Two friends A and B are standing a distance x apart in an open field and wind is blowing from A to B.A beats a drum and B hears the sound `t_1` timed after he sees the event. A and B interchange their positions and the experiment is repeated. This time B hears the drum `t_2` time after he sees the event. Suppose A and B in the previous problem change their positions in such a way that the line joining them becomes perpendicular to the direction of wind while maintaining the separation x. What will be the time lag B finds betweenseeing and hearing the drum beating by A?

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Conditions
- Friends A and B are standing a distance \( x \) apart.
- Wind is blowing from A to B.
- When A beats the drum, B hears it after a time \( t_1 \).

### Step 2: Establish the Relationship for \( t_1 \)
- The speed of sound in still air is \( v \).
- The speed of wind is \( u \).
- The effective speed of sound from A to B, with the wind aiding it, is \( v + u \).
- The time taken for the sound to reach B is given by:
\[ t_1 = \frac{x}{v + u} \]

### Step 3: Interchange Positions of A and B
- Now A and B switch positions.
- B beats the drum, and A hears it after time \( t_2 \).

### Step 4: Establish the Relationship for \( t_2 \)
- The effective speed of sound from B to A, against the wind, is \( v - u \).
- The time taken for the sound to reach A is given by:
\[ t_2 = \frac{x}{v - u} \]

### Step 5: Relate \( t_1 \) and \( t_2 \)
- We can express \( t_1 \) and \( t_2 \) as:
\[ t_1 = \frac{x}{v + u} \quad \text{and} \quad t_2 = \frac{x}{v - u} \]

### Step 6: Find the Relationship Between \( t_1 \) and \( t_2 \)
- Rearranging gives:
\[ \frac{1}{t_1} = \frac{v + u}{x} \quad \text{and} \quad \frac{1}{t_2} = \frac{v - u}{x} \]
- Subtracting these two equations:
\[ \frac{1}{t_1} - \frac{1}{t_2} = \frac{(v + u) - (v - u)}{x} = \frac{2u}{x} \]

### Step 7: Solve for \( u \)
- Rearranging gives:
\[ 2u = \frac{x}{t_1 t_2} \implies u = \frac{x}{2(t_1 t_2)} \]

### Step 8: Determine the Time Lag When Positions are Perpendicular
- When A and B are perpendicular to the wind direction, the effective speed of sound between them is given by:
\[ v_{\text{effective}} = \sqrt{v^2 - u^2} \]
- The time lag \( t \) for B to hear the drum when A beats it is:
\[ t = \frac{x}{\sqrt{v^2 - u^2}} \]

### Step 9: Substitute for \( u \)
- Substitute \( u \) into the equation:
\[ t = \frac{x}{\sqrt{v^2 - \left(\frac{x}{2(t_1 t_2)}\right)^2}} \]

### Step 10: Final Expression for Time Lag
- The final expression for the time lag \( t \) becomes:
\[ t = \sqrt{t_1 t_2} \]

### Conclusion
Thus, the time lag \( t \) that B finds between seeing and hearing the drum beating by A when they are perpendicular to the wind direction is:
\[ t = \sqrt{t_1 t_2} \]