Which of the following represents a standing wave ?

Options

A. `y=A sin ( omega t- kx) `
B. `A = Ae^(-bx) sin ( omega t - kx+ alpha ) `
C. `y=A sin kx sin ( omega t= theta ) `
D. `y=( ax + b) sin ( omega t - kx )`

Answer

To determine which of the given equations represents a standing wave, we need to analyze the nature of each equation based on the principles of wave interference.

### Step-by-Step Solution:

1. **Understanding Standing Waves**:
- A standing wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. The resulting wave does not propagate but oscillates in place.

2. **General Form of Standing Wave**:
- The general equation for a standing wave can be expressed as:
\[ y(x, t) = 2A \sin(kx) \cos(\omega t) \]
- Here, \(2A \sin(kx)\) represents the amplitude of the standing wave, which varies with position \(x\), and \(\cos(\omega t)\) represents the time-dependent oscillation.

3. **Analyzing the Given Equations**:
- Let's examine each of the four equations provided in the question:

**Option 1**: \(y = A \sin(\omega t - kx)\)
- This represents a traveling wave, not a standing wave, as the amplitude \(A\) is constant.

**Option 2**: \(y = A e^{-bx} \sin(\omega t - kx + \alpha)\)
- This equation has an exponential term \(e^{-bx}\) which modifies the amplitude, making it non-sinusoidal. Thus, it cannot represent a standing wave.

**Option 3**: \(y = A \sin(kx) \sin(\omega t + \theta)\)
- This equation can be rewritten using the product-to-sum identities, and it resembles the form of a standing wave. Here, \(A \sin(kx)\) acts as the amplitude which varies with position, and \(\sin(\omega t + \theta)\) represents the oscillation in time.

**Option 4**: \(y = Ax + B \sin(\omega t - x)\)
- The term \(Ax + B\) indicates a linear variation in amplitude, which is not sinusoidal. Therefore, this cannot represent a standing wave.

4. **Conclusion**:
- After analyzing all four options, the only equation that fits the criteria for a standing wave is:
\[ y = A \sin(kx) \sin(\omega t + \theta) \]
- Thus, the correct answer is **Option 3**.