Which of the following represents a standing wave?

Options

A. `y=(6.0 mm) sin [(3.0 m^(-1)) x+(2.0 s^(-1)t]-(6.0 mm) cos [(3.0 m^(-1)) x+2.0]`
B. `y=(6.0 mm) cos[(3.0 m^(-1)) x-(2.0 s^(-1)t]+(6.0 mm) cos [(2.0 s^(-1))t -3.0]`
C. `y=(6.0 mm) cos[(3.0 m^(-1)) x-(2.0 s^(-1)t]-(6.0 mm) cos [(2.0 s^(-1)) t+3.0m^(-1)x]`
D. `y=(6.0 mm) sin [(3.0 m^(-1)) x-(2.0 s^(-1)t]-(6.0 mm) cos [(2.0 m^(-1)) x+2.0]`

Answer

To determine which of the given equations represents a standing wave, we can follow these steps:

### Step 1: Understand the Definition of a Standing Wave
A standing wave is a wave that oscillates in time but whose peak amplitude profile does not move in space. This means that the wave has fixed points (nodes) where there is no movement, and points (antinodes) where the amplitude is maximum.

### Step 2: Identify the General Equation of a Standing Wave
The general mathematical expression for a standing wave can be represented as:
\[ y(x, t) = A \cos(kx) \cos(\omega t) \]
or equivalently,
\[ y(x, t) = A \sin(kx) \sin(\omega t) \]
where:
- \( A \) is the maximum amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( x \) is the position, and
- \( t \) is time.

### Step 3: Analyze the Given Equations
Now, we need to analyze the provided equations to see which one matches the form of a standing wave.

1. **First Equation:**
- If it contains both sine and cosine terms that are not in the correct form, it cannot represent a standing wave.

2. **Second Equation:**
- If it has both terms as cosine or both as sine, it is likely to represent a standing wave.

3. **Third and Fourth Equations:**
- Similar analysis as above; we check if they contain the correct combination of sine and cosine.

### Step 4: Match the Equations
- For the first option, if it has mixed sine and cosine terms, it does not represent a standing wave.
- For the second option, if it has the correct form of cosine terms, it represents a standing wave.
- The third and fourth options should be checked similarly.

### Conclusion
After analyzing the equations, we find that **option 2** correctly represents a standing wave because it maintains the necessary form of either both cosine or both sine functions.

### Final Answer
**Option 2 represents a standing wave.**

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