A wave travelling along the x-axis is described by the equation `v(x, t) = 0.005 cos(alpha x - betat)`. If the wavelength and the time period of the wave are `0.08m` and `2.0s`, respectively, then `alpha and beta` in appropriate units are

Options

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

Explanation

To solve the problem, we need to find the values of \(\alpha\) and \(\beta\) from the wave equation given as:

\[ v(x, t) = 0.005 \cos(\alpha x - \beta t) \]

We are provided with the wavelength (\(\lambda\)) and the time period (\(T\)) of the wave:

- Wavelength, \(\lambda = 0.08 \, \text{m}\)
- Time period, \(T = 2.0 \, \text{s}\)

### Step 1: Identify the relationship between \(\alpha\) and the wave number \(k\)

From the standard wave equation, we know that:

\[ v(x, t) = A \cos(kx - \omega t) \]

Here, \(k\) is the wave number, and it is related to the wavelength by the formula:

\[ k = \frac{2\pi}{\lambda} \]

Since we have \(\alpha = k\), we can substitute the value of \(\lambda\) to find \(\alpha\).

### Step 2: Calculate \(\alpha\)

Substituting the given wavelength into the equation for \(k\):

\[ \alpha = \frac{2\pi}{\lambda} = \frac{2\pi}{0.08} \]

Calculating this gives:

\[ \alpha = \frac{2\pi}{0.08} = \frac{2\pi}{\frac{8}{100}} = \frac{2\pi \cdot 100}{8} = 25\pi \, \text{m}^{-1} \]

### Step 3: Identify the relationship between \(\beta\) and the angular frequency \(\omega\)

The angular frequency \(\omega\) is related to the time period \(T\) by the formula:

\[ \omega = \frac{2\pi}{T} \]

Since \(\beta = \omega\), we can substitute the value of \(T\) to find \(\beta\).

### Step 4: Calculate \(\beta\)

Substituting the given time period into the equation for \(\omega\):

\[ \beta = \frac{2\pi}{T} = \frac{2\pi}{2.0} \]

Calculating this gives:

\[ \beta = \frac{2\pi}{2} = \pi \, \text{s}^{-1} \]

### Final Answers

Thus, we have:

\[ \alpha = 25\pi \, \text{m}^{-1} \quad \text{and} \quad \beta = \pi \, \text{s}^{-1} \]

### Summary

The values of \(\alpha\) and \(\beta\) are:

- \(\alpha = 25\pi \, \text{m}^{-1}\)
- \(\beta = \pi \, \text{s}^{-1}\)