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 will follow these steps:

### Step-by-Step Solution:

1. **Understand the Doppler Effect Formula**:
The Doppler effect describes how the observed frequency (or wavelength) of a wave changes due to the relative motion between the source and the observer. The formula for the observed wavelength (\(\lambda'\)) when the observer is moving towards a stationary source is given by:
\[ \lambda' = \lambda_0 \frac{c + v_o}{c - v_s} \]
where:
- \(\lambda_0\) is the original wavelength,
- \(c\) is the speed of sound,
- \(v_o\) is the speed of the observer,
- \(v_s\) is the speed of the source.

2. **Substitute Known Values**:
In this scenario:
- The observer is moving towards the source, so \(v_o = v\) (the speed of the observer).
- The source is stationary, so \(v_s = 0\).
Therefore, the formula simplifies to:
\[ \lambda' = \lambda_0 \frac{c + v}{c} \]

3. **Calculate the Apparent Wavelength**:
Now, substituting the values into the formula:
\[ \lambda' = \lambda_0 \frac{c + v}{c} = \lambda_0 \left(1 + \frac{v}{c}\right) \]

4. **Determine the Wavelength Shift**:
The wavelength shift (\(\Delta \lambda\)) is defined as the difference between the observed wavelength and the original wavelength:
\[ \Delta \lambda = \lambda' - \lambda_0 \]
Substituting \(\lambda'\) from the previous step:
\[ \Delta \lambda = \lambda_0 \left(1 + \frac{v}{c}\right) - \lambda_0 \]
Simplifying this gives:
\[ \Delta \lambda = \lambda_0 \frac{v}{c} \]

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