A wave travelling along the x-axis is described by the equation y (x, t) = 0.005 sin (`alphax - betat`). If the wavelength and time period of the wave are 0.08 m and 2 s respectively, then `alpha, beta` in appropriate units are

Options

A. `alpha = 25.00 pi, beta = pi`
B. `alpha = (0.08)/(pi), (2.0)/(pi)`
C. `alpha = (0.04)/(pi), beta = (1.0)/(pi)`
D. `alpha = 12.50 pi, beta = (pi)/(2.0)`

Answer

To find the values of \( \alpha \) and \( \beta \) in the wave equation \( y(x, t) = 0.005 \sin(\alpha x - \beta t) \), we will use the relationships between the wave parameters and the wave equation.

### Step-by-Step Solution:

1. **Identify the wave parameters**:
The wave is described by the equation \( y(x, t) = 0.005 \sin(\alpha x - \beta t) \). Here, \( \alpha \) corresponds to the wave number \( k \) and \( \beta \) corresponds to the angular frequency \( \omega \).

2. **Relate wave number \( k \) to wavelength \( \lambda \)**:
The wave number \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Given that the wavelength \( \lambda = 0.08 \, \text{m} \), we can substitute this value into the equation:
\[ k = \frac{2\pi}{0.08} \]

3. **Calculate \( k \)**:
\[ k = \frac{2\pi}{0.08} = \frac{2\pi}{\frac{8}{100}} = \frac{2\pi \times 100}{8} = \frac{200\pi}{8} = 25\pi \]
Thus, we find:
\[ \alpha = 25\pi \]

4. **Relate angular frequency \( \omega \) to time period \( T \)**:
The angular frequency \( \omega \) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Given that the time period \( T = 2 \, \text{s} \), we can substitute this value into the equation:
\[ \omega = \frac{2\pi}{2} = \pi \]
Thus, we find:
\[ \beta = \pi \]

5. **Final values**:
Therefore, the values of \( \alpha \) and \( \beta \) are:
\[ \alpha = 25\pi, \quad \beta = \pi \]

### Summary of the Solution:
The values of \( \alpha \) and \( \beta \) are:
\[ \alpha = 25\pi, \quad \beta = \pi \]