Question

A sound wave is travelling with a frequency of $50\ \text{Hz}$. The phase difference between the two points in the path of a wave is $\frac{\pi}{3}$. The distance between those two points is (Velocity of sound in air $= 330\ \text{m/s}$)

• A. $1.1\ \text{m}$
• B. $0.6\ \text{m}$
• C. $2.2\ \text{m}$
• D. $1.7\ \text{m}$

Hint:

Think of phase difference as a fractional portion of a full wave cycle. A full cycle of $2\pi$ radians corresponds to exactly one full wavelength ($\lambda = 6.6\ \text{m}$). The given phase difference is $\frac{\pi}{3}$, which is exactly $\frac{1}{6}\text{th}$ of a full $2\pi$ cycle ($\frac{\pi/3}{2\pi} = \frac{1}{6}$). Therefore, the distance must simply be $\frac{1}{6}\text{th}$ of the wavelength: $\frac{6.6}{6} = 1.1\ \text{m}$. This mental shortcut completely bypasses formal algebraic rearrangements!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A progressive sound wave travels through air with a known frequency and speed. We are given the angular phase difference ($\Delta \phi$) between two distinct points along its propagation path and need to calculate the actual physical distance (path difference, $\Delta x$) separating them.

Step 2: Key Formula or Approach:
1. Relate the velocity ($v$), frequency ($f$), and wavelength ($\lambda$) of the wave using the standard wave equation: $$v = f \cdot \lambda \implies \lambda = \frac{v}{f}$$ 2. Connect the phase difference ($\Delta \phi$) to the path difference ($\Delta x$) using the fundamental wave relation: $$\Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta x \implies \Delta x = \frac{\lambda}{2\pi} \cdot \Delta \phi$$

Step 3: Detailed Explanation:
Let's first determine the wavelength ($\lambda$) of the sound wave from the provided data ($v = 330\ \text{m/s}$ and $f = 50\ \text{Hz}$): $$\lambda = \frac{330}{50} = 6.6\ \text{m}$$ Now, substitute this wavelength and the given phase difference $\Delta \phi = \frac{\pi}{3}$ into our path difference relationship formula: $$\Delta x = \frac{6.6}{2\pi} \times \frac{\pi}{3}$$ The constant $\pi$ cancels out perfectly from both the numerator and denominator: $$\Delta x = \frac{6.6}{2 \times 3} = \frac{6.6}{6} = 1.1\ \text{m}$$ Therefore, the physical spatial distance separating the two points is exactly $1.1\ \text{m}$.

Step 4: Final Answer:
The distance between the two points is $1.1\ \text{m}$, which matches option (A).