Question

A wave of frequency 500 Hz is travelling with a velocity 1000 m/s. How far are two points situated in wave whose displacement differ in phase by \( \frac{\pi}{3} \)?

• A. 0.50 cm
• B. 2.5 m
• C. 0.25 m
• D. 0.50 m

Hint:

When your calculated answer doesn't match any options, double-check your calculations. If they are correct, consider the possibility of a typo in the question. You can work backwards from the answers to see which simple change in the input data would lead to one of the options. For waves, phase differences of \( \pi/2 \), \( \pi \), and \( 2\pi \) are very common.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The phase difference between two points on a wave is related to the physical distance between them (the path difference) and the wavelength of the wave. We first need to calculate the wavelength from the given frequency and velocity.
Step 2: Key Formula or Approach:
1. Wavelength (\( \lambda \)): \( \lambda = \frac{v}{f} \), where \( v \) is the wave velocity and \( f \) is the frequency.
2. Relationship between phase difference (\( \Delta \phi \)) and path difference (\( \Delta x \)): \( \Delta \phi = \frac{2\pi}{\lambda} \Delta x \).
Step 3: Detailed Explanation:
1. Calculate the wavelength (\( \lambda \)):
Given \( f = 500 \) Hz and \( v = 1000 \) m/s.
\[ \lambda = \frac{1000 \, \text{m/s}}{500 \, \text{Hz}} = 2 \, \text{m} \] 2. Calculate the path difference (\( \Delta x \)):
Given the phase difference \( \Delta \phi = \frac{\pi}{3} \).
Rearranging the formula: \( \Delta x = \frac{\lambda}{2\pi} \Delta \phi \).
\[ \Delta x = \frac{2 \, \text{m}}{2\pi} \times \left(\frac{\pi}{3}\right) = \frac{1}{3} \, \text{m} \approx 0.333 \, \text{m} \] This calculated value (0.333 m) does not exactly match any of the options. This suggests a possible typo in the question's given values (frequency, velocity, or phase difference). Let's check what phase difference would lead to the given options.
If we assume the intended phase difference was \( \Delta \phi = \frac{\pi}{2} \), a common value in wave problems:
\[ \Delta x = \frac{2 \, \text{m}}{2\pi} \times \left(\frac{\pi}{2}\right) = \frac{2}{4} \, \text{m} = 0.5 \, \text{m} \] This result matches option (D). It is highly probable that the phase difference was intended to be \( \frac{\pi}{2} \) instead of \( \frac{\pi}{3} \).
Step 4: Final Answer:
Assuming a typo in the question where the phase difference should be \( \frac{\pi}{2} \), the path difference is 0.50 m.