Which of the following equations represents a wave

Options

A. `Y=A(omega t-kx)`
B. `Y= A cos kx`
C. `Y=A sin omega t`
D. `Y=A sin (at-bx+c)`

Answer

To determine which of the given equations represents a wave, we need to analyze each option based on the wave equation characteristics. The general form of a wave equation is given by:

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

where \( v \) is the wave speed.

Let's analyze each option step by step:

### Step 1: Analyze Option A
- The equation is given in a form where \( y \) is a linear function of \( t \).
- When we differentiate \( y \) with respect to \( t \) twice, we get:
\[ \frac{\partial^2 y}{\partial t^2} = 0 \]
- Since the second derivative with respect to time is zero, it does not satisfy the wave equation.

**Conclusion**: Option A does not represent a wave.

### Step 2: Analyze Option B
- The equation has a term involving \( \sin(kx) \).
- The first derivative with respect to \( x \) gives:
\[ \frac{\partial^2 y}{\partial x^2} = -k^2 \sin(kx) \]
- However, when we differentiate with respect to \( t \), we find:
\[ \frac{\partial^2 y}{\partial t^2} = 0 \]
- Since one side of the wave equation is zero while the other is not, this option also does not satisfy the wave equation.

**Conclusion**: Option B does not represent a wave.

### Step 3: Analyze Option C
- This option involves both sine and cosine functions, but it is structured such that:
\[ \frac{\partial^2 y}{\partial t^2} = 0 \]
- Similar to the previous options, this does not satisfy the wave equation since one side is zero.

**Conclusion**: Option C does not represent a wave.

### Step 4: Analyze Option D
- The equation is given as:
\[ y = a \cos(at - bx + c) \]
- When we differentiate this with respect to \( t \) twice, we get:
\[ \frac{\partial^2 y}{\partial t^2} = -a^2 \cos(at - bx + c) \]
- Differentiating with respect to \( x \) twice gives:
\[ \frac{\partial^2 y}{\partial x^2} = -b^2 \cos(at - bx + c) \]
- Substituting these into the wave equation:
\[ -a^2 \cos(at - bx + c) = v^2 (-b^2 \cos(at - bx + c)) \]
- This simplifies to a valid wave equation, confirming that this option represents a wave.

**Conclusion**: Option D represents a wave.

### Final Answer
The equation that represents a wave is **Option D**.

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