Two identical speakers emit sound waves of frequency `680 H_(Z)` uniformly in all directions with a total audio output of `1 mW` each. The speed of sound in air is `340 m//s`. A point `P` is a distance `2.00 m` from one speaker and `3.00 m` from the other.
(a) Finf the intensity `I_(1)` and `I_(2)` from rach speaker at point `p` separately.
(b) If the speakers are driven coherently and in phase, what is the intensity at point `p`?
(c ) If they driven coherently but of phase by `180^(@)`, what is the intensity at point `P`?
(d) If the speakers are incoherent, what is the intensity at point `p`?

Explanation

To solve the problem step by step, we will break it down according to the parts of the question.

### Given Data:
- Frequency of sound waves, \( f = 680 \, \text{Hz} \)
- Total audio output from each speaker, \( P = 1 \, \text{mW} = 1 \times 10^{-3} \, \text{W} \)
- Speed of sound in air, \( v = 340 \, \text{m/s} \)
- Distance from Speaker 1 to point P, \( r_1 = 2.00 \, \text{m} \)
- Distance from Speaker 2 to point P, \( r_2 = 3.00 \, \text{m} \)

### (a) Finding the Intensity \( I_1 \) and \( I_2 \)

1. **Calculate the intensity \( I_1 \) from Speaker 1:**
\[ I_1 = \frac{P}{A} = \frac{P}{4 \pi r_1^2} \]
Substituting the values:
\[ I_1 = \frac{1 \times 10^{-3}}{4 \pi (2.00)^2} \]
\[ I_1 = \frac{1 \times 10^{-3}}{4 \pi \times 4} = \frac{1 \times 10^{-3}}{16 \pi} \]
\[ I_1 \approx 19.5 \times 10^{-6} \, \text{W/m}^2 \]

2. **Calculate the intensity \( I_2 \) from Speaker 2:**
\[ I_2 = \frac{P}{A} = \frac{P}{4 \pi r_2^2} \]
Substituting the values:
\[ I_2 = \frac{1 \times 10^{-3}}{4 \pi (3.00)^2} \]
\[ I_2 = \frac{1 \times 10^{-3}}{4 \pi \times 9} = \frac{1 \times 10^{-3}}{36 \pi} \]
\[ I_2 \approx 8.84 \times 10^{-6} \, \text{W/m}^2 \]

### (b) Intensity at Point P when Speakers are Coherent and In Phase

1. **Calculate the resultant intensity \( I_P \):**
\[ I_P = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting the values:
\[ I_P = (\sqrt{19.5 \times 10^{-6}} + \sqrt{8.84 \times 10^{-6}})^2 \]
\[ I_P \approx (4.42 \times 10^{-3} + 2.97 \times 10^{-3})^2 \]
\[ I_P \approx (7.39 \times 10^{-3})^2 \approx 54.6 \times 10^{-6} \, \text{W/m}^2 \]

### (c) Intensity at Point P when Speakers are Coherent but Out of Phase by 180°

1. **Calculate the resultant intensity \( I_P \):**
\[ I_P = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values:
\[ I_P = (\sqrt{19.5 \times 10^{-6}} - \sqrt{8.84 \times 10^{-6}})^2 \]
\[ I_P \approx (4.42 \times 10^{-3} - 2.97 \times 10^{-3})^2 \]
\[ I_P \approx (1.45 \times 10^{-3})^2 \approx 2.1 \times 10^{-6} \, \text{W/m}^2 \]

### (d) Intensity at Point P when Speakers are Incoherent

1. **Calculate the resultant intensity \( I_P \):**
\[ I_P = I_1 + I_2 \]
Substituting the values:
\[ I_P = 19.5 \times 10^{-6} + 8.84 \times 10^{-6} \]
\[ I_P \approx 28.34 \times 10^{-6} \, \text{W/m}^2 \]

### Summary of Results:
- \( I_1 \approx 19.5 \times 10^{-6} \, \text{W/m}^2 \)
- \( I_2 \approx 8.84 \times 10^{-6} \, \text{W/m}^2 \)
- \( I_P \) (coherent and in phase) \( \approx 55.3 \times 10^{-6} \, \text{W/m}^2 \)
- \( I_P \) (coherent and out of phase) \( \approx 2.2 \times 10^{-6} \, \text{W/m}^2 \)
- \( I_P \) (incoherent) \( \approx 28.34 \times 10^{-6} \, \text{W/m}^2 \)