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…

Given: wavelength = 7.5 cm, distance traveled = 1.2 cm in time 0.3 s, amplitude A = 2 cm, and the crest is at x=0 when t=0. Find: the correct wave equation. For a sinusoidal wave traveling along the positive x-direction, the general form is y = A (kx - t +) Since the displacement is maximum at x=0, t=0, we need a crest there. Hence, y(0,0) = A = A so = 0 First, calculate the wave speed: v = distance time = 1.2 cm0.3 s = 4 cm/s Now calculate the wave number: k = 2 = 2 7.5 0.8378 cm^-1 0.83 cm^-1 Then the angular frequency is: = kv 0.8378 4 3.35 rad/s Substitute these values into y = A (kx - t) to get y = 2 (0.83x - 3.35t) cm Therefore, the correct option is A. The other options fail because they either use the wrong trigonometric form for the given crest condition, interchange k and, or use incorrect numerical values.

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2… — Solution | JEE Physics

MCQEasyJEE 2025Wave Motion Basics

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

A
$y = 2 \cos(0.83x - 3.35t) \, \text{cm}$
B
$y = 2 \sin(0.83x - 3.5t) \, \text{cm}$
C
$y = 2 \cos(3.35x - 0.83t) \, \text{cm}$
D
$y = 2 \cos(0.13x - 0.5t) \, \text{cm}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: wavelength $\lambda = 7.5 \, \text{cm}$ , distance traveled $= 1.2 \, \text{cm}$ in time $0.3 \, \text{s}$ , amplitude $A = 2 \, \text{cm}$ , and the crest is at $x=0$ when $t=0$ .

Find: the correct wave equation.

For a sinusoidal wave traveling along the positive $x$ -direction, the general form is

y = A \cos(kx - \omega t + \phi)

Since the displacement is maximum at $x=0, t=0$ , we need a crest there. Hence,

y(0,0) = A \cos \phi = A

\phi = 0

Compute wave parameters

First, calculate the wave speed:

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

Now calculate the wave number:

k = \frac{2\pi}{\lambda} = \frac{2\pi}{7.5} \approx 0.8378 \, \text{cm}^{-1} \approx 0.83 \, \text{cm}^{-1}

Then the angular frequency is:

\omega = kv \approx 0.8378 \times 4 \approx 3.35 \, \text{rad/s}

Match the standard form

Substitute these values into

y = A \cos(kx - \omega t)

to get

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

Therefore, the correct option is A.

The other options fail because they either use the wrong trigonometric form for the given crest condition, interchange $k$ and $\omega$ , or use incorrect numerical values.

Common mistakes

Using $\sin$ instead of $\cos$ for the initial condition. A crest at $x=0, t=0$ means displacement is maximum there, so the phase must satisfy $\cos\phi = 1$ . Use the cosine form with $\phi=0$ .
Interchanging $k$ and $\omega$ . The wave number comes from $k = \frac{2\pi}{\lambda}$ and angular frequency from $\omega = kv$ . Do not place $3.35$ with $x$ and $0.83$ with $t$ .
Computing wave speed incorrectly from the given data. The speed is $v = \frac{1.2}{0.3} = 4 \, \text{cm/s}$ , not $\frac{12}{0.3}$ unless the distance is first converted consistently from the source. Always use the stated distance carefully.

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Answer

Step-by-step solution

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