Apparent frequency will increase if

Options

A. Source is at rest, listener is moving towards source.
B. Source and listener are moving towards each other.
C. Source is moving towards listener at rest.
D. Source and listener are moving in same direction and with same velocity

Answer

To determine when the apparent frequency will increase, we can analyze different scenarios based on the Doppler effect. The apparent frequency \( f \) can be calculated using the formula:

\[ f = f_0 \frac{v + v_o}{v - v_s} \]

where:
- \( f_0 \) is the original frequency of the source,
- \( v \) is the speed of sound in the medium,
- \( v_o \) is the speed of the observer (listener),
- \( v_s \) is the speed of the source.

Now, let's analyze each scenario step by step:

### Step 1: Source at Rest, Listener Moving Towards Source
1. **Situation**: The source is stationary, and the listener is moving towards the source.
2. **Formula**: Since the source is at rest, \( v_s = 0 \). The formula becomes:
\[ f = f_0 \frac{v + v_o}{v} \]
3. **Analysis**: Here, \( v + v_o > v \), which means \( f > f_0 \). Thus, the apparent frequency increases.

### Step 2: Source and Listener Moving Towards Each Other
1. **Situation**: Both the source and listener are moving towards each other.
2. **Formula**: In this case, both \( v_o \) and \( v_s \) are positive. The formula becomes:
\[ f = f_0 \frac{v + v_o}{v - v_s} \]
3. **Analysis**: Since both \( v + v_o > v \) and \( v - v_s < v \), we have \( f > f_0 \). The apparent frequency increases.

### Step 3: Source Moving Towards Listener, Listener at Rest
1. **Situation**: The source is moving towards a stationary listener.
2. **Formula**: The formula becomes:
\[ f = f_0 \frac{v}{v - v_s} \]
3. **Analysis**: Here, \( v > v - v_s \) since \( v_s > 0 \), which means \( f > f_0 \). The apparent frequency increases.

### Step 4: Source and Listener Moving in the Same Direction with Same Velocity
1. **Situation**: Both the source and listener are moving in the same direction with the same velocity.
2. **Formula**: The formula becomes:
\[ f = f_0 \frac{v - v_s}{v - v_s} \]
3. **Analysis**: Here, the numerator and denominator are equal, leading to \( f = f_0 \). The apparent frequency does not increase.

### Conclusion
From the analysis of the four cases, we find that the apparent frequency increases in the first three cases but remains the same in the fourth case.

### Summary of Results:
- **Apparent frequency increases**:
1. Source at rest, listener moving towards the source.
2. Source and listener moving towards each other.
3. Source moving towards a stationary listener.
- **Apparent frequency does not increase**:
4. Source and listener moving in the same direction with the same velocity.