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 options represents a wave, we can use the wave equation, which is given by:

\[ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \]

where \(y\) is the displacement, \(t\) is time, \(x\) is position, and \(c\) is the speed of the wave.

Let's analyze each option step by step:

### Step 1: Analyze Option A
Assume the form of the wave is given by:
\[ y = A \sin(\omega t - kx) \]

1. **Calculate \(\frac{\partial^2 y}{\partial t^2}\)**:
- First derivative:
\[ \frac{\partial y}{\partial t} = A \omega \cos(\omega t - kx) \]
- Second derivative:
\[ \frac{\partial^2 y}{\partial t^2} = -A \omega^2 \sin(\omega t - kx) \]

2. **Calculate \(\frac{\partial^2 y}{\partial x^2}\)**:
- First derivative:
\[ \frac{\partial y}{\partial x} = -A k \cos(\omega t - kx) \]
- Second derivative:
\[ \frac{\partial^2 y}{\partial x^2} = -A k^2 \sin(\omega t - kx) \]

3. **Substitute into the wave equation**:
- LHS:
\[ -A \omega^2 \sin(\omega t - kx) \]
- RHS:
\[ c^2 (-A k^2 \sin(\omega t - kx)) \]
- Setting LHS = RHS gives:
\[ A \omega^2 = c^2 A k^2 \implies \frac{\omega^2}{k^2} = c^2 \implies c = \frac{\omega}{k} \]
- This satisfies the wave equation.

### Step 2: Analyze Option B
Assume:
\[ y = A \cos^2(\theta) + A \sin^2(\theta) \]
Using the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\):
\[ y = A \]
- This is a constant and does not vary with time or position, hence it does not represent a wave.

### Step 3: Analyze Option C
Assume:
\[ y = A \]
- This is also a constant value and does not depend on time or position, hence it does not represent a wave.

### Step 4: Analyze Option D
Assume:
\[ y = 0 \]
- This is a trivial case where there is no displacement at all, hence it does not represent a wave.

### Conclusion
After analyzing all options, we find that only **Option A** satisfies the wave equation and represents a wave.

### Final Answer
The correct option that represents a wave is **Option A**.

---