A wave travelling in the `+ve` x-direction having displacement along y-direction as 1`m`, wavelength `2pi` m and frequency of `1//pi` Hz is represented by

Options

A. `Y=sin (xx -2t)`
B. `Y=sin (2pixx - 2t)`
C. `Y=sin (10pixx - 20pit)`
D. `Y=sin (2pixx - 2pit)`

Answer

To solve the problem, we need to represent the wave mathematically using the given parameters: amplitude, wavelength, and frequency.

### Step-by-Step Solution:

1. **Identify the wave equation**:
The general form of a wave traveling in the positive x-direction is given by:
\[ y(x, t) = A \sin(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

2. **Substitute the amplitude**:
The problem states that the displacement along the y-direction (amplitude) is \( 1 \, \text{m} \). Therefore, we have:
\[ A = 1 \]
So, the equation becomes:
\[ y(x, t) = 1 \sin(kx - \omega t) = \sin(kx - \omega t) \]

3. **Calculate the wave number \( k \)**:
The wave number \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
where \( \lambda \) is the wavelength. Given that \( \lambda = 2\pi \, \text{m} \), we substitute:
\[ k = \frac{2\pi}{2\pi} = 1 \]

4. **Calculate the angular frequency \( \omega \)**:
The angular frequency \( \omega \) is given by:
\[ \omega = 2\pi f \]
where \( f \) is the frequency. Given that \( f = \frac{1}{\pi} \, \text{Hz} \), we substitute:
\[ \omega = 2\pi \left(\frac{1}{\pi}\right) = 2 \]

5. **Combine the values into the wave equation**:
Now we can substitute \( k \) and \( \omega \) back into the wave equation:
\[ y(x, t) = \sin(1 \cdot x - 2 \cdot t) = \sin(x - 2t) \]

6. **Final answer**:
Thus, the wave can be represented as:
\[ y(x, t) = \sin(x - 2t) \]