Question

The wave described by \( y = 0.35 \sin (2\pi t - 10\pi x) \), where \( x \) and \( y \) are in metre and \( t \) in second, is a wave travelling along the:

• A. negative x-direction with amplitude 0.35 m and wavelength \( \lambda = 0.5 \, \text{m} \)
• B. negative x-direction with frequency \( \pi \, \text{Hz} \) and wavelength \( \lambda = 0.5 \, \text{m} \)
• C. positive x-direction with frequency 1 Hz and amplitude 3.5 m
• D. positive x-direction with frequency 1 Hz and wavelength \( \lambda = 0.2 \, \text{m} \)

Hint:

In a wave equation of the form \( y = A \sin(kx - \omega t) \), the wave moves in the positive x-direction, and for \( y = A \sin(kx + \omega t) \), it moves in the negative x-direction.

Answer 1

The Correct Option is B

Step 1: Understanding the Wave Equation.
The given wave equation is of the form: \[ y = A \sin (kx - \omega t) \] where \( k = 10\pi \) and \( \omega = 2\pi \). The wavelength \( \lambda \) is related to \( k \) by: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = 0.5 \, \text{m} \] The frequency \( f \) is related to \( \omega \) by: \[ f = \frac{\omega}{2\pi} = \frac{2\pi}{2\pi} = 1 \, \text{Hz} \] Since the wave is of the form \( \sin (kx - \omega t) \), it is moving in the positive x-direction, but the negative sign inside the sine function indicates motion in the opposite direction, i.e., negative x-direction. Step 2: Final Answer.
Thus, the wave travels in the negative x-direction with frequency \( \pi \, \text{Hz} \) and wavelength \( \lambda = 0.5 \, \text{m} \).