Question

Consider the wave represented by the equation \( y = (0.02) \sin \left( \pi x + 8\pi t \right) \), where all quantities are in SI units. The wavelength and speed of this wave respectively are \( y = (0.02) \sin \left( \pi x + 8\pi t \right) \).

• A. \( 2 \, \text{m}, \, 8 \, \text{ms}^{-1} \)
• B. \( 2 \, \text{m}, \, 0.02 \, \text{ms}^{-1} \)
• C. \( 0.02 \, \text{m}, \, 8 \, \text{ms}^{-1} \)
• D. \( 4 \, \text{m}, \, 8 \, \text{ms}^{-1} \)

Hint:

Remember that the wave speed can be derived from the angular frequency and wave number relations. Always verify the units of each quantity.

Answer 1

The Correct Option is A

The equation of the wave is given by \( y = A \sin (kx + \omega t) \), where \( A = 0.02 \) m, \( k \) is the wave number, and \( \omega \) is the angular frequency. The general relation for the wave number is \( k = \frac{2\pi}{\lambda} \), and the relation for the angular frequency is \( \omega = 2\pi f \), where \( \lambda \) is the wavelength and \( f \) is the frequency. 

Step 1: From the given equation, we have \( k = \pi \) and \( \omega = 8\pi \). Now, using the relation \( k = \frac{2\pi}{\lambda} \), we can find \( \lambda \): \[ \pi = \frac{2\pi}{\lambda} \quad \Rightarrow \quad \lambda = 2 \, \text{m}. \] 

Step 2: Next, we use the relation \( v = f\lambda \), where \( v \) is the speed of the wave. We can find the speed using the angular frequency relation \( \omega = 2\pi f \): \[ \omega = 8\pi \quad \Rightarrow \quad f = 4 \, \text{Hz}. \] Now, the speed \( v = f \lambda \) is \[ v = 4 \, \text{Hz} \times 2 \, \text{m} = 8 \, \text{ms}^{-1}. \]