.
If the positive maxima of the first wave coincide with the positive maxima of the second wave, then the amplitude of the resultant wave will be more than the constituent waves. This is known as constructive interference.

Production of Beats

Beats are produced when two or more sound waves with different frequencies are superimposed. Beats are the louder sound that is produced after a fixed interval of time. The frequency of the beat remains the same and depends on the frequencies of the constituent sounds.

Beat Frequency

When two sources superimpose, then the beat occurs after a fixed interval of time. The beat frequency is defined as the number of times the beat is heard in one second.
The time interval between the two successive beats is known as the time period of the beat.
The beat frequency is related to the time period of the beats as,
\(f = \frac{1}{T}\)
Where,
\(f\) is the frequency of the beat.
\(T\) is the time interval between the two successive beats.

Derivation of Beat Frequency

Let the equation sound wave from the first source be,
\({y_1} = A\sin \left( {{\omega _1}t – kx} \right) = A\sin \left( {2\pi {f_1}t – kx} \right)\)
Where,
\({y_1}\) is the amplitude of the displacement of the medium particle located at a point.
\({{\omega _1}}\) is the angular frequency of the wave.
\({f_1}\) is the frequency of the first sound wave.
\(t\) is the time.
A is the maximum displacement of the medium particle from the mean position.
Similarly, the equation of the sound wave from the second source is,
\({y_2} = A\sin \left( {{\omega _2}t – kx} \right) = A\sin \left( {2\pi {f_2}t – kx} \right)\)
\({y_2}\) is the amplitude of the displacement of the medium particle located at a point.
\({{\omega _2}}\) is the angular frequency of the second sound wave.
\({f_2}\) is the frequency of the wave
When the sound waves from the two sources overlap, the resultant wave is given by,
\({y_{net}} = {y_1} + {y_2}\)
Putting in the values we get,
\({y_{net}} = A\sin \left( {{\omega _2}t – kx} \right) + A\sin \left( {{\omega _2}t – kx} \right)\)
\( \Rightarrow {y_{net}} = A\sin \left( {2\pi {f_1}t – kx} \right) + A\sin \left( {2\pi {f_2}t – kx} \right)\)
Using the trigonometric property,
\(\sin (A) + \sin (B) = 2\cos \left( {\frac{{A – B}}{2}} \right)\sin \left( {\frac{{A + B}}{2}} \right)\)
\( \Rightarrow {y_{net}} = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\sin \left( {\left( {{f_1} + {f_2}} \right)\pi t – kx} \right)\)
On comparing with the equation of wave,
\(y = A\sin (\omega t – kx) = A\sin (2\pi ft – kx)\)
The amplitude of the resultant wave is given by,
\(A = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\)
Thus, the amplitude of the resultant will vary with time. Therefore, for the beat to occur, the amplitude of the wave should be maximum-
\({A_{net}} = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\) should be maximum.
\(\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\) should be maximum.
\(\left( {{f_1} – {f_2}} \right)\pi {t_{{\rm{beats }}}} = k\pi \)
\(\frac{1}{{{f_{{\rm{beats }}}}}} = {t_{{\rm{beats }}}}\)
\( \Rightarrow {f_{{\rm{beats }}}} = \left| {{f_1} – {f_2}} \right|\)

Sample Problems

Q.1. Two sound sources having frequencies \({\rm{200}}\,{\rm{Hz}}\) and \(205\;{\rm{Hz}}\) are present near each other Find the beat frequency of the beats produced by the overlap of the two sound waves.
Ans: Given,
The frequency of the first sound wave is,
\({f_1} = 200\;{\rm{Hz}}\)
The frequency of the second wave is,
\({f_2} = 205\;{\rm{Hz}}\)
We know that the beat frequency is given by,
\( \Rightarrow {f_{{\rm{beats }}}} = \left| {{f_1} – {f_2}} \right|\)
Where,
\({f_1}\) is the frequency of the first sound wave.
\({f_2}\) is the frequency of the second sound wave.
\( \Rightarrow {f_{{\rm{beats }}}} = 5\;{\rm{Hz}}\)
Therefore the beat frequency comes out to be \(5\;{\rm{Hz}}{\rm{.}}\)

Q.2. Suppose the frequency of the two instruments in a concert is \(170\,{\rm{Hz}}\) and \(172\,{\rm{Hz}}.\) Find the time period of the beats observed.
Ans: Given,
The frequency two instruments in a concert are,
\({f_1} = 170\;{\rm{Hz}}\)
And,
\({f_2} = 172\;{\rm{Hz}}\)
The beat frequency is given by,
\( \Rightarrow {f_{{\rm{beats }}}} = \left| {{f_1} – {f_2}} \right|\)
Where,
f1 is the frequency of the first sound wave.
f2 is the frequency of the second sound wave.
\( \Rightarrow {f_{{\rm{beats}}}} = |172 – 170| = 2\;{\rm{Hz}}\)
The relation between the beat frequency and the time period of the beat is given by,
\(\frac{1}{{{f_{{\rm{beats }}}}}} = {t_{{\rm{beats }}}}\)
Therefore, the time period of the beat is \({\rm{0}}{\rm{.5}}\,{\rm{secs}}{\rm{.}}\)

Summary

In the given article, we discussed sound waves and a special phenomenon known as beats that occur due to sound waves’ interference. Interference is the property of waves. The phenomenon such as Young’s double-slit experiment is also due to interference. Beats are sounds that are produced at regular intervals, which is louder than any of the constituent sounds. The amplitude of the resultant wave is a function of time. The beats are the result of constructive interference of the waves.

Frequestly Asked Questions on Beats

Q.1. Is sound a wave?
Ans: Yes, the sound is a longitudinal wave. The particles of the medium vibrate in the direction of wave propagation and therefore require a medium to propagate.

Q.2. What is interference?
Ans: Interference is the superposition of two or more waves to form a resultant wave whose amplitude can be more than the constituent waves.

Q.3. Will beats be produced if there are more than two waves?
Ans: Yes, the beats will be produced if there are more than two waves. There can be more than one beat possible.

Q.4. Will beats occur inside water?
Ans: Yes, the beats can occur in any medium where sound can travel.

Q.5. On what factor does the beat frequency depend?
Ans: Beat frequency depends on the frequencies of the constituent sound waves.
\({f_{{\rm{beats }}}} = \left| {{f_1} – {f_2}} \right|\)
Where,
\({{f_1}}\) is the frequency of the first sound wave.
\({{f_2}}\) is the frequency of the second sound wave.

Q.6. Will the beat frequency change if the medium in which the sound wave is propagating?
Ans: No, the beat frequency will remain the same even if the medium as the frequency does not change with medium and the beat frequency depends on the frequencies of the constituent waves in the medium. Thus the beat frequency will remain the same.

Q.7. Does interference occur only in the sound wave?
Ans: No, interference occurs in both sound waves and light waves. In light waves, the young’s double-slit experiment. For the double-slit experiment, we use coherent sources (the frequency of the two sources are the same). Therefore, the resultant maxima remain constant with respect to time at a particular point in space. For sound waves, we use the sources with different frequencies and thus the Beats; that is, the resultant maxima occurs at some time interval at a time particular point.

Study about Speed of Sound here

We hope you find this article on ‘Beats’ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.