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(2pix-2pit )`
B. `y=sin(10 pi x- 20 pit )`
C. `y=sin(2pi x+ 2pi t`
D. `y=sin(x-2t)`

Answer

To solve the problem, we will derive the wave equation step by step using the given parameters: amplitude, wavelength, and frequency.

### Step-by-Step Solution:

1. **Identify the given parameters:**
- Amplitude (A) = 1 m (displacement along y-direction)
- Wavelength (λ) = 2π m
- Frequency (f) = 1/π Hz

2. **Write the standard wave equation:**
The standard equation for a wave traveling in the positive x-direction is:
\[ y = A \sin(kx - \omega t) \]
where:
- \(y\) is the displacement,
- \(A\) is the amplitude,
- \(k\) is the wave number,
- \(x\) is the position,
- \(\omega\) is the angular frequency,
- \(t\) is the time.

3. **Calculate the angular frequency (ω):**
The angular frequency is related to the frequency by the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \left(\frac{1}{\pi}\right) = 2 \text{ rad/s} \]

4. **Calculate the wave number (k):**
The wave number is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the given wavelength:
\[ k = \frac{2\pi}{2\pi} = 1 \text{ rad/m} \]

5. **Substitute the values into the wave equation:**
Now that we have \(A\), \(k\), and \(\omega\), we can substitute these values into the wave equation:
\[ y = 1 \sin(1 \cdot x - 2 \cdot t) \]
This simplifies to:
\[ y = \sin(x - 2t) \]

6. **Final wave equation:**
Therefore, the wave traveling in the positive x-direction can be represented as:
\[ y = \sin(x - 2t) \]

### Final Answer:
The wave is represented by the equation:
\[ y = \sin(x - 2t) \]