There will be no change in frequency or no Doppler effect if

Options

A. Both source and listener are at rest and the medium is moving
B. Source and listener move in same direction with same speed.
C. When source is at centre of a circle and listener is moving along that circle with uniform speed.
D. When listener is at centre of a circle and source is moving along the circle with uniform speed.

Answer

To determine the conditions under which there will be no change in frequency or no Doppler effect, we can analyze the given scenarios step by step.

### Step-by-Step Solution

1. **Understanding the Doppler Effect**:
- The Doppler effect refers to the change in frequency (or wavelength) of a wave in relation to an observer moving relative to the wave source. The formula for the apparent frequency \( f' \) when the source and observer are in motion is given by:
\[ f' = f_0 \frac{v - v_o}{v - v_s} \]
where:
- \( f' \) = apparent frequency,
- \( f_0 \) = original frequency,
- \( v \) = speed of sound,
- \( v_o \) = speed of the observer,
- \( v_s \) = speed of the source.

2. **Condition 1: Both Source and Observer at Rest**:
- If both the source and observer are at rest, then \( v_o = 0 \) and \( v_s = 0 \). The formula simplifies to:
\[ f' = f_0 \frac{v - 0}{v - 0} = f_0 \]
- Thus, there is no change in frequency.

**Hint**: Consider what happens to the velocities when both the source and observer are stationary.

3. **Condition 2: Source and Observer Moving in the Same Direction with Same Speed**:
- If both the source and observer are moving in the same direction with the same speed, then \( v_o = v_s \). The formula becomes:
\[ f' = f_0 \frac{v - v}{v - v} = f_0 \]
- Again, there is no change in frequency.

**Hint**: Think about how relative motion affects frequency when both are moving together.

4. **Condition 3: Source at the Center of a Circle, Observer Moving Along the Circle**:
- In this scenario, the observer moves in a circular path around the stationary source. The distance between the source and observer remains constant, and the angle between their positions is always changing but does not affect the line of sight. Thus, the apparent frequency remains:
\[ f' = f_0 \]

**Hint**: Visualize the circular motion and how it affects the line of sight between the source and observer.

5. **Condition 4: Observer at the Center of a Circle, Source Moving Along the Circle**:
- Similar to the previous case, if the observer is at the center and the source moves along a circular path, the distance remains constant, and the angle does not lead to a change in frequency:
\[ f' = f_0 \]

**Hint**: Consider how the circular motion keeps the distance constant and affects the frequency.

### Conclusion
In summary, there will be no change in frequency or Doppler effect under the following conditions:
1. Both source and observer are at rest.
2. Source and observer are moving in the same direction with the same speed.
3. The source is at the center of a circle while the observer moves along the circle.
4. The observer is at the center of a circle while the source moves along the circle.

Each of these scenarios results in the apparent frequency remaining equal to the original frequency \( f_0 \).