A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x = 0 at t = 0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Physics Q34 – JEE-MAIN 2025

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Home Physics WavesQuestion #632

A sinusoidal wave of wavelength

7.5

cm travels a distance of

1.2

cm along the x-direction in

0.3

sec. The crest P is at

x = 0

t = 0

sec and maximum displacement of the wave is

2

cm. Which equation correctly represents this wave ?

Clue 1 of 3: Wave Equation Form

Identify the general form of a sinusoidal wave equation and determine the appropriate trigonometric function based on the initial conditions.

🥷

Ninja Strategy•Initial Condition and Parameter Check

First, use the initial condition (crest at $x = 0, t = 0$ ) to determine the correct trigonometric function (cosine) and eliminate options. Then, calculate the wave number $k$ and angular frequency $ω$ and match them to the remaining options.

🧠 Full Solution Path

Step 1: Determine Amplitude and Wave Function Type✦ Active

The maximum displacement of the wave is $2$ cm, so the amplitude $A = 2$ cm. A crest is at $x = 0$ at $t = 0$ . This means the displacement is maximum at the origin at $t = 0$ . This condition is satisfied by a cosine function with a phase constant of zero. For a wave traveling in the positive x-direction, the general form is $y (x, t) = A \cos (k x - ω t)$ . This eliminates option 3, which uses a sine function.

Step 2: Calculate Wave Number (k)○ Expand

The wavelength is $λ = 7.5$ cm. The wave number $k$ is given by the formula $k = \frac{2 π}{λ}$ .

k = \frac{2 π}{7.5 cm} \approx \frac{6.283}{7.5} \approx 0.8377 {cm}^{- 1}

Rounding to two decimal places, $k \approx 0.84 {cm}^{- 1}$ . Among the given options, $0.83$ is the closest value for $k$ .

💡 Teacher's Secret Hint

Remember to use the correct units for wavelength when calculating k.

Step 3: Calculate Angular Frequency (ω) and Form the Equation○ Expand

The wave travels a distance of $1.2$ cm in $0.3$ sec. The wave speed $v$ is calculated as:

v = \frac{distance}{time} = \frac{1.2 cm}{0.3 s} = 4 cm/s

The angular frequency $ω$ is related to wave speed and wave number by $v = \frac{ω}{k}$ , so $ω = v k$ .

ω = (4 cm/s) \times (0.8377 {cm}^{- 1}) \approx 3.3508 rad/s

Rounding to two decimal places, $ω \approx 3.35 rad/s$ . Substituting $A = 2$ , $k \approx 0.83$ , and $ω \approx 3.35$ into the wave equation $y (x, t) = A \cos (k x - ω t)$ gives:

y (x, t) = 2 \cos (0.83 x - 3.35 t) cm

This matches option 4.

💡 Teacher's Secret Hint

Ensure consistent units throughout the calculation. The wave is traveling in the positive x-direction, hence the minus sign in $(k x - ω t)$ .

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Physics Question 34 – JEE-MAIN 2025

Clue 1 of 3: Wave Equation Form

🧠 Full Solution Path