A source of sound produces waves of wave length `48cm`. This source is moving towards north with speed `V//4` where V is speed of sound. The apparent wavelength of the waves to an observer standing south of the moving source will be

Options

A. `48cm`
B. `60cm`
C. `72cm`
D. `96cm`

Explanation

To solve the problem, we need to determine the apparent wavelength of the sound waves as perceived by an observer located to the south of a moving sound source. The source is moving northward at a speed of \( \frac{V}{4} \), where \( V \) is the speed of sound, and the original wavelength of the sound produced by the source is \( 48 \, \text{cm} \).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Original wavelength (\( \lambda \)) = \( 48 \, \text{cm} \)
- Speed of sound (\( V \))
- Speed of the source (\( V_s \)) = \( \frac{V}{4} \)
- Observer is located to the south of the source.

2. **Calculate the Frequency of the Source:**
The frequency (\( f \)) of the sound can be calculated using the formula:
\[ f = \frac{V}{\lambda} \]
Substituting the values:
\[ f = \frac{V}{48 \, \text{cm}} = \frac{V}{0.48 \, \text{m}} \quad \text{(converting cm to m)} \]

3. **Determine the Apparent Frequency:**
Since the source is moving towards the observer, the apparent frequency (\( f' \)) can be calculated using the Doppler effect formula for a source moving towards a stationary observer:
\[ f' = f \left( \frac{V}{V - V_s} \right) \]
Substituting \( V_s = \frac{V}{4} \):
\[ f' = f \left( \frac{V}{V - \frac{V}{4}} \right) = f \left( \frac{V}{\frac{3V}{4}} \right) = f \left( \frac{4}{3} \right) \]

4. **Substituting the Frequency into the Apparent Frequency Formula:**
Now, substituting the frequency \( f \) into the equation:
\[ f' = \left( \frac{V}{48 \, \text{cm}} \right) \left( \frac{4}{3} \right) = \frac{4V}{144 \, \text{cm}} = \frac{V}{36 \, \text{cm}} \]

5. **Calculate the Apparent Wavelength:**
The apparent wavelength (\( \lambda' \)) can be calculated using the relationship between speed, frequency, and wavelength:
\[ \lambda' = \frac{V}{f'} \]
Substituting \( f' \):
\[ \lambda' = \frac{V}{\frac{V}{36 \, \text{cm}}} = 36 \, \text{cm} \]

### Final Answer:
The apparent wavelength of the waves to an observer standing south of the moving source is \( 36 \, \text{cm} \).

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