Two harmonic waves moving in the same direction superimpose to form a… | JEE Main 2025 (7 Apr, Shift 1)

Q18JEE Main 2025 · 7 APR · SHIFT 1Medium

Beats

Two harmonic waves moving in the same direction superimpose to form a wave $x = a\cos(1.5t)\cos(50.5t)$ where t is in seconds. Find the period with which they beat. (close to nearest integer)

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Answer: (c)

$a\cos(1.5t)\cos(50.5t) = \dfrac{a}{2}[\cos 52t + \cos 49t]$ , so $\omega_1 = 52$ , $\omega_2 = 49\text{ rad/s}$ .
Beat frequency $f_b = \dfrac{\omega_1 - \omega_2}{2\pi} = \dfrac{3}{2\pi}\text{ Hz}$ .
Beat period $T_b = \dfrac{2\pi}{3} \approx 2.09\text{ s} \approx 2\text{ s}$
So the answer is (c) 2 s.

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