A simple harmonic wave has the equation `y_1 = 0.3 sin (314 t - 1.57 x)` and another wave has equation `y_2= 0.1 sin (314t - 1.57x+1.57)` where `x,y_1` and `y_2` are in metre and t is in second.

Options

A. `v_1 = v_2 = 50Hz`
B. `lamda_1 = lamda_2 = 4m`
C. Ratio of intensity is 9
D. `y_2` leads `y_1` by a phase angle of `pi/2`

Answer

To solve the problem, we will analyze the two given wave equations step by step.

### Given Wave Equations:
1. \( y_1 = 0.3 \sin(314t - 1.57x) \)
2. \( y_2 = 0.1 \sin(314t - 1.57x + 1.57) \)

### Step 1: Identify the Parameters of the Waves
From the equations, we can identify the following parameters:
- **Angular Frequency (\( \omega \))**: Both waves have \( \omega = 314 \) rad/s.
- **Wave Number (\( k \))**: Both waves have \( k = 1.57 \) rad/m.
- **Amplitude**:
- For wave 1: \( A_1 = 0.3 \) m
- For wave 2: \( A_2 = 0.1 \) m

### Step 2: Calculate the Phase Difference
The phase of the first wave is \( \phi_1 = 314t - 1.57x \) and the phase of the second wave is \( \phi_2 = 314t - 1.57x + 1.57 \).

The phase difference (\( \Delta \phi \)) is given by:
\[ \Delta \phi = \phi_2 - \phi_1 = (314t - 1.57x + 1.57) - (314t - 1.57x) = 1.57 \text{ radians} \]

### Step 3: Convert Phase Difference to Degrees
To express the phase difference in degrees:
\[ \Delta \phi = 1.57 \text{ radians} = \frac{1.57}{\pi} \times 180 \approx 90^\circ \]
This means that wave \( y_2 \) leads wave \( y_1 \) by \( 90^\circ \) or \( \frac{\pi}{2} \) radians.

### Step 4: Calculate the Ratio of Intensities
The intensity of a wave is proportional to the square of its amplitude. Therefore, the ratio of intensities \( \frac{I_1}{I_2} \) is given by:

\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} = \frac{(0.3)^2}{(0.1)^2} = \frac{0.09}{0.01} = 9 \]

### Step 5: Calculate the Frequency
The frequency \( f \) can be calculated from the angular frequency \( \omega \) using the formula:
\[ f = \frac{\omega}{2\pi} \]
Substituting \( \omega = 314 \):
\[ f = \frac{314}{2\pi} \approx 50 \text{ Hz} \]

### Step 6: Calculate the Wavelength
The wavelength \( \lambda \) can be calculated using the relationship \( v = f \lambda \), where \( v \) is the wave speed. The wave speed \( v \) can also be expressed as:
\[ v = \frac{\omega}{k} \]
Substituting \( \omega = 314 \) and \( k = 1.57 \):
\[ v = \frac{314}{1.57} \approx 200 \text{ m/s} \]
Now, using \( v = f \lambda \):
\[ \lambda = \frac{v}{f} = \frac{200}{50} = 4 \text{ m} \]

### Summary of Results:
- Phase difference: \( \frac{\pi}{2} \) radians (or \( 90^\circ \))
- Ratio of intensities: \( 9 \)
- Frequency: \( 50 \text{ Hz} \)
- Wavelength: \( 4 \text{ m} \)