Question

When beats are produced by two progressive waves of nearly the same frequency, which one of the following is correct?
A. The particles vibrate simply harmonically with the frequency equal to the difference in the component frequencies.
B. The amplitude of vibration at any point changes simply harmonically with a frequency equal to the difference in the frequencies of the two waves.
C. The frequency of beats depends upon the position where the observer is.
D. The frequency of beats changes as the time progresses.

Answer 1

Hint: In the case, if a problem is based on progressive waves, we know that there are three kinds of waves – longitudinal, transverse, and orbital with different aspects hence, analyze every option with the scientific approach and check which option seems to be more appropriate for the given situation to present the answer with proper explanation.

Complete step by step solution:
We know that, displacement relation in progressive wave is given as:
$y = {A_b}\sin (2\pi {n_{av}}t)$ … (1)
where, ${A_b} = 2A\cos (2\pi {n_A}t)$ = amplitude of vibration
where, ${n_A} = \dfrac{{{n_1} - {n_2}}}{2}$
where ${n_1}$= frequency of 1st progressive wave
and, ${n_2}$= frequency of 2nd progressive wave
Then, eq. (1) becomes
$y = 2A\cos (\pi ({n_1} - {n_2})t)\sin (2\pi {n_{av}}t)$
Thus, the amplitude of vibration at any point changes simply harmonically with a frequency equal to the difference in the frequencies of the two waves.

Hence, the correct option is B.

Note: Since this is a problem of multiple-choice questions (theory-based) hence, it is essential that given options are to be analysed very carefully to give a precise explanation. While writing an explanation of this kind of conceptual problem, always keep in mind to provide the exact reasons in support of your explanation.