Three waves due to three coherent sources meet at one point. Their amplitudes are `sqrt2A_0 , 3A_0` and `sqrt2A_0` . Intensity corresponding to `A_0` is `l_0`. Phasse difference between first and second is `45^@`. Path difference between first and third is `lambda/4`. In phase angle, first wave lags behind from the other two waves. Find resultant intensity at this point.

Explanation

To solve the problem step-by-step, we will analyze the three waves and calculate the resultant intensity at the point where they meet.

### Step 1: Identify the Given Data
- Amplitudes of the waves:
- \( A_1 = \sqrt{2} A_0 \)
- \( A_2 = 3 A_0 \)
- \( A_3 = \sqrt{2} A_0 \)
- Intensity corresponding to \( A_0 \) is \( I_0 \).
- Phase difference between the first and second wave \( \Delta \phi_{12} = 45^\circ \).
- Path difference between the first and third wave corresponds to a phase difference of \( \Delta \phi_{13} = \frac{\lambda}{4} = 90^\circ \).

### Step 2: Convert Phase Differences to Radians
- Convert degrees to radians for calculations:
- \( 45^\circ = \frac{\pi}{4} \) radians
- \( 90^\circ = \frac{\pi}{2} \) radians

### Step 3: Represent the Waves as Phasors
- The first wave can be represented as:
\[ \text{Wave 1: } A_1 e^{i(0)} = \sqrt{2} A_0 \]
- The second wave leads the first by \( 45^\circ \):
\[ \text{Wave 2: } A_2 e^{i(\frac{\pi}{4})} = 3 A_0 e^{i(\frac{\pi}{4})} \]
- The third wave leads the first by \( 90^\circ \):
\[ \text{Wave 3: } A_3 e^{i(\frac{\pi}{2})} = \sqrt{2} A_0 e^{i(\frac{\pi}{2})} \]

### Step 4: Calculate the Resultant Amplitude
To find the resultant amplitude, we will combine the first and third waves first, since they have the same amplitude and a phase difference of \( 90^\circ \).

- The resultant amplitude \( A_{13} \) of waves 1 and 3:
\[ A_{13} = \sqrt{A_1^2 + A_3^2} = \sqrt{(\sqrt{2} A_0)^2 + (\sqrt{2} A_0)^2} = \sqrt{2 A_0^2 + 2 A_0^2} = \sqrt{4 A_0^2} = 2 A_0 \]

- Now, add the second wave:
\[ A_{\text{resultant}} = A_{13} + A_2 = 2 A_0 + 3 A_0 = 5 A_0 \]

### Step 5: Calculate the Resultant Intensity
The intensity \( I \) is proportional to the square of the amplitude:
\[ I \propto A^2 \]
Given that the intensity corresponding to \( A_0 \) is \( I_0 \):
\[ I_{\text{resultant}} = k (A_{\text{resultant}})^2 = k (5 A_0)^2 = k \cdot 25 A_0^2 \]
Since \( I_0 = k A_0^2 \), we can express \( I_{\text{resultant}} \) in terms of \( I_0 \):
\[ I_{\text{resultant}} = 25 I_0 \]

### Final Answer
The resultant intensity at the point where the three waves meet is:
\[ \boxed{25 I_0} \]