The equation `y=a sin 2 pi//lamda (vt -x)` is expression for :-

Options

A. Stationary wave of single frequency along x-axis
B. A simple harmonic motion
C. A progressive wave of single frequency along x-axis
D. The resultant of two SHM's of slightly different frequencies

Explanation

To solve the question, we need to analyze the given equation:

**Given Equation:**
\[ y = a \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]

### Step 1: Identify the Components of the Equation
The equation can be rewritten in a more recognizable form:
\[ y = a \sin(kx - \omega t) \]
where:
- \( k = \frac{2\pi}{\lambda} \) (wave number)
- \( \omega = \frac{2\pi}{T} \) (angular frequency)
- \( v = \frac{\lambda}{T} \) (wave speed)

### Step 2: Determine the Type of Wave
The equation \( y = a \sin(kx - \omega t) \) represents a **progressive wave** traveling in the positive x-direction. This is because the term \( (vt - x) \) indicates that the wave is moving with speed \( v \).

### Step 3: Analyze the Options
Now, let’s analyze the options provided in the question:

1. **Stationary Wave**: A stationary wave is represented by a different form, typically involving both sine and cosine terms, such as \( y = A \cos(kx) \sin(\omega t) \). Therefore, this option is incorrect.

2. **Simple Harmonic Motion (SHM)**: SHM is represented by equations like \( y = A \sin(\omega t) \) or \( y = A \cos(\omega t) \). The presence of the \( x \) term in the given equation indicates that it is not purely SHM. Thus, this option is also incorrect.

3. **Progressive Wave**: As established in Step 2, the equation represents a progressive wave traveling along the x-axis. This option is correct.

4. **Resultant of Two SHMs**: The resultant of two SHMs with slightly different frequencies does not include a term involving \( x \) in the same way as the given equation. Therefore, this option is incorrect.

### Conclusion
The correct answer is that the equation \( y = a \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \) represents a **progressive wave of single frequency along the x-axis**.

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