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: Conceptual Foundation: The phase difference between two wave points correlates with their spatial separation (path difference) and the wave's wavelength. First, determine the wavelength using the provided frequency and velocity.
Step 2: Governing Equations: 1. Wavelength (\( \lambda \)): \( \lambda = \frac{v}{f} \), where \( v \) represents wave velocity and \( f \) represents frequency.
2. Phase difference (\( \Delta \phi \)) to path difference (\( \Delta x \)) relation: \( \Delta \phi = \frac{2\pi}{\lambda} \Delta x \).
Step 3: Calculation and Analysis: 1. Wavelength Calculation (\( \lambda \)): Given \( f = 500 \) Hz and \( v = 1000 \) m/s.
\( \lambda = \frac{1000 \, \text{m/s}}{500 \, \text{Hz}} = 2 \, \text{m} \)
2. Path Difference Calculation (\( \Delta x \)): With a given phase difference \( \Delta \phi = \frac{\pi}{3} \).
Rearranging the formula yields: \( \Delta x = \frac{\lambda}{2\pi} \Delta \phi \).
\( \Delta x = \frac{2 \, \text{m}}{2\pi} \times \left(\frac{\pi}{3}\right) = \frac{1}{3} \, \text{m} \approx 0.333 \, \text{m} \)
The computed value (0.333 m) does not align with any provided options, suggesting a potential error in the input parameters (frequency, velocity, or phase difference). Let's investigate the phase difference that would yield the given options.
Assuming the intended phase difference was a common value in wave problems, \( \Delta \phi = \frac{\pi}{2} \):
\( \Delta x = \frac{2 \, \text{m}}{2\pi} \times \left(\frac{\pi}{2}\right) = \frac{2}{4} \, \text{m} = 0.5 \, \text{m} \)
This result corresponds to option (D). It is highly probable that the intended phase difference was \( \frac{\pi}{2} \) instead of \( \frac{\pi}{3} \).
Step 4: Conclusion: Under the assumption of a typo in the problem statement, where the phase difference should be \( \frac{\pi}{2} \), the calculated path difference is 0.50 m.