Question

A transverse harmonic wave on a string is given by y(x,t) = 5 sin(6t + 0.003x) where x and y are in cm and t in sec. The wave velocity is _______ ms-1 

Hint:

The wave velocity is the ratio of the angular frequency to the wave number: \(v = \frac{ω }{k}\). Make sure your units are consistent (e.g., both in meters or both in cen timeters)

Answer 1

Correct Answer: 20

Step 1: General Equation for a Transverse Harmonic Wave

The general equation is:

\( y(x, t) = A \sin(kx \pm \omega t) \)

• \( A \): Amplitude
• \( k \): Wave number, \( k = \frac{2\pi}{\lambda} \)
• \( \omega \): Angular frequency, \( \omega = 2\pi f \)
• \( \lambda \): Wavelength
• \( f \): Frequency

Step 2: Given Equation

Compare the given equation \( y(x, t) = 5 \sin(6t + 0.003x) \) with the general equation:

• \( A = 5 \, \text{cm} \)
• \( \omega = 6 \, \text{rad/s} \)
• \( k = 0.003 \, \text{rad/cm} = 0.3 \, \text{rad/m} \) (converted cm to m)

Step 3: Formula for Wave Velocity

The wave velocity (\( v \)) is related to \( \omega \) and \( k \) by:

\( v = \frac{\omega}{k} \)

Step 4: Substitute the Values

Substitute \( \omega = 6 \, \text{rad/s} \) and \( k = 0.3 \, \text{rad/m} \):

\[ v = \frac{6}{0.3} = 20 \, \text{m/s} \]

Final Answer:

The wave velocity is 20 m/s.