Consider the superposition of `N` harmonic waves of equal amplitude and frequency. If `N` is a very large number determine the resultant intensity in terms of the intensity `(I_(0))` of each component wave for the conditions when the component wave have identical phases.

Explanation

To solve the problem step by step, we will analyze the superposition of `N` harmonic waves of equal amplitude and frequency, specifically when they have identical phases.

### Step-by-Step Solution:

1. **Understanding the Waves**:
We have `N` harmonic waves, each described by the equation:
\[ y_n = A \sin(\omega t - kx + \phi) \]
Given that the waves have identical phases, we can set the phase difference \(\phi = 0\).

2. **Wave Equations**:
For `N` waves with identical phases, the equations can be written as:
\[ y_1 = A \sin(\omega t - kx) \]
\[ y_2 = A \sin(\omega t - kx) \]
\[ \vdots \]
\[ y_N = A \sin(\omega t - kx) \]

3. **Superposition of Waves**:
The resultant wave \(y_R\) from the superposition of these `N` waves is given by:
\[ y_R = y_1 + y_2 + \ldots + y_N = N \cdot A \sin(\omega t - kx) \]
This is because all the waves are in phase and can be added directly.

4. **Amplitude of Resultant Wave**:
The amplitude of the resultant wave is:
\[ A_R = N \cdot A \]

5. **Intensity of Waves**:
The intensity \(I\) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]
If the intensity of each individual wave is \(I_0\), we can express it as:
\[ I_0 = k \cdot A^2 \]
where \(k\) is a proportionality constant.

6. **Intensity of Resultant Wave**:
The intensity \(I_R\) of the resultant wave can be expressed as:
\[ I_R = k \cdot (A_R)^2 = k \cdot (N \cdot A)^2 = k \cdot N^2 \cdot A^2 \]

7. **Substituting for \(A^2\)**:
We know from the individual wave intensity that:
\[ A^2 = \frac{I_0}{k} \]
Substituting this into the expression for \(I_R\):
\[ I_R = k \cdot N^2 \cdot \left(\frac{I_0}{k}\right) = N^2 \cdot I_0 \]

8. **Final Result**:
Thus, the resultant intensity \(I_R\) in terms of the intensity \(I_0\) of each component wave is:
\[ I_R = N^2 \cdot I_0 \]