An observer is approaching with a speed v , towards a stationary source emitting sound waves of wavelength `lambda_0` . The wavelength shift detected by the observer is ( Take c= speed of sound )

Options

A. `(lambda_0 v)/(c) `
B. `(lambda_0 c)/(v) `
C. `(lambda_0 v^(2))/(c^(2))`
D. Zero

Answer

To solve the problem of finding the wavelength shift detected by an observer approaching a stationary sound source, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a stationary sound source emitting sound waves with a wavelength \( \lambda_0 \).
- An observer is moving towards the source with a speed \( v \).
- The speed of sound in the medium is denoted as \( c \).

2. **Using the Doppler Effect**:
- The Doppler effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave.
- The formula for the observed frequency \( f' \) when the observer is moving towards a stationary source is given by:
\[ f' = f_s \left( \frac{c + v}{c} \right) \]
where \( f_s \) is the frequency emitted by the source.

3. **Relating Frequency and Wavelength**:
- The frequency of a wave is related to its wavelength by the equation:
\[ f = \frac{c}{\lambda} \]
- Therefore, we can express the emitted frequency \( f_s \) in terms of the wavelength \( \lambda_0 \):
\[ f_s = \frac{c}{\lambda_0} \]

4. **Substituting into the Doppler Effect Formula**:
- Substitute \( f_s \) into the observed frequency equation:
\[ f' = \frac{c}{\lambda_0} \left( \frac{c + v}{c} \right) = \frac{c + v}{\lambda_0} \]

5. **Finding the Observed Wavelength**:
- Now, we can find the observed wavelength \( \lambda \) using the relationship between frequency and wavelength:
\[ f' = \frac{c}{\lambda} \]
- Setting the two expressions for \( f' \) equal gives:
\[ \frac{c + v}{\lambda_0} = \frac{c}{\lambda} \]

6. **Rearranging for the Observed Wavelength**:
- Rearranging the equation to solve for \( \lambda \):
\[ \lambda = \frac{c \lambda_0}{c + v} \]

7. **Calculating the Wavelength Shift**:
- The wavelength shift \( \Delta \lambda \) is defined as the difference between the emitted wavelength \( \lambda_0 \) and the observed wavelength \( \lambda \):
\[ \Delta \lambda = \lambda_0 - \lambda \]
- Substituting the expression for \( \lambda \):
\[ \Delta \lambda = \lambda_0 - \frac{c \lambda_0}{c + v} \]
- Factoring out \( \lambda_0 \):
\[ \Delta \lambda = \lambda_0 \left( 1 - \frac{c}{c + v} \right) \]
- Simplifying the expression:
\[ \Delta \lambda = \lambda_0 \left( \frac{(c + v) - c}{c + v} \right) = \lambda_0 \left( \frac{v}{c + v} \right) \]

### Final Result:
Thus, the wavelength shift detected by the observer is:
\[ \Delta \lambda = \frac{v \lambda_0}{c + v} \]