Waves traveling in same medium having equations: `y_(1)=Asin(lamdat-betax)andy_(2)=Acos[alphat+betax-(pi//4)]` have different

Options

A. speeds
B. direction
C. wavelength
D. frequencies

Answer

To solve the problem, we need to analyze the two wave equations provided and compare their parameters: speed, direction, wavelength, and frequency.

### Step-by-Step Solution:

1. **Identify the Wave Equations**:
The two wave equations given are:
- \( y_1 = A \sin(\lambda t - \beta x) \)
- \( y_2 = A \cos(\alpha t + \beta x - \frac{\pi}{4}) \)

2. **Convert the Second Wave Equation**:
To compare both equations, we need to express \( y_2 \) in terms of sine. We can use the identity \( \cos(x) = \sin\left(x + \frac{\pi}{2}\right) \):
\[ y_2 = A \cos\left(\alpha t + \beta x - \frac{\pi}{4}\right) = A \sin\left(\alpha t + \beta x - \frac{\pi}{4} + \frac{\pi}{2}\right) \]
This simplifies to:
\[ y_2 = A \sin\left(\alpha t + \beta x + \frac{\pi}{4}\right) \]

3. **Identify Parameters**:
From the equations, we can identify the wave numbers and angular frequencies:
- For \( y_1 \):
- \( k_1 = \beta \)
- \( \omega_1 = \lambda \)
- For \( y_2 \):
- \( k_2 = \beta \)
- \( \omega_2 = \alpha \)

4. **Calculate Speed**:
The speed of a wave is given by the formula \( v = \frac{\omega}{k} \):
- For \( y_1 \):
\[ v_1 = \frac{\lambda}{\beta} \]
- For \( y_2 \):
\[ v_2 = \frac{\alpha}{\beta} \]
Since \( \lambda \) and \( \alpha \) can be different, \( v_1 \) and \( v_2 \) are different.

5. **Determine Wavelength**:
The wave number \( k \) is related to wavelength \( \lambda \) by the formula \( k = \frac{2\pi}{\lambda} \):
- Since \( k_1 = k_2 = \beta \), we conclude:
\[ \lambda_1 = \lambda_2 \]
Thus, the wavelengths are the same.

6. **Calculate Frequency**:
The relationship between speed, frequency, and wavelength is given by \( v = f \lambda \):
- Since the wavelengths are the same, if the speeds are different, the frequencies must also be different:
\[ f_1 \neq f_2 \]

7. **Determine Direction**:
The direction of wave propagation can be inferred from the sign of the terms in the wave equations:
- The first equation \( y_1 \) has a negative sign in front of \( \beta x \), indicating it travels in one direction.
- The second equation \( y_2 \) has a positive sign in front of \( \beta x \), indicating it travels in the opposite direction.
Therefore, the directions are different.

### Summary of Findings:
- **Speed**: Different
- **Wavelength**: Same
- **Frequency**: Different
- **Direction**: Different

### Final Answer:
The parameters that are different are speed, frequency, and direction.