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

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

where:

• \( A \) is the amplitude of the wave.
• \( k \) is the wave number (measured in rad/m).
• \( \omega \) is the angular frequency (measured in rad/s).
• \( y(x, t) \) is the displacement at position \( x \) and time \( t \).

From the given equation, we identify:

• Amplitude, \( A = 4.0 \) mm.
• Wave number, \( k = 20 \times 10^{-3} \) rad/mm \( = 20 \) rad/m (since there are 1000 mm in a meter).
• Angular frequency, \( \omega = 600 \) rad/s.

The wave velocity \( v \) is given by the formula:

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

Substituting the values found:

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

The solution here gives us the magnitude of the wave velocity as 30 m/s. However, the wave travels in the negative x-direction, as indicated by the positive sign in the wave equation (\(kx + \omega t\)). This means the direction of propagation is negative.

Therefore, the velocity of the wave is -30 m/s, matching with the correct option -30 m/s.

Thus, the correct answer is -30 m/s.

Answer 2

The general equation for a wave is given by: \[ y(x, t) = A \sin(kx + \omega t), \] where \( k \) is the wave number and \( \omega \) is the angular frequency. From the given equation, we identify: \[ k = 20 \times 10^{-3} \, \text{m}^{-1}, \quad \omega = 600 \, \text{s}^{-1}. \] The velocity of the wave \( v \) is related to the angular frequency and wave number by: \[ v = \frac{\omega}{k}. \] Substitute the values of \( \omega \) and \( k \): \[ v = \frac{600}{20 \times 10^{-3}} = -30 \, \text{m/s}. \]