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:
Show Hint
- +30 m/s
- -60 m/s
- -30 m/s
- +60 m/s
The Correct Option is C
Approach Solution - 1
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.
Approach Solution -2
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