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

Given: The resultant wave is x = a (1.5t) (50.5t). Find: The beat period. Use the identity A B = 12[ (A+B) + (A-B)] Applying it, x = a2[ (1.5t + 50.5t) + (50.5t - 1.5t)] x = a2[ (52t) + (49t)] So the two component waves have angular frequencies 1 = 52 rad/s and 2 = 49 rad/s. Convert angular frequencies to ordinary frequencies: f1 = 522 Hz, f2 = 492 Hz Hence the beat frequency is f beat = |f1 - f2| = |52-49|2 = 32 Hz The beat period is T beat = 1f beat = 2 3 s Numerically, T beat 2.0944 s The nearest integer is 2 s, so the correct option is D.

Two harmonic waves moving in the same direction superimpose to fo… — Solution | JEE Physics

MCQMediumJEE 2025Beats & Resonance

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 the nearest integer):

A
$6 \, \text{s}$
B
$4 \, \text{s}$
C
$1 \, \text{s}$
D
$2 \, \text{s}$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The resultant wave is $x = a \cos(1.5t) \cos(50.5t)$ .

Find: The beat period.

Use the identity

\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]

Applying it,

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

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

Beat Frequency Calculation

So the two component waves have angular frequencies $\omega_1 = 52 \, \text{rad/s}$ and $\omega_2 = 49 \, \text{rad/s}$ .

Convert angular frequencies to ordinary frequencies:

f_1 = \frac{52}{2\pi} \, \text{Hz}, \qquad f_2 = \frac{49}{2\pi} \, \text{Hz}

Hence the beat frequency is

f_{\text{beat}} = |f_1 - f_2| = \frac{|52-49|}{2\pi} = \frac{3}{2\pi} \, \text{Hz}

Direct Beat Period

The beat period is

T_{\text{beat}} = \frac{1}{f_{\text{beat}}} = \frac{2\pi}{3} \, \text{s}

Numerically,

T_{\text{beat}} \approx 2.0944 \, \text{s}

The nearest integer is $2 \, \text{s}$ , so the correct option is D.

Common mistakes

Using the modulation factor $\cos(1.5t)$ directly as the beat frequency is incorrect because beat frequency must be obtained from the two actual component waves after expanding the product. First rewrite the expression as the sum of two cosine waves.
Taking beat frequency as $|52-49| = 3$ directly in hertz is wrong because $52$ and $49$ are angular frequencies in $\text{rad/s}$ . Convert to hertz by dividing by $2\pi$ .
Confusing beat frequency with beat period leads to selecting the wrong option. After finding $f_{\text{beat}}$ , take its reciprocal to get $T_{\text{beat}}$ .

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