## Beats – Class 11 | Chapter – 15 | Physics Short Notes Series PDF for NEET & JEE

Beats: Beats are a phenomenon that occur when two waves with slightly different frequencies are combined. Beats result in a periodic variation in amplitude, producing a pulsing sound or light. The frequency of the beats is equal to the difference between the frequencies of the two waves.

For example, if two guitar strings are tuned slightly differently and are played at the same time, the resulting sound will have a pulsing quality as the crests and troughs of the two waves interact. The frequency of the pulsing is equal to the difference between the frequencies of the two strings.

In general, beats are most noticeable when the two waves have similar amplitudes and are close in frequency. They provide a useful tool for determining the frequency of a sound or for tuning musical instruments.

In conclusion, beats are a phenomenon that occur when two waves with slightly different frequencies are combined. Beats result in a periodic variation in amplitude, producing a pulsing sound or light, and the frequency of the beats is equal to the difference between the frequencies of the two waves.

## When do Beats Happen?

Beats occur when two waves with slightly different frequencies are combined. When the two waves are combined, the crests and troughs of the two waves interact, resulting in a periodic variation in amplitude. The frequency of the variation is equal to the difference between the frequencies of the two waves, and is known as the beat frequency.

Beats are most noticeable when the two waves have similar amplitudes and are close in frequency. They are used in a variety of applications, including tuning musical instruments, determining the frequency of a sound, and demonstrating the principle of superposition of waves.

In conclusion, beats occur when two waves with slightly different frequencies are combined, resulting in a periodic variation in amplitude with a frequency equal to the difference between the frequencies of the two waves. Beats are most noticeable when the two waves have similar amplitudes and are close in frequency.

## Frequency of Beats

Beat frequency is the frequency at which two waves with slightly different frequencies combine to produce a periodic variation in amplitude. In other words, beat frequency is the difference between the frequencies of the two waves.

When two waves with slightly different frequencies are combined, the crests and troughs of the two waves interact, producing a pulsing pattern in the combined wave. The frequency of the pulsing is equal to the beat frequency. For example, if two guitar strings are tuned slightly differently and are played at the same time, the resulting sound will have a pulsing quality with a frequency equal to the beat frequency between the two strings.

The beat frequency is proportional to the difference between the frequencies of the two waves and inversely proportional to the sum of the frequencies. As the difference between the frequencies of the two waves increases, the beat frequency also increases.

In conclusion, beat frequency is the frequency at which two waves with slightly different frequencies combine to produce a periodic variation in amplitude, equal to the difference between the frequencies of the two waves. The beat frequency is proportional to the difference between the frequencies and inversely proportional to the sum of the frequencies of the two waves.

## Beat Frequency Formula & Derivation

The beat frequency can be derived by considering the superposition of two waves with frequencies f1 and f2. The combined wave can be represented as:

y(t) = A1 * cos(2 * pi * f1 * t) + A2 * cos(2 * pi * f2 * t)

where A1 and A2 are the amplitudes of the two waves, and t is time. The beat frequency, fb, is given by the difference between the two frequencies:

fb = |f1 – f2|

The expression for the combined wave can be expanded using the identity cos(a + b) = cos(a)cos(b) – sin(a)sin(b):

y(t) = A1 * cos(2 * pi * f1 * t) + A2 * cos(2 * pi * f2 * t)

y(t) = (A1 + A2)cos(2 * pi * (f1 – f2) * t/2)cos(2 * pi * (f1 + f2) * t/2) – (A1 + A2)sin(2 * pi * (f1 – f2) * t/2)sin(2 * pi * (f1 + f2) * t/2)

As a result, the combined wave consists of two waves with frequencies (f1 + f2)/2 and (f1 – f2)/2. The frequency (f1 – f2)/2 is the beat frequency, fb, and the other frequency (f1 + f2)/2 is the average frequency.

In conclusion, the beat frequency, fb, is equal to the difference between the frequencies of the two waves, |f1 – f2|. This result can be derived by considering the superposition of two waves with frequencies f1 and f2, and expanding the expression for the combined wave using the identity cos(a + b) = cos(a)cos(b) – sin(a)sin(b).

## Application of Beats

There are various applications of beats, though the three are discussed in detail below.

• Beats are used in determining the unknown frequency
• Beats are used in determining the existence of poisonous gases in mines.

To identify the presence of poisonous gases in the mine, the following experiment is carried out. Two pipes equal in size are taken, and one is filled with the pure ai. The other pipe is filled with the air in the mine. The pipes are blown together. The pipe that consists of pure air does not produce any sound whereas if some amount of sound is produced by the pipe that consists of mine air, then it indicates the presence of poisonous gases. If the beats are produced, then the mine air is not pure.

• Beats are used in determining the tone of musical instruments.

Two musical instruments are sounded together to notice if two instruments vibrate in a similar tone. If the tones of both the instruments are similar, then no beats will be heard. In such a way, the tones of various musical instruments can be adjusted according to the beats.

• Beats are used to adjust the vibrating length between two bridges in the sonometer experiment.

• Based on the phenomenon of the beats, the doppler ultrasonography and echocardiography works.

• The aeroplane speed can be determined by the Doppler RADAR. This is based on the phenomenon of beats.

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