The apparent frequency and real frequency of sound are the same
(A) If source of sound and observer are moving in the same direction with same speed
(B) If source of sound and observer are moving in the opposite direction with the same speed

Options

A. A is true, B is false
B. A is false B is true
C. Both A & B are true
D. Both A & B are false

Answer

To solve the question regarding the conditions under which the apparent frequency and real frequency of sound are the same, we need to analyze the Doppler effect.

### Step-by-Step Solution:

1. **Understanding Apparent Frequency**:
The apparent frequency (f_a) is the frequency of sound perceived by an observer. The real frequency (f_o) is the frequency emitted by the source of sound. The Doppler effect describes how the frequency changes when the source and observer are in motion relative to each other.

2. **Doppler Effect Formula**:
The formula for apparent frequency when both the source and observer are moving is given by:
\[ f_a = f_o \frac{v + v_o}{v - v_s} \]
where:
- \( f_a \) = apparent frequency
- \( f_o \) = real frequency
- \( v \) = speed of sound in the medium
- \( v_o \) = speed of the observer (positive if moving towards the source)
- \( v_s \) = speed of the source (positive if moving away from the observer)

3. **Case A: Source and Observer Moving in the Same Direction with Same Speed**:
- If both the source and observer are moving in the same direction at the same speed, then:
- \( v_o = v_s \)
- Substituting into the formula:
\[ f_a = f_o \frac{v + v_s}{v - v_s} \]
- Since \( v_o = v_s \), the formula simplifies to:
\[ f_a = f_o \frac{v + v_s}{v - v_s} = f_o \frac{v + v_o}{v - v_o} \]
- Here, the numerator and denominator will effectively cancel out the relative motion, leading to:
\[ f_a = f_o \]
- Thus, the apparent frequency equals the real frequency.

4. **Case B: Source and Observer Moving in Opposite Directions with Same Speed**:
- If the source and observer are moving towards each other with the same speed:
- \( v_o \) is positive (observer moving towards the source)
- \( v_s \) is negative (source moving towards the observer)
- The formula becomes:
\[ f_a = f_o \frac{v + v_o}{v - (-v_s)} = f_o \frac{v + v_o}{v + v_s} \]
- Since both are moving towards each other, the apparent frequency will be higher than the real frequency:
\[ f_a > f_o \]
- Therefore, the apparent frequency does not equal the real frequency.

5. **Conclusion**:
- The apparent frequency and real frequency of sound are the same only when the source and observer are moving in the same direction with the same speed (Option A). In contrast, when they are moving in opposite directions with the same speed (Option B), the apparent frequency is greater than the real frequency.

### Final Answer:
(A) If the source of sound and observer are moving in the same direction with the same speed.