Question

The equation of a transverse wave travelling along a string is \( y(x, t) = 4.0 \sin \left( 20 \times 10^{-3} x + 600t \right) \) mm, where \( x \) is in mm and \( t \) is in seconds. The velocity of the wave is:

• A. +30 m/s
• B. -60 m/s
• C. -30 m/s
• D. +60 m/s

Hint:

The velocity of a wave is calculated using the relation \( v = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number.

Answer 1

The Correct Option is C

To determine the velocity of the transverse wave described by the equation \( y(x, t) = 4.0 \sin \left( 20 \times 10^{-3} x + 600t \right) \) mm, we analyze the standard wave equation:

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

where:

• \( A \) represents the wave's amplitude.
• \( k \) is the wave number in rad/m.
• \( \omega \) is the angular frequency in rad/s.
• \( y(x, t) \) denotes the displacement at position \( x \) and time \( t \).

Comparing the given equation with the standard form, we identify the following parameters:

• Amplitude, \( A = 4.0 \) mm.
• Wave number, \( k = 20 \times 10^{-3} \) rad/mm. Converting to rad/m: \( k = 20 \times 10^{-3} \times 10^3 \) rad/m \( = 20 \) rad/m.
• Angular frequency, \( \omega = 600 \) rad/s.

The wave velocity \( v \) is calculated using the formula:

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

Substituting the derived values:

\(v = \frac{600 \, \text{rad/s}}{20 \, \text{rad/m}} = 30 \, \text{m/s}\)

This calculation yields the magnitude of the wave velocity. The wave propagates in the negative x-direction, as indicated by the positive sign between the \( kx \) and \( \omega t \) terms in the wave equation (\(kx + \omega t\)). Therefore, the velocity is negative.

Consequently, the velocity of the wave is -30 m/s.

The correct answer is -30 m/s.