Which one of the following represents a wave ?

Options

A. `y= A sin ( omega t- kx)`
B. `y=A cos^(2)(at-bx+c) +A sin^(2) (at-bx+c)`
C. `y=A sin kx `
D. `y=Asin omega t `

Answer

To determine which of the given equations represents a wave, we need to analyze each equation based on the characteristics of wave equations. A wave equation typically depends on both position (x) and time (t). Let's go through the options step by step.

### Step-by-Step Solution:

1. **Understanding Wave Equations**:
- A wave equation generally has the form \( y(x, t) = f(x - vt) \) or \( y(x, t) = f(kt - \omega t) \), where \( f \) is a function that can be sinusoidal (like sine or cosine), \( x \) is the position, \( t \) is time, \( v \) is the wave speed, \( k \) is the wave number, and \( \omega \) is the angular frequency.

2. **Analyzing the First Option**:
- Equation: \( y = a \sin(\omega t - kx) \)
- This equation contains both \( x \) (position) and \( t \) (time). Therefore, it is a valid wave equation.

3. **Analyzing the Second Option**:
- Equation: \( y = a \cos^2(\theta) + a \sin^2(\theta) \)
- Here, if we let \( \theta \) be a function of \( x \) and \( t \), we find that \( \cos^2(\theta) + \sin^2(\theta) = 1 \). Thus, this reduces to \( y = a \), which is a constant and does not depend on \( x \) or \( t \). Therefore, this is not a wave equation.

4. **Analyzing the Third Option**:
- Equation: \( y = a \sin(kx) \)
- This equation depends only on \( x \) and does not include \( t \). Thus, it is not a wave equation.

5. **Analyzing the Fourth Option**:
- Equation: \( y = a \sin(\omega t) \)
- This equation depends only on \( t \) and does not include \( x \). Therefore, it is also not a wave equation.

6. **Conclusion**:
- Among the four options, only the first equation \( y = a \sin(\omega t - kx) \) represents a wave.

### Final Answer:
The equation that represents a wave is \( y = a \sin(\omega t - kx) \).