Question

A wave travelling in a stretched string is described by the equation \( y = A\sin(kx - \omega t) \). The maximum particle velocity is :

• A. \(A\omega\)
• B. \( \omega/k \)
• C. \( d\omega/dk \)
• D. \( x/t \)

Hint:

Maximum particle velocity in wave motion is \(A\omega\), not wave speed.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Particle velocity (\( v_p \)) is the rate of change of displacement of a particle in the medium with respect to time. It is obtained by differentiating the wave equation with respect to \( t \).
: Key Formula or Approach:
\[ v_p = \frac{\partial y}{\partial t} \]
Step 2: Detailed Explanation:
Given wave equation: \( y = A\sin(kx - \omega t) \).
Differentiating \( y \) partially with respect to \( t \):
\[ v_p = \frac{\partial}{\partial t} [A\sin(kx - \omega t)] \]
\[ v_p = A \cos(kx - \omega t) \cdot (-\omega) \]
\[ v_p = -A\omega \cos(kx - \omega t) \]
The maximum value of particle velocity (\( v_{p,max} \)) occurs when \( |\cos(kx - \omega t)| = 1 \).
\[ v_{p,max} = A\omega \]
Step 3: Final Answer:
The maximum particle velocity is \( A\omega \).