Two sources of sound placed close to each other are wmitting progressive waves given by `y_1=4sin600pit` and `y_2=5sin608pit`. An observer located near these two sources of sound will hear:

Options

A. 4 beat/s with internsity ratio 25: 16 between waxing and waning
B. 8 beat/s with intensity ratio 25 :16 between waxing and waning
C. 8 beat/s with intensity ratio 81:1 between waxing and waning
D. 4 beat/s with intensity ratio 81: 1 between waxing and waning

Answer

To solve the problem step by step, we will analyze the given wave equations and determine the number of beats and the intensity ratio.

### Step 1: Identify the wave equations
The two wave equations provided are:
- \( y_1 = 4 \sin(600 \pi t) \)
- \( y_2 = 5 \sin(608 \pi t) \)

### Step 2: Determine the frequencies
The general form of a wave equation is given by \( y = a \sin(2 \pi f t) \).

From the equations:
- For \( y_1 \):
- \( 600 \pi = 2 \pi f_1 \)
- Therefore, \( f_1 = \frac{600 \pi}{2 \pi} = 300 \, \text{Hz} \)

- For \( y_2 \):
- \( 608 \pi = 2 \pi f_2 \)
- Therefore, \( f_2 = \frac{608 \pi}{2 \pi} = 304 \, \text{Hz} \)

### Step 3: Calculate the number of beats
The number of beats per second is given by the difference in frequencies:
\[ \text{Number of beats} = |f_2 - f_1| = |304 - 300| = 4 \, \text{beats per second} \]

### Step 4: Calculate the intensity ratio
The intensity ratio of two waves can be calculated using the formula:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \left( \frac{A_1 + A_2}{A_1 - A_2} \right)^2 \]
where \( A_1 \) and \( A_2 \) are the amplitudes of the two waves.

Here, \( A_1 = 4 \) and \( A_2 = 5 \).

Substituting the values:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \left( \frac{4 + 5}{4 - 5} \right)^2 = \left( \frac{9}{-1} \right)^2 = 81 \]

### Step 5: Conclusion
The observer will hear:
- **Number of beats**: 4 beats per second
- **Intensity ratio**: 81:1

Thus, the final answer is:
The observer will hear 4 beats per second with an intensity ratio of 81:1.

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