Physics Q34 – JEE-MAIN 2025

Two harmonic waves moving in the same direction superimpose to form a wave

x = a \cos (1.5 t) \cos (50.5 t)

where

t

is in seconds. Find the period with which they beat. (close to nearest integer)

Clue 1 of 3: Beat Phenomenon

When two waves of slightly different frequencies superimpose, they produce beats, which are periodic variations in the amplitude of the resultant wave.

🥷

Ninja Strategy•Estimate Beat Period

Recognize that the beat angular frequency is $2 \times 1.5 = 3$ rad/s. The beat period is $2 π / 3$ , which is approximately $6.28 / 3 \approx 2.09$ seconds. This quickly points to 2s as the closest integer.

🧠 Full Solution Path

Step 1: Relate to Standard Beat Equation✦ Active

The given wave equation is $x = a \cos (1.5 t) \cos (50.5 t)$ . This form is equivalent to the superposition of two waves $y = A_{0} \cos (ω_{1} t) + A_{0} \cos (ω_{2} t)$ , which results in $y = 2 A_{0} \cos (\frac{ω_{1} + ω_{2}}{2} t) \cos (\frac{ω_{1} - ω_{2}}{2} t)$ . Comparing the given equation with this standard form:

\frac{ω_{1} - ω_{2}}{2} = 1.5 ⟹ ω_{1} - ω_{2} = 3 rad/s

\frac{ω_{1} + ω_{2}}{2} = 50.5 ⟹ ω_{1} + ω_{2} = 101 rad/s

Step 2: Calculate Beat Angular Frequency and Period○ Expand

The angular beat frequency, $ω_{b e a t}$ , is the absolute difference between the angular frequencies of the two component waves.

ω_{b e a t} = | ω_{1} - ω_{2} | = 3 rad/s

The beat period, $T_{b e a t}$ , is given by $T_{b e a t} = \frac{2 π}{ω_{b e a t}}$ .

T_{b e a t} = \frac{2 π}{3} s

💡 Teacher's Secret Hint

Remember that the beat period is the reciprocal of the beat frequency, and for angular frequencies, it's $2 π$ divided by the angular beat frequency.

Step 3: Calculate Numerical Value and Round○ Expand

Substitute the value of $π \approx 3.14159$ into the expression for $T_{b e a t}$ :

T_{b e a t} = \frac{2 \times 3.14159}{3} \approx \frac{6.28318}{3} \approx 2.094 s

Rounding to the nearest integer, the beat period is $2 s$ .

💡 Teacher's Secret Hint

Always pay attention to rounding instructions, especially 'close to nearest integer'.

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Physics Question 34 – JEE-MAIN 2025

Clue 1 of 3: Beat Phenomenon

🧠 Full Solution Path