Two simple harmonic waves are represented by the equations given as
`y_(1) = 0.3 sin(314 t - 1.57 x)`
`y_(2) = 0.1 sin(314 t - 1.57x + 1.57)`
where `x, y_(1)` and `y_(2)` are in metre and t is in second, then we have

Options

A. `v_(1) = v_(2)` = 50 Hz
B. `lambda_(1) = lambda_(2)` = 4 m
C. Ratio of intensity is 9
D. `y_(2)` leads` y_(1)` by a phase of `(pi)/(2)`

Explanation

To solve the problem, we need to analyze the two simple harmonic wave equations given:

1. \( y_1 = 0.3 \sin(314t - 1.57x) \)
2. \( y_2 = 0.1 \sin(314t - 1.57x + 1.57) \)

### Step 1: Determine the frequency of both waves

The general form of a wave equation is given by:
\[ y = A \sin(\omega t - kx + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

From the equations, we can see that both waves have the same angular frequency \( \omega = 314 \, \text{rad/s} \).

To find the frequency \( f \), we use the relation:
\[ \omega = 2\pi f \]
Thus,
\[ f = \frac{\omega}{2\pi} = \frac{314}{2\pi} \approx 50 \, \text{Hz} \]

### Step 2: Determine the wavelength of both waves

The wave number \( k \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
From the equations, we see that \( k = 1.57 \, \text{m}^{-1} \).

Using the relationship for wavelength:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{1.57} \approx 4 \, \text{m} \]

### Step 3: Calculate the ratio of intensities of the two waves

The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]
Thus, the ratio of intensities \( \frac{I_1}{I_2} \) can be calculated as:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} = \frac{(0.3)^2}{(0.1)^2} = \frac{0.09}{0.01} = 9 \]

### Step 4: Determine the phase difference between the two waves

The phase difference \( \Delta \phi \) between the two waves can be calculated from the phase constants:
- For \( y_1 \): \( \phi_1 = -1.57 \)
- For \( y_2 \): \( \phi_2 = -1.57 + 1.57 = 0 \)

Thus, the phase difference is:
\[ \Delta \phi = \phi_2 - \phi_1 = 0 - (-1.57) = 1.57 \, \text{radians} \]

### Summary of Results
1. Frequency of both waves: \( 50 \, \text{Hz} \)
2. Wavelength of both waves: \( 4 \, \text{m} \)
3. Ratio of intensities: \( 9 \)
4. Phase difference: \( 1.57 \, \text{radians} \)