Question

There are three sources of equal intensity with frequencies 401, 402, 403 vibrations/sec. The number of beats heard per second is
A) 1
B) 2
C) 3
D) 0

Answer 1

Hint
Use the principle of superposition and add the waves differing in their frequencies. The amplitude is maximum when $\cos 2\pi t$ is max which is possible when $t = 0,1,2...$. As the time interval between is one second the beat frequency is one second.

Complete step-by-step answer
Formation of beats is adding waves that differ in frequencies using the superposition principle. It is given that frequencies of three sources are 401, 402, 403 vibrations/sec.
The equation of the corresponding sources is given by,
${y_1} = A\sin 2\pi 400t$
${y_2} = A\sin 2\pi 401t$
${y_3} = A\sin 2\pi 403t$
According to the principle of superposition,
$y = {y_1} + {y_2} + {y_3}$
$y = A\sin (802\pi t)[1 + \cos 2\pi t]$
Here, the amplitude is maximum when $\cos 2\pi t$ is max.
Therefore,
$\cos 2\pi t = 1\,$ At $t = 0,1,2...$
Maximum sound is heard when $t = 0,1,2...$
And the time interval between each of them is one second so the beat frequency is 1s.
The correct option is (A).

Note
Principle of superposition: The displacement at any time due to a number of waves meeting simultaneously at a point in a medium is the vector sum of the individual displacements due to each one of the waves at that point at the same time.