Question

A sound wave of frequency $160\text{ Hz}$ has a velocity of $320\text{ m/s}$. When it travels through air, the particles having a phase difference of $90^\circ$ are separated by a distance of

• A. $50\text{ cm}$
• B. $1\text{ cm}$
• C. $25\text{ cm}$
• D. $75\text{ cm}$

Hint:

Think of phase difference as a fraction of a full wave cycle. A phase difference of $90^\circ$ is exactly one-quarter of a full $360^\circ$ cycle. Therefore, the distance separation is simply one-quarter of the wavelength: $\Delta x = \frac{\lambda}{4} = \frac{200\text{ cm}}{4} = 50\text{ cm}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a acoustic wave propagating through air with a frequency $f = 160\text{ Hz}$ and a velocity $v = 320\text{ m/s}$. We need to calculate the physical path separation distance $\Delta x$ between two points that oscillate with a constant phase difference of $\Delta \phi = 90^\circ$.

Step 2: Key Formula or Approach:
1. First, calculate the wavelength $\lambda$ of the sound wave using the wave speed relation: $$\lambda = \frac{v}{f}$$ 2. Next, use the fundamental relationship connecting phase difference and path separation distance: $$\Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta x$$ Note that the angles must be in radians, so convert $90^\circ$ to $\frac{\pi}{2}$ radians.

Step 3: Detailed Explanation:
First, calculate the spatial wavelength $\lambda$: $$\lambda = \frac{320\text{ m/s}}{160\text{ Hz}} = 2\text{ meters}$$ Convert the wavelength to centimeters to match the units in the options: $$\lambda = 2 \times 100\text{ cm} = 200\text{ cm}$$ Now, convert the phase difference from degrees to radians: $$\Delta \phi = 90^\circ = \frac{\pi}{2}\text{ radians}$$ Substitute these values into the path separation distance formula to isolate $\Delta x$: $$\frac{\pi}{2} = \frac{2\pi}{200} \cdot \Delta x$$ $$\frac{\pi}{2} = \frac{\pi}{100} \cdot \Delta x$$ Cancel out $\pi$ from both sides: $$\frac{1}{2} = \frac{\Delta x}{100} \implies \Delta x = \frac{100}{2} = 50\text{ cm}$$

Step 4: Final Answer:
The distance separating the particles is $50\text{ cm}$, which corresponds to option (A).