A sound wave of frequency 100 Hz is travelling in air. The speed of sound in air is `350 ms^-1` (a) By how much is the phase changed at a given point in 2.5 ms ? (b) What is the phase difference at a given instant between two points separated by a distance of 10.0 cm along the direction of propagation ?

Explanation

To solve the problem step by step, we will break it down into two parts as per the question.

### Part (a): Phase Change at a Given Point in 2.5 ms

**Step 1: Calculate the distance travelled by the sound wave in 2.5 ms.**
- Given:
- Speed of sound (v) = 350 m/s
- Time (t) = 2.5 ms = 2.5 × 10^-3 s
- Distance (d) = Speed × Time
\[ d = v \times t = 350 \, \text{m/s} \times 2.5 \times 10^{-3} \, \text{s} = 0.875 \, \text{m} \]

**Step 2: Calculate the wavelength (λ) using the frequency (f).**
- Given:
- Frequency (f) = 100 Hz
- The relationship between speed, frequency, and wavelength is:
\[ v = f \times \lambda \implies \lambda = \frac{v}{f} = \frac{350 \, \text{m/s}}{100 \, \text{Hz}} = 3.5 \, \text{m} \]

**Step 3: Calculate the phase change (Δφ) using the formula.**
- The formula for phase change is:
\[ \Delta \phi = \frac{2\pi}{\lambda} \times d \]
- Substitute the values:
\[ \Delta \phi = \frac{2\pi}{3.5} \times 0.875 \]
- Simplifying:
\[ \Delta \phi = \frac{2\pi \times 0.875}{3.5} = \frac{2\pi \times 0.875}{3.5} = \frac{2\pi \times 0.875}{3.5} = \frac{1.75\pi}{3.5} = \frac{\pi}{2} \]

### Part (b): Phase Difference Between Two Points Separated by 10.0 cm

**Step 1: Convert the distance from cm to m.**
- Given:
- Distance (d) = 10.0 cm = 0.1 m

**Step 2: Calculate the phase difference (Δφ) using the same formula.**
- Using the same formula for phase difference:
\[ \Delta \phi = \frac{2\pi}{\lambda} \times d \]
- Substitute the values:
\[ \Delta \phi = \frac{2\pi}{3.5} \times 0.1 \]
- Simplifying:
\[ \Delta \phi = \frac{2\pi \times 0.1}{3.5} = \frac{0.2\pi}{3.5} = \frac{0.2\pi}{3.5} = \frac{2\pi}{35} \]

### Final Answers:
(a) The phase change at a given point in 2.5 ms is \( \frac{\pi}{2} \).

(b) The phase difference at a given instant between two points separated by a distance of 10.0 cm is \( \frac{2\pi}{35} \).