Which of the following functions represent stationary wave. Where a,b,c are constants
(a) `y= a cos (bx) sin (ct) ` (b) `y= a sin (bx) cos (ct)`
(c) `y=a sin (bx+ct)`
(d) `y= a sin (bx+ct)+a sin (bx-ct)`

Options

A. a & b
B. a,b & c
C. b,c & d
D. a,b, & d

Answer

To determine which of the given functions represents a stationary wave, we need to analyze each option based on the characteristics of stationary waves. A stationary wave is formed by the interference of two waves traveling in opposite directions.

### Step-by-Step Solution:

1. **Understanding Stationary Waves**:
- A stationary wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. The general form of a stationary wave can be expressed as the product of a function of position and a function of time.

2. **Analyzing Option (a)**:
- The function is given as \( y = a \cos(bx) \sin(ct) \).
- Here, \( \cos(bx) \) is a function of position and \( \sin(ct) \) is a function of time.
- This can be expressed as a product of two separate functions: one depending on \( x \) and the other on \( t \).
- **Conclusion**: This represents a stationary wave.

3. **Analyzing Option (b)**:
- The function is given as \( y = a \sin(bx) \cos(ct) \).
- Similar to option (a), \( \sin(bx) \) is a function of position and \( \cos(ct) \) is a function of time.
- This also can be expressed as a product of two separate functions.
- **Conclusion**: This represents a stationary wave.

4. **Analyzing Option (c)**:
- The function is given as \( y = a \sin(bx + ct) \).
- This function cannot be separated into a product of a function of position and a function of time. It is a single wave function that represents a traveling wave, not a stationary wave.
- **Conclusion**: This does not represent a stationary wave.

5. **Analyzing Option (d)**:
- The function is given as \( y = a \sin(bx + ct) + a \sin(bx - ct) \).
- This can be simplified using the principle of superposition. The two sine functions represent waves traveling in opposite directions.
- Using the trigonometric identity, we find that this can be expressed as:
\[ y = 2a \sin(bx) \cos(ct) \]
- This shows that it can be expressed as a product of a function of position and a function of time.
- **Conclusion**: This represents a stationary wave.

### Final Answer:
The functions that represent stationary waves are (a), (b), and (d).