In a large room , a person receives direct sound waves from a source `120 m` away . He also receives waves from the same source which reach him after being reflected from the ` 5m` high ceiling at a point halfway between them . For which wavelengths will these two sound waves interfere constructively ?

Options

A. `10, 10//2, 10//3, 10//4`
B. `20, 20//3, 20//5, 20//7`………
C. `30, 20, 10`,……..
D. `10, 10//3, 10//5, 10//7`…….

Explanation

To solve the problem, we need to determine the wavelengths for which the direct sound waves and the reflected sound waves interfere constructively.

### Step-by-Step Solution:

1. **Identify the Setup**:
- The source of sound (S) is located 120 m away from the person (A).
- The sound waves also reach the person after reflecting off a ceiling that is 5 m high. The reflection occurs at a point halfway between the source and the person.

2. **Calculate the Distance to the Reflection Point**:
- The distance from the source (S) to the reflection point (O) is half of 120 m, which is 60 m.
- The height of the ceiling is 5 m.

3. **Determine the Path of the Reflected Sound Wave**:
- The path of the reflected sound wave is from the source (S) to the reflection point (O) and then to the person (A).
- The distance AO (from the reflection point to the person) is also 60 m.

4. **Calculate the Total Distance for the Reflected Wave**:
- The total distance for the reflected sound wave (S to O to A) can be calculated using the Pythagorean theorem:
\[ \text{Distance SO} = \sqrt{(60^2 + 5^2)} = \sqrt{3600 + 25} = \sqrt{3625} \approx 60.41 \text{ m} \]
- The total distance for the reflected wave is:
\[ \text{Total Distance} = \text{Distance SO} + \text{Distance OA} = 60.41 \text{ m} + 60 \text{ m} = 120.41 \text{ m} \]

5. **Calculate the Path Difference**:
- The path difference (Δx) between the direct sound wave and the reflected sound wave is:
\[ \Delta x = \text{Total Distance (Reflected)} - \text{Direct Distance} = 120.41 \text{ m} - 120 \text{ m} = 0.41 \text{ m} \]

6. **Condition for Constructive Interference**:
- For constructive interference, the path difference must be an integral multiple of the wavelength (λ):
\[ \Delta x = n \lambda \quad \text{where } n \text{ is an integer} \]

7. **Calculate Wavelengths**:
- Rearranging gives:
\[ \lambda = \frac{\Delta x}{n} = \frac{0.41}{n} \]
- For different values of n (1, 2, 3, ...):
- For \( n = 1 \): \( \lambda = 0.41 \text{ m} \)
- For \( n = 2 \): \( \lambda = 0.205 \text{ m} \)
- For \( n = 3 \): \( \lambda = 0.1367 \text{ m} \)
- And so on.

### Final Answer:
The wavelengths for which the two sound waves will interfere constructively are given by:
\[ \lambda = \frac{0.41}{n} \text{ for } n = 1, 2, 3, \ldots \]