$$n$$ identical waves each of intensity $$I_0$$ interfere with each other. The ratio of maximum intensities if the interference is (i) coherent and (ii) incoherent is :

We are given $$n$$ identical waves, each of intensity $$I_0$$. We need to find the ratio of the maximum intensities for two cases: (i) coherent interference and (ii) incoherent interference.

First, recall that intensity is proportional to the square of the amplitude. For a single wave, intensity $$I_0$$ corresponds to an amplitude of $$\sqrt{I_0}$$. Let the amplitude of each wave be $$a$$, so $$I_0 = a^2$$.

Now, consider case (i): coherent interference. In coherent interference, the waves have a constant phase difference. The maximum intensity occurs when all waves are in phase, meaning their phase difference is zero. In this case, the amplitudes add up linearly.

The resultant amplitude for $$n$$ waves in phase is:

$$A_{\text{coherent}} = a + a + \cdots + a \quad (n \text{ times}) = n a$$

Since $$a = \sqrt{I_0}$$, we substitute:

$$A_{\text{coherent}} = n \sqrt{I_0}$$

The maximum intensity for coherent interference is proportional to the square of the resultant amplitude:

$$I_{\text{max, coherent}} = \left(A_{\text{coherent}}\right)^2 = \left(n \sqrt{I_0}\right)^2 = n^2 I_0$$

So, $$I_{\text{max, coherent}} = n^2 I_0$$.

Now, consider case (ii): incoherent interference. In incoherent interference, the phase differences between the waves are random and vary rapidly with time. Therefore, the waves do not maintain a constant phase relationship, and the cross terms in the interference average to zero. As a result, the total intensity is simply the sum of the individual intensities.

For $$n$$ waves, each of intensity $$I_0$$, the total intensity is:

$$I_{\text{incoherent}} = I_0 + I_0 + \cdots + I_0 \quad (n \text{ times}) = n I_0$$

Since the intensity is constant and does not vary with position (no interference pattern), this is also the maximum intensity for incoherent interference. So, $$I_{\text{max, incoherent}} = n I_0$$.

The ratio of the maximum intensities for coherent to incoherent interference is:

$$\text{Ratio} = \frac{I_{\text{max, coherent}}}{I_{\text{max, incoherent}}} = \frac{n^2 I_0}{n I_0} = \frac{n^2}{n} = n$$

Hence, the ratio is $$n$$.

Comparing with the options:

A. $$n^2$$

B. $$\frac{1}{n}$$

C. $$\frac{1}{n^2}$$

D. $$n$$

The correct option is D.

Hence, the correct answer is Option D.