Question

The equation of wave is given by \( Y = 10^2 \sin 2 \pi \left( (60t - 0.5x + \frac{\pi}{4}) \right) \) where \( x \) and \( Y \) are in m and t in s. The speed of the wave is _____ km h\(^{-1}\).

Hint:

To find the speed of a wave, use the relation \( v = \frac{\omega}{k} \), and convert the units from m/s to km/h by multiplying by \( \frac{18}{5} \).

Answer 1

Correct Answer: 1152

The equation of the wave is given as: \[ Y = 10^2 \sin 2 \pi \left( (60t - 0.5x + \frac{\pi}{4}) \right) \] The general form of the wave equation is: \[ Y = A \sin \left( 2 \pi \left( \frac{x}{\lambda} - \frac{t}{T} \right) \right) \] Here, \( \lambda \) is the wavelength, \( T \) is the period, and the wave number \( k = \frac{2\pi}{\lambda} \) and the angular frequency \( \omega = \frac{2\pi}{T} \). From the given equation: \[ \omega = 60 \quad \text{and} \quad k = 0.5 \] The speed of the wave is given by: \[ v = \frac{\omega}{k} \] Substituting the values: \[ v = \frac{60}{0.5} = 120 \, \text{m/s} \] Now, converting the speed into km/h: \[ v = 120 \times \frac{18}{5} = 1152 \, \text{km/h} \] Thus, the speed of the wave is \( 1152 \, \text{km/h} \).