Two loudspeakers radiate in phase at `170 H_(Z)`. An observer sits at `8 m` from one speaker and `11 m` from the other . The intensity level from either speaker acting alone is `60 dB`. The speed of sound is `340 m//s`. Find the observer intensity when both speakers are on together.

Explanation

To find the observer's intensity when both speakers are on together, we can follow these steps:

### Step 1: Calculate the wavelength (λ)
The speed of sound (v) is given as 340 m/s, and the frequency (f) is given as 170 Hz. The wavelength can be calculated using the formula:

\[ \lambda = \frac{v}{f} \]

Substituting the values:

\[ \lambda = \frac{340 \, \text{m/s}}{170 \, \text{Hz}} = 2 \, \text{m} \]

### Step 2: Calculate the path difference (Δx)
The observer is 8 m from one speaker and 11 m from the other. The path difference (Δx) is:

\[ \Delta x = |d_2 - d_1| = |11 \, \text{m} - 8 \, \text{m}| = 3 \, \text{m} \]

### Step 3: Calculate the phase difference (Δφ)
The phase difference (Δφ) can be calculated using the formula:

\[ \Delta \phi = \frac{2\pi \Delta x}{\lambda} \]

Substituting the values:

\[ \Delta \phi = \frac{2\pi \cdot 3 \, \text{m}}{2 \, \text{m}} = 3\pi \]

### Step 4: Calculate the intensity from each speaker (I1 and I2)
The intensity level (L) from either speaker acting alone is given as 60 dB. We can convert this to intensity (I) using the formula:

\[ L = 10 \log_{10} \left(\frac{I}{I_0}\right) \]

Where \(I_0 = 10^{-12} \, \text{W/m}^2\) (the reference intensity). Rearranging gives:

\[ I = I_0 \cdot 10^{\frac{L}{10}} = 10^{-12} \cdot 10^{\frac{60}{10}} = 10^{-12} \cdot 10^6 = 10^{-6} \, \text{W/m}^2 \]

### Step 5: Calculate the total intensity when both speakers are on
When both speakers are on, the total intensity (I) can be calculated using the formula:

\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\Delta \phi) \]

Since \(I_1 = I_2 = I\):

\[ I = I + I + 2I \cos(3\pi) = 2I + 2I(-1) = 2I - 2I = 0 \]

### Conclusion
The observer's intensity when both speakers are on together is:

\[ I = 0 \, \text{W/m}^2 \]