Which of the following represents a stationary wave ? (a, b, c are constants in the equation)

Options

A. `y = a cos (2bx) sin (ct)`
B. `y = a sin (cx) sin (2bt)`
C. `y = a cos (bx + ct)`
D. `y = a sin (bx - ct)`

Answer

To determine which of the given equations represents a stationary wave, we need to analyze the characteristics of stationary waves compared to traveling waves.

### Step-by-Step Solution:

1. **Understanding Stationary Waves**:
- A stationary wave (or standing wave) is characterized by fixed points (nodes) where the amplitude is always zero, and points (antinodes) where the amplitude varies with time but remains fixed in position. The amplitude of a stationary wave depends only on the position and not on time.

2. **General Equation of a Stationary Wave**:
- The general form of a stationary wave can be expressed as:
\[ y(x, t) = A(x) \sin(kx) \cos(\omega t) \]
- Here, \(A(x)\) is the amplitude that varies with position \(x\), while \(\sin(kx)\) and \(\cos(\omega t)\) describe the spatial and temporal variations, respectively.

3. **Analyzing the Given Options**:
- Let's denote the options as follows:
- Option 1: \(y_1 = a \cos(2bx) \sin(ct)\)
- Option 2: \(y_2 = a \sin(cx) \sin(2bt)\)
- Option 3: \(y_3 = a \sin(bx - ct)\)
- Option 4: \(y_4 = a \sin(bx + ct)\)

4. **Identifying the Stationary Wave**:
- **Option 1**: \(y_1 = a \cos(2bx) \sin(ct)\)
- Here, the amplitude \(A(x) = a \cos(2bx)\) depends only on \(x\) and not on \(t\). Thus, this represents a stationary wave.
- **Option 2**: \(y_2 = a \sin(cx) \sin(2bt)\)
- Similar to option 1, the amplitude \(A(x) = a \sin(cx)\) depends only on \(x\), making this also a stationary wave.
- **Option 3**: \(y_3 = a \sin(bx - ct)\)
- In this case, the amplitude is constant and does not depend solely on \(x\) or \(t\). This represents a traveling wave.
- **Option 4**: \(y_4 = a \sin(bx + ct)\)
- Like option 3, this equation also represents a traveling wave as the amplitude varies with both \(x\) and \(t\).

5. **Conclusion**:
- The equations that represent stationary waves are **Option 1** and **Option 2**. Options 3 and 4 represent traveling waves.

### Final Answer:
- **Option 1** and **Option 2** represent stationary waves.