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