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) [2025]

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Solution

Q.
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) [2025]

1 6 s

2 4 s

3 1 s

4 2 s

Ans.
(4)

The given equation can be written as

$x = \frac{a}{2} \cos [1.5 + 50.5] t + \frac{a}{2} \cos [50.5 - 1.5] t$

$x = \frac{a}{2} \cos [52 t] + \frac{a}{2} \cos [49 t]$

Here, $2 π f_{1} = 52 and 2 π f_{2} = 49$

$f_{1} = \frac{52}{2 π}, f_{2} = \frac{49}{2 π}$

$\Rightarrow △ f = f_{Beat} = f_{1} - f_{2} = \frac{3}{2 π} Hz$

$∴ T_{Beat} = \frac{1}{f_{Beat}} = \frac{2 π}{3} \sec = 2.09 \sec$

$\Rightarrow T_{Beat} \approx 2 \sec$

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