Question

4. What are beats Obtain an expression
for the beat frequency. Where and how
are beats made use of​

Answer 1

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Answer 2

A) Definition of beats

Because a wave travels not only in space but also through time, if two waves can produce interference by overlapping in space, they should also produce an interference pattern when they overlap in time.

This phenomenon is known as beats.

B) To obtain an expression for the beat frequency-

Let two soundwaves of frequencies $n_{1}$ and $n_{2}$ are propagating simultaneously in the same direction.

Then, at $x=0$

$y_{1} = A sin2\pi n_{1} t$

$y_{2} = A sin2\pi n_{2} t$

Assume that, amplitude of both waves are same.

Using the principle of superposition, the resultant displacement is-

$y=y_{1} +y_{2 } =2Acos2\pi n_{A}t sin2\pi n_{av}t$

where,

$n_{av} =\frac{n_{1}+n_{2} }{2}$,  $n_{A} =\frac{n_{1}-n_{2} }{2}$

$y=A_{beat} sin2\pi n_{av}t$

$A_{beat} =2Acos2\pi n_{A}t$

Also, amplitude of the resultant wave varies periodically as frequency.

Time interval between two consecutive beats,

$\Delta t=t_{n} -t_{n-1}=\frac{1}{2n_{A} }$

Number of beats per second is given as,

$n_{beat} =\frac{1}{\Delta t} =2n_{A} -\frac{2(n_{1}-n_{2}) }{2} =n_{1}-n_{2}$

Thus, beat frequency is equal to the difference in frequency of the two interferring waves. 

C) Application of beats

• Musical instruments are tuned using beats.
• In the Sonometer Experiment, beats can be used to adjust the vibrating length between the two bridges.

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