A source of sound produces waves of wave length `48cm`. This source is moving towards north with speed `1//4` th that of sound the apparent wave length of the waves to an observer standing south of the moving source will be

Options

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

Explanation

To solve the problem, we need to determine the apparent wavelength of sound waves produced by a moving source. The source is moving towards the north with a speed that is one-fourth of the speed of sound. The observer is located to the south of the moving source. We will use the Doppler effect to find the apparent wavelength.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength of sound produced by the source, \( \lambda = 48 \, \text{cm} \)
- Speed of sound, \( v \) (let's denote it as \( v \))
- Speed of the source, \( v_s = \frac{1}{4} v \)
- The observer is stationary and located south of the source.

2. **Determine the Apparent Frequency:**
According to the Doppler effect, when the source is moving towards the observer, the apparent frequency \( n' \) can be calculated using the formula:
\[ n' = n \left( \frac{v}{v - v_s} \right) \]
where \( n \) is the actual frequency of the source. Since \( v_s = \frac{1}{4}v \), we can substitute this into the equation:
\[ n' = n \left( \frac{v}{v - \frac{1}{4}v} \right) = n \left( \frac{v}{\frac{3}{4}v} \right) = n \left( \frac{4}{3} \right) \]

3. **Relate Frequency and Wavelength:**
The relationship between frequency and wavelength is given by:
\[ \lambda = \frac{v}{n} \]
Therefore, the apparent wavelength \( \lambda' \) can be expressed as:
\[ \lambda' = \frac{v}{n'} \]
Substituting \( n' \) from the previous step:
\[ \lambda' = \frac{v}{n \left( \frac{4}{3} \right)} = \frac{3v}{4n} \]

4. **Substituting for Initial Wavelength:**
We know that \( \lambda = \frac{v}{n} \), so substituting this into the equation for \( \lambda' \):
\[ \lambda' = \frac{3}{4} \cdot \lambda \]

5. **Calculating the Apparent Wavelength:**
Now substituting the given wavelength \( \lambda = 48 \, \text{cm} \):
\[ \lambda' = \frac{3}{4} \cdot 48 \, \text{cm} = 36 \, \text{cm} \]

### Final Answer:
The apparent wavelength of the waves to an observer standing south of the moving source is **36 cm**.