A wave travelling along the x-axis is described by the equation `y(x,t) = 0.005cos (alpha x - beta t)` . If the 1wavelength and the time period of the wave are 0.08m and 2.0s, respectively, then `alpha/beta = n xx 5`, then the value of n is____(where `alpha, beta` are in appropriate units).

Answer

To solve the problem, we need to find the value of \( n \) in the equation \( \frac{\alpha}{\beta} = n \times 5 \), given the wave equation \( y(x,t) = 0.005 \cos(\alpha x - \beta t) \) and the values of wavelength and time period.

### Step-by-Step Solution:

1. **Identify the parameters**:
- Wavelength \( \lambda = 0.08 \, \text{m} \)
- Time period \( T = 2.0 \, \text{s} \)

2. **Calculate the wave vector \( \alpha \)**:
- The wave vector \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
- Substitute the value of \( \lambda \):
\[ k = \frac{2\pi}{0.08} = \frac{2\pi}{0.08} = 25\pi \, \text{m}^{-1} \]
- Therefore, \( \alpha = k = 25\pi \).

3. **Calculate the angular frequency \( \beta \)**:
- The angular frequency \( \beta \) is given by the formula:
\[ \beta = \frac{2\pi}{T} \]
- Substitute the value of \( T \):
\[ \beta = \frac{2\pi}{2} = \pi \, \text{s}^{-1} \]

4. **Calculate the ratio \( \frac{\alpha}{\beta} \)**:
- Now, we can find \( \frac{\alpha}{\beta} \):
\[ \frac{\alpha}{\beta} = \frac{25\pi}{\pi} = 25 \]

5. **Relate \( \frac{\alpha}{\beta} \) to \( n \)**:
- According to the problem, we have:
\[ \frac{\alpha}{\beta} = n \times 5 \]
- Substitute the value we found:
\[ 25 = n \times 5 \]
- Solve for \( n \):
\[ n = \frac{25}{5} = 5 \]

### Final Answer:
The value of \( n \) is \( 5 \).

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