Two waves of intensity ration `1 : 9` cross eachother at a point. Calculate the ratio of resultant intensities at a point, when (a) waves are incoherent (b) waves are coherent and differ in phase by `60^(@)`.

Explanation

To solve the problem, we will break it down into two parts: (a) when the waves are incoherent and (b) when the waves are coherent and differ in phase by \(60^\circ\).

### Given:
- Intensity ratio of two waves: \(I_1 : I_2 = 1 : 9\)

Let:
- \(I_1 = I\)
- \(I_2 = 9I\)

### Step 1: Calculate the Amplitudes
The intensity of a wave is proportional to the square of its amplitude. Therefore, we can write:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Substituting the values, we have:
\[ \frac{1}{9} = \frac{A_1^2}{A_2^2} \]
Taking the square root:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]
Let \(A_1 = A\) and \(A_2 = 3A\).

### Part (a): Incoherent Waves
For incoherent waves, the resultant intensity \(I_R\) is simply the sum of the individual intensities:
\[ I_R = I_1 + I_2 = I + 9I = 10I \]

### Step 2: Resultant Intensity Ratio for Incoherent Waves
The ratio of the resultant intensity to the intensity of one of the waves can be expressed as:
\[ \text{Ratio} = \frac{I_R}{I_1} = \frac{10I}{I} = 10 \]

### Part (b): Coherent Waves with Phase Difference of \(60^\circ\)
For coherent waves, the resultant intensity \(I_R\) is given by:
\[ I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
where \(\phi\) is the phase difference. Substituting the values:
\[ I_R = I + 9I + 2\sqrt{I \cdot 9I} \cos(60^\circ) \]
Calculating \(\cos(60^\circ) = \frac{1}{2}\):
\[ I_R = 10I + 2\sqrt{9I^2} \cdot \frac{1}{2} \]
\[ = 10I + 3I = 13I \]

### Step 3: Resultant Intensity Ratio for Coherent Waves
The ratio of the resultant intensity to the intensity of one of the waves can be expressed as:
\[ \text{Ratio} = \frac{I_R}{I_1} = \frac{13I}{I} = 13 \]

### Final Answers:
(a) The ratio of resultant intensities for incoherent waves is \(10:1\).

(b) The ratio of resultant intensities for coherent waves with a phase difference of \(60^\circ\) is \(13:1\).