Four monochromatic and coherent sources of light emitting waves in phase at placed on y axis at y = 0, a, 2a and 3a. If the intensity of wave reaching at point P far away on y axis from each of the source is almost the same and equal to `I_(0)`, then the resultant intensity at P for `a=(lambda)/(8)` is `nI_(0)`. The value of `[n]` is.
Here [] is greatest integer funciton.

Answer

To solve the problem, we need to find the resultant intensity at point P due to the four coherent sources of light placed at y = 0, a, 2a, and 3a, where \( a = \frac{\lambda}{8} \). The intensity of the waves reaching point P from each source is \( I_0 \).

### Step-by-Step Solution:

1. **Identify the Phase Difference**:
The sources are at positions \( y = 0 \), \( y = a \), \( y = 2a \), and \( y = 3a \). The distance between consecutive sources is \( a \). The phase difference \( \Delta \phi \) between two consecutive sources can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta y \]
where \( \Delta y = a \). Thus,
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot a = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{8} = \frac{\pi}{4} \]

2. **Calculate the Phase Differences**:
- Between source 1 (at \( y = 0 \)) and source 2 (at \( y = a \)): \( \phi_2 - \phi_1 = \frac{\pi}{4} \)
- Between source 2 (at \( y = a \)) and source 3 (at \( y = 2a \)): \( \phi_3 - \phi_2 = \frac{\pi}{4} \)
- Between source 3 (at \( y = 2a \)) and source 4 (at \( y = 3a \)): \( \phi_4 - \phi_3 = \frac{\pi}{4} \)

3. **Resultant Amplitude Calculation**:
The resultant amplitude \( E \) at point P can be calculated using the formula for the resultant of coherent sources:
\[ E = E_1 + E_2 + E_3 + E_4 \]
Each source contributes an amplitude \( E_0 \) (where \( I_0 \propto E_0^2 \)). The amplitudes can be represented as:
- \( E_1 = E_0 \)
- \( E_2 = E_0 e^{i\frac{\pi}{4}} \)
- \( E_3 = E_0 e^{i\frac{\pi}{2}} \)
- \( E_4 = E_0 e^{i\frac{3\pi}{4}} \)

4. **Calculate the Resultant Amplitude**:
The resultant amplitude can be calculated by adding these vectors in the complex plane:
\[ E = E_0 \left( 1 + e^{i\frac{\pi}{4}} + e^{i\frac{\pi}{2}} + e^{i\frac{3\pi}{4}} \right) \]
This simplifies to:
\[ E = E_0 \left( 1 + \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} + i + \frac{-\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right) \]
The real part sums to \( 1 + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 1 \) and the imaginary part sums to \( i + i\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} = i\sqrt{2} \).

5. **Calculate Resultant Intensity**:
The resultant intensity \( I \) is given by:
\[ I = |E|^2 = \left( \sqrt{1^2 + (\sqrt{2})^2} \right)^2 = (1 + 2) = 3 \]
Thus, the resultant intensity at point P is:
\[ I = 3 I_0 \]

6. **Determine \( n \)**:
The problem states that the resultant intensity is \( n I_0 \). Therefore, \( n = 3 \).

### Final Answer:
The value of \( [n] \) is:
\[ [n] = 3 \]