Which of the following functions represent a stationary wave ? Here `a , b` and `c` are constants:

Options

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)`

Explanation

To determine which of the given functions represent a stationary wave, we need to analyze the characteristics of stationary waves. A stationary wave is formed by the superposition of two waves traveling in opposite directions, and it is characterized by nodes and antinodes. The particles of the medium oscillate in simple harmonic motion (SHM) at fixed positions, with the amplitude of oscillation varying along the length of the wave.

Let's denote the functions as follows:
1. \( f_1(x, t) = A(x) \sin(ct) \)
2. \( f_2(x, t) = B(x) \cos(ct) \)
3. \( f_3(x, t) = C \sin(kx) \)

### Step 1: Identify the form of the functions
The general form of a stationary wave can be expressed as:
\[ y(x, t) = A(x) \cos(\omega t + \phi) \]
or
\[ y(x, t) = A(x) \sin(\omega t + \phi) \]
where \( A(x) \) is the amplitude that varies with position \( x \).

### Step 2: Analyze each function
1. **Function 1**: \( f_1(x, t) = A(x) \sin(ct) \)
- This function has an amplitude \( A(x) \) that varies with position \( x \) and oscillates with time \( t \) in a sine form. This is characteristic of a stationary wave.

2. **Function 2**: \( f_2(x, t) = B(x) \cos(ct) \)
- Similar to the first function, this function also has a varying amplitude \( B(x) \) and oscillates with time \( t \) in a cosine form. This is also characteristic of a stationary wave.

3. **Function 3**: \( f_3(x, t) = C \sin(kx) \)
- This function does not have a time-dependent component. It only varies with position \( x \) and does not represent oscillation over time. Therefore, it does not represent a stationary wave.

### Conclusion
The functions that represent stationary waves are:
- \( f_1(x, t) = A(x) \sin(ct) \)
- \( f_2(x, t) = B(x) \cos(ct) \)

### Final Answer
The functions that represent stationary waves are \( f_1 \) and \( f_2 \).