If two sound waves `y_(1) = 0.3 sin 596 pi (t - (x)/(330)) and y_(2) = 0.5 sin 604 pi (t - (x)/(330))`. The frequency at which beats are produced and the ratio of maximum and minimum intensities of beats are

Options

A. 4 and 16
B. 2 and 4
C. 4 and 8
D. 2 and 16

Explanation

To solve the problem, we need to find the frequency at which beats are produced and the ratio of maximum and minimum intensities of the beats generated by the two sound waves given by:

1. \( y_1 = 0.3 \sin(596 \pi t - \frac{596}{330} \pi x) \)
2. \( y_2 = 0.5 \sin(604 \pi t - \frac{604}{330} \pi x) \)

### Step 1: Identify the frequencies of the sound waves

The general form of a wave is given by:
\[ y = A \sin(\omega t - kx) \]
where:
- \( \omega = 2\pi f \) (angular frequency)
- \( f \) is the frequency
- \( A \) is the amplitude

From the equations of the waves:
- For \( y_1 \), the angular frequency \( \omega_1 = 596 \pi \), thus:
\[ f_1 = \frac{\omega_1}{2\pi} = \frac{596 \pi}{2\pi} = 298 \text{ Hz} \]

- For \( y_2 \), the angular frequency \( \omega_2 = 604 \pi \), thus:
\[ f_2 = \frac{\omega_2}{2\pi} = \frac{604 \pi}{2\pi} = 302 \text{ Hz} \]

### Step 2: Calculate the beat frequency

The beat frequency \( f_b \) is given by the absolute difference between the two frequencies:
\[ f_b = |f_2 - f_1| = |302 - 298| = 4 \text{ Hz} \]

### Step 3: Calculate the maximum and minimum intensities of the beats

The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]

For the two waves:
- Amplitude of \( y_1 \) is \( A_1 = 0.3 \)
- Amplitude of \( y_2 \) is \( A_2 = 0.5 \)

**Maximum Intensity \( I_{max} \)** occurs when the waves interfere constructively:
\[ I_{max} \propto (A_1 + A_2)^2 = (0.3 + 0.5)^2 = (0.8)^2 = 0.64 \]

**Minimum Intensity \( I_{min} \)** occurs when the waves interfere destructively:
\[ I_{min} \propto (A_1 - A_2)^2 = (0.3 - 0.5)^2 = (-0.2)^2 = 0.04 \]

### Step 4: Calculate the ratio of maximum to minimum intensities

The ratio of maximum intensity to minimum intensity is:
\[ \text{Ratio} = \frac{I_{max}}{I_{min}} = \frac{0.64}{0.04} = 16 \]

### Final Answers:
- Frequency of beats produced: \( 4 \text{ Hz} \)
- Ratio of maximum to minimum intensities: \( 16 \)