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 wave equations given and determine their characteristics. The two waves are represented as:

1. \( y_1 = A \sin(\alpha t - \beta x) \)
2. \( y_2 = A \cos(\beta x + \alpha t - \frac{\pi}{4}) \)

### Step-by-Step Solution:

**Step 1: Identify the wave parameters for \( y_1 \)**
The wave equation \( y_1 = A \sin(\alpha t - \beta x) \) can be compared to the general wave equation \( y = A \sin(kx - \omega t) \).
- Here, \( \omega = \alpha \) (angular frequency)
- \( k = \beta \) (wave number)

**Step 2: Identify the wave parameters for \( y_2 \)**
The wave equation \( y_2 = A \cos(\beta x + \alpha t - \frac{\pi}{4}) \) can be rewritten using the identity \( \cos(x) = \sin(x + \frac{\pi}{2}) \):
\[ y_2 = A \sin\left(\beta x + \alpha t - \frac{\pi}{4} + \frac{\pi}{2}\right) \]
This simplifies to:
\[ y_2 = A \sin\left(\beta x + \alpha t + \frac{\pi}{4}\right) \]
- Here, we can identify \( \omega = \alpha \) and \( k = \beta \) as well.

**Step 3: Compare the wave directions**
- The term \( \alpha t - \beta x \) in \( y_1 \) indicates that the wave is traveling in the positive x-direction.
- The term \( \beta x + \alpha t \) in \( y_2 \) indicates that the wave is traveling in the negative x-direction (since it can be rewritten as \( -\beta x - \alpha t \)).

**Step 4: Determine the speed, wavelength, and frequency**
- The speed of both waves can be calculated using the formula \( v = \frac{\omega}{k} \).
For both waves, since \( \omega = \alpha \) and \( k = \beta \), we find:
\[ v = \frac{\alpha}{\beta} \]
Thus, the speed is the same for both waves.

- The wavelength \( \lambda \) can be calculated using \( k = \frac{2\pi}{\lambda} \).
Since \( k_1 = k_2 = \beta \), we find that the wavelengths are the same.

- The frequency \( f \) can be calculated using \( f = \frac{\omega}{2\pi} \).
Since \( \omega \) is the same for both waves, the frequencies are also the same.

### Conclusion:
The two waves have the same speed, wavelength, and frequency, but they are traveling in opposite directions. Therefore, the only difference between the two waves is their direction of propagation.

### Final Answer:
The two waves have different directions of propagation.

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