Question

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?

• 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. \( y = 2 \cos(0.13x - 0.5t) \, \text{cm} \)

Hint:

Remember, the general form of the wave equation is \( y = A \cos(kx - \omega t) \), where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency. Use these relationships to derive the wave equation.

Answer 1

The Correct Option is A

The velocity \( v \) of the wave is given by: \[ v = \frac{\text{distance}}{\text{time}} = \frac{12 \, \text{cm}}{0.3 \, \text{s}} = 4 \, \text{cm/s} \] Next, the wave number \( k \) and angular frequency \( \omega \) are related to the wavelength \( \lambda \) and frequency \( f \) as: \[ k = \frac{2 \pi}{\lambda} = \frac{2 \pi}{7.5} = 0.83 \, \text{cm}^{-1} \] \[ \omega = v k = 4 \times 0.83 = 3.35 \, \text{rad/s} \] Thus, the wave equation is: \[ y = A \cos(kx - \omega t) = A \cos(0.83x - 3.35t) \] Given that the amplitude \( A \) is 2 cm (maximum displacement), the equation becomes: \[ y = 2 \cos(0.83x - 3.35t) \, \text{cm} \]

Answer 2

We are given a sinusoidal wave traveling along +x with wavelength \( \lambda = 7.5\,\text{cm} \), it covers \( 1.2\,\text{cm} \) in \( 0.3\,\text{s} \), the crest is at \( x=0 \) at \( t=0 \), and amplitude \( A=2\,\text{cm} \). We need the correct wave equation.

Concept Used:

A right-moving sinusoidal wave can be written as \( y(x,t)=A\cos(kx-\omega t+\phi) \). Here \( k=\dfrac{2\pi}{\lambda} \), \( \omega=kv \), and the initial phase \( \phi \) is set by the initial condition. If the displacement is maximum (crest) at \( x=0,t=0 \), then \( \cos(\phi)=1 \Rightarrow \phi=0 \).

Step-by-Step Solution:

Step 1: Compute the wave speed from the given travel distance and time:

\[ v=\frac{1.2\,\text{cm}}{0.3\,\text{s}}=4\,\text{cm s}^{-1}. \]

Step 2: Compute the wave number \(k\) from the wavelength:

\[ k=\frac{2\pi}{\lambda}=\frac{2\pi}{7.5\,\text{cm}}\approx 0.8378\,\text{cm}^{-1}\ (\text{rounds to }0.83\,\text{cm}^{-1}). \]

Step 3: Compute the angular frequency \( \omega \):

\[ \omega=kv\approx(0.8378)(4)\approx 3.351\,\text{s}^{-1}\ (\text{rounds to }3.35\,\text{s}^{-1}). \]

Step 4: Use the crest-at-origin condition to fix phase:

\[ y(0,0)=A\cos(\phi)=\text{maximum}=A\ \Rightarrow\ \phi=0. \]

Step 5: Write the equation with amplitude \(A=2\,\text{cm}\):

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

Final Computation & Result

The correct representation is \( y=2\cos(0.83x-3.35t)\,\text{cm} \) (first option).