A plane progressive wave is represented by the equation `y= 0.25 cos (2 pi t - 2 pi x)` The equation of a wave is with double the amplitude and half frequency but travelling in the opposite direction will be :-

Options

A. `y = 0.5 cos (pi t - pi x)`
B. `y = 0.5 cos (2 pi t + 2 pi x)`
C. `y = 0.25 cos (pi t + 2 pi x)`
D. `y = 0.5 cos (pi t +pi x)`

Explanation

To solve the problem, we need to derive the equation of a wave that has double the amplitude, half the frequency, and travels in the opposite direction compared to the given wave equation \( y = 0.25 \cos(2 \pi t - 2 \pi x) \).

### Step-by-Step Solution:

1. **Identify the given parameters from the wave equation:**
The given wave equation is:
\[ y = 0.25 \cos(2 \pi t - 2 \pi x) \]
From this equation, we can identify:
- Amplitude \( a = 0.25 \)
- Angular frequency \( \omega = 2 \pi \)
- Wave number \( k = 2 \pi \)

2. **Calculate the initial frequency:**
The angular frequency \( \omega \) is related to frequency \( f \) by the equation:
\[ \omega = 2 \pi f \]
Substituting \( \omega = 2 \pi \):
\[ 2 \pi = 2 \pi f \implies f = 1 \text{ Hz} \]

3. **Determine the new frequency:**
The problem states that the new frequency is half of the original frequency:
\[ f' = \frac{1}{2} \text{ Hz} \]

4. **Calculate the new angular frequency:**
Using the relationship \( \omega' = 2 \pi f' \):
\[ \omega' = 2 \pi \left( \frac{1}{2} \right) = \pi \]

5. **Determine the new amplitude:**
The new amplitude is double the original amplitude:
\[ a' = 2 \times 0.25 = 0.5 \]

6. **Calculate the new wavelength:**
The wave number \( k \) is related to the wavelength \( \lambda \) by:
\[ k = \frac{2 \pi}{\lambda} \]
For the original wave:
\[ 2 \pi = \frac{2 \pi}{\lambda} \implies \lambda = 1 \]
Since the frequency is halved, the wavelength will double:
\[ \lambda' = 2 \times 1 = 2 \]

7. **Calculate the new wave number:**
The new wave number \( k' \) is:
\[ k' = \frac{2 \pi}{\lambda'} = \frac{2 \pi}{2} = \pi \]

8. **Write the equation for the wave traveling in the opposite direction:**
A wave traveling in the opposite direction has the form:
\[ y = a' \cos(\omega' t + k' x) \]
Substituting the values we found:
\[ y = 0.5 \cos(\pi t + \pi x) \]

### Final Answer:
The equation of the wave with double the amplitude, half the frequency, and traveling in the opposite direction is:
\[ y = 0.5 \cos(\pi t + \pi x) \]