The equation of a progressive wave moving in +ve X-direction is given by -

Options

A. `Y = A sin 2pi ((Vt)/(lambda) - X)`
B. `Y = A sin (2pi)/(lambda) (Vt - (X)/(lambda))`
C. `Y = A sin (2pi)/(lambda) (Vt - X) `
D. `Y = A sin (2pi)/(lambda) ( X - Vt)`

Answer

To solve the question regarding the equation of a progressive wave moving in the positive x-direction, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the General Equation of a Wave**:
The general form of a wave moving in the positive x-direction is given by:
\[ y = A \sin(\omega t - kx) \]
where:
- \(y\) is the displacement,
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency,
- \(k\) is the wave number,
- \(x\) is the position,
- \(t\) is the time.

2. **Identify the Parameters**:
- The amplitude \(A\) represents the maximum displacement from the rest position.
- The angular frequency \(\omega\) relates to the frequency \(f\) of the wave by the equation:
\[ \omega = 2\pi f \]
- The wave number \(k\) is defined as:
\[ k = \frac{2\pi}{\lambda} \]
where \(\lambda\) is the wavelength.

3. **Substituting for Angular Frequency and Wave Number**:
The equation can be rewritten by substituting \(\omega\) and \(k\):
\[ y = A \sin\left(2\pi f t - \frac{2\pi}{\lambda} x\right) \]

4. **Using the Wave Speed Relation**:
The speed \(V\) of the wave is given by the relationship:
\[ V = f \lambda \]
From this, we can express the frequency \(f\) as:
\[ f = \frac{V}{\lambda} \]

5. **Substituting Frequency into the Equation**:
Now substituting \(f\) back into the wave equation:
\[ y = A \sin\left(2\pi \left(\frac{V}{\lambda}\right) t - \frac{2\pi}{\lambda} x\right) \]

6. **Simplifying the Equation**:
This can be simplified further:
\[ y = A \sin\left(\frac{2\pi}{\lambda} (Vt - x)\right) \]
This is the final form of the wave equation for a progressive wave moving in the positive x-direction.

7. **Final Result**:
Thus, the equation of the progressive wave is:
\[ y = A \sin\left(\frac{2\pi}{\lambda} (Vt - x)\right) \]