In a large room , a person receives direct sound waves from a source 120 metres away from him. He also receives waves from the same source which reach , being reflected from the `25 m` high ceiling at a point halfway between them . The two waves interfere constructively for a wavelength of

Options

A. `20 , 20/3 , 20/5` , etc.
B. `10 , 5 , 2.5 `, etc.
C. `10 , 20 , 30 `, etc.
D. ` 15 , 25 , 35 `, etc.

Explanation

To solve the problem, we need to determine the wavelength for which the direct sound waves and the reflected sound waves interfere constructively. Here's a step-by-step solution:

### Step 1: Understand the setup
We have a sound source located 120 meters away from a person. The sound waves travel directly to the person and also reflect off the ceiling (which is 25 meters high) before reaching the person.

### Step 2: Determine the geometry of the situation
The distance from the sound source to the person is 120 meters. The sound reflects off the ceiling at a point halfway between the source and the person. Therefore, the distance from the source to the halfway point is:
\[ \text{Halfway distance} = \frac{120}{2} = 60 \text{ meters} \]

### Step 3: Calculate the path of the reflected wave
The reflected wave travels from the source to the halfway point, then reflects off the ceiling, and finally travels to the person. To find the total distance traveled by the reflected wave, we can use the Pythagorean theorem.

1. The horizontal distance from the source to the halfway point is 60 meters.
2. The vertical distance from the halfway point to the ceiling is 25 meters.

Using the Pythagorean theorem:
\[ d = \sqrt{(60^2) + (25^2)} \]
\[ d = \sqrt{3600 + 625} \]
\[ d = \sqrt{4225} \]
\[ d = 65 \text{ meters} \]

Since the wave travels to the halfway point and back down to the person, the total distance for the reflected wave is:
\[ \text{Total distance} = 2 \times 65 = 130 \text{ meters} \]

### Step 4: Determine the path difference
The path difference between the direct wave and the reflected wave is:
\[ \text{Path difference} = \text{Distance of reflected wave} - \text{Distance of direct wave} \]
\[ \text{Path difference} = 130 \text{ meters} - 120 \text{ meters} = 10 \text{ meters} \]

### Step 5: Apply the condition for constructive interference
For constructive interference, the path difference must be an integral multiple of the wavelength (\( n \lambda \)):
\[ \text{Path difference} = n \lambda \]
Thus, we have:
\[ 10 = n \lambda \]

### Step 6: Solve for the wavelength
From the equation \( 10 = n \lambda \), we can express the wavelength as:
\[ \lambda = \frac{10}{n} \]

### Step 7: Find possible wavelengths
We can find different wavelengths for different integer values of \( n \):
- For \( n = 1 \): \( \lambda = 10 \text{ meters} \)
- For \( n = 2 \): \( \lambda = 5 \text{ meters} \)
- For \( n = 3 \): \( \lambda \approx 3.33 \text{ meters} \)
- For \( n = 4 \): \( \lambda = 2.5 \text{ meters} \)

### Conclusion
Among the possible wavelengths, the correct answer is \( \lambda = 5 \text{ meters} \) when \( n = 2 \).

### Final Answer
The wavelength for which the waves interfere constructively is \( \lambda = 5 \text{ meters} \).

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