Two waves travelling in a medium in the x-direction are represented by `y_(1) = A sin (alpha t - beta x)` and `y_(2) = A cos (beta x + alpha t - (pi)/(4))`, where `y_(1)` and `y_(2)` are the displacements of the particles of the medium `t` is time and `alpha` and `beta` constants. The two have different :-

Options

A. speeds
B. directions of propagation
C. wavelengths
D. frequencies

Explanation

To solve the problem, we need to analyze the two given wave equations and determine their characteristics.

### Step-by-Step Solution:

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

2. **Convert the Cosine Wave to Sine Form**:
To compare both waves, we can rewrite \(y_2\) in sine form. We know that:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Therefore, we can express \(y_2\) as:
\[ y_2 = A \sin\left(\beta x + \alpha t - \frac{\pi}{4} + \frac{\pi}{2}\right) = A \sin\left(\beta x + \alpha t + \frac{\pi}{4}\right) \]

3. **Determine the Wave Numbers and Angular Frequencies**:
From the equations, we can identify:
- For \(y_1\):
- Angular frequency \(\omega_1 = \alpha\)
- Wave number \(k_1 = \beta\)
- For \(y_2\):
- Angular frequency \(\omega_2 = \alpha\)
- Wave number \(k_2 = \beta\)

4. **Calculate the Speed of the Waves**:
The speed \(v\) of a wave is given by:
\[ v = \frac{\omega}{k} \]
Since both waves have the same \(\alpha\) and \(\beta\):
\[ v_1 = \frac{\alpha}{\beta}, \quad v_2 = \frac{\alpha}{\beta} \]
Thus, the speeds of both waves are the same.

5. **Determine the Wavelengths**:
The wavelength \(\lambda\) is related to the wave number by:
\[ k = \frac{2\pi}{\lambda} \]
Since \(k_1 = k_2 = \beta\), it follows that:
\[ \lambda_1 = \lambda_2 \]
Therefore, the wavelengths of both waves are the same.

6. **Determine the Frequencies**:
The frequency \(f\) is given by:
\[ f = \frac{\omega}{2\pi} \]
Since \(\omega_1 = \omega_2 = \alpha\), we conclude:
\[ f_1 = f_2 \]
Hence, the frequencies of both waves are the same.

7. **Analyze the Direction of Propagation**:
The wave \(y_1\) travels in the positive x-direction (since it has a negative x term), while \(y_2\) travels in the negative x-direction (since it has a positive x term). Thus, the two waves are traveling in opposite directions.

### Conclusion:
The two waves have the same speed, the same wavelength, and the same frequency, but they are traveling in opposite directions.