Question

The velocity of a transverse wave propagating on a stretched string represented by the equation, $y = 0.5sin(x)(\frac{\pi}2 t + \frac{\pi}3 x)$ is (where x and y are in metres and / in seconds)

• A. 0.5 $ms^{-1}$
• B. 1.0 $ms^{-1}$
• C. 2 $ms^{-1}$
• D. 3 $ms^{-1}$
• E. 1.5 $ms^{-1}$

Answer 1

The Correct Option is C

The wave equation is given as: 

\[ y = 0.5 \sin\left(x\right)\left(\frac{\pi}{2} t + \frac{\pi}{3} x\right) \] This represents a transverse wave, where: - \( y \) is the displacement of the wave, - \( x \) is the position along the string, - \( t \) is the time. For a general wave equation of the form: \[ y(x, t) = A \sin(kx - \omega t) \] The velocity of the wave, \( v \), is given by: \[ v = \frac{\omega}{k} \] From the given equation, we identify: - The coefficient of \( x \) in the argument of the sine function gives \( k = \frac{\pi}{3} \), - The coefficient of \( t \) in the argument gives \( \omega = \frac{\pi}{2} \). Now, using the formula for wave velocity: \[ v = \frac{\omega}{k} = \frac{\frac{\pi}{2}}{\frac{\pi}{3}} = \frac{3}{2} = 2 \, \text{m/s} \] Thus, the velocity of the wave is \( 2 \, \text{m/s} \).

Correct Answer:

Correct Answer: (C) 2 \( \text{m/s} \)