A stationary source sends forth monochromatic sound. A wall approaches it with velocity 33 cm/s. The propagation velocity of sound in the medium is `c=330(m)/(s)`. How much, in per cent , does the wavelength of sound change on reflection from the wall?

Explanation

To solve the problem, we need to determine how much the wavelength of sound changes when it reflects off a wall that is approaching the sound source. We will use the Doppler effect principles to find the change in wavelength and express it as a percentage.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Velocity of sound in the medium, \( c = 330 \, \text{m/s} \)
- Velocity of the wall (approaching the source), \( u = 33 \, \text{cm/s} = 0.33 \, \text{m/s} \)

2. **Calculate the Effective Velocity of Sound Towards the Wall**:
Since the wall is moving towards the stationary source, the effective velocity of sound towards the wall is:
\[ V' = c + u = 330 \, \text{m/s} + 0.33 \, \text{m/s} = 330.33 \, \text{m/s} \]

3. **Determine the Frequency of Sound When it Reaches the Wall**:
The frequency \( f \) of the sound remains constant since the source is stationary. The wavelength \( \lambda \) is given by:
\[ \lambda = \frac{c}{f} \]
When the sound reflects off the wall, it behaves as a new source of sound.

4. **Calculate the Frequency Observed After Reflection**:
The frequency \( f' \) observed after reflection can be calculated using the Doppler effect formula:
\[ f' = f \frac{c + u}{c - u} \]
Here, \( c - u \) is used because the wall is moving towards the sound source.

5. **Determine the New Wavelength After Reflection**:
The new wavelength \( \lambda' \) after reflection can be calculated using:
\[ \lambda' = \frac{c}{f'} \]
Substituting \( f' \):
\[ \lambda' = \frac{c}{f \frac{c + u}{c - u}} = \lambda \frac{c - u}{c + u} \]

6. **Calculate the Change in Wavelength**:
The change in wavelength can be expressed as:
\[ \frac{\lambda' - \lambda}{\lambda} = \frac{\lambda \frac{c - u}{c + u} - \lambda}{\lambda} = \frac{c - u - c - u}{c + u} = \frac{-2u}{c + u} \]

7. **Calculate the Percentage Change in Wavelength**:
The percentage change in wavelength is given by:
\[ \text{Percentage Change} = \left( \frac{-2u}{c + u} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left( \frac{-2 \times 0.33}{330 + 0.33} \right) \times 100 \approx \left( \frac{-0.66}{330.33} \right) \times 100 \approx -0.2\% \]

### Final Answer:
The percentage change in the wavelength of sound on reflection from the wall is approximately **-0.2%** (indicating a decrease in wavelength).