Which of the following combinations can give standing wave.

Options

A. `y_(1)=A sin^(2)(omegat-kx),y_(2)=-A sin^(2)(omega t+kx)`
B. `y_(1)=A sin(kx-omegat),y_(2)=A cos ( omega t +kx)`
C. `y_(1)=2A cos^(2)(omega t-kx+pi),y_(2)=A[sin 2 ( omega t +kx)-1]`
D. `y_(1)=A sin (kx-omegat + 30^(@)), y_(2)=A cos ( omega t +kx - 60^(@)).`

Answer

To determine which combinations can give standing waves, we need to analyze the general form of the standing wave equation. The standard form of a standing wave can be expressed as:

\[ y(x, t) = 2A \cos(kx) \sin(\omega t) \]

or equivalently,

\[ y(x, t) = 2A \sin(kx) \cos(\omega t) \]

Where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( x \) is the position,
- \( t \) is the time.

### Step-by-step Solution:

1. **Identify the Form of the Equations**:
We need to check if the given combinations can be expressed in the form of the standing wave equation. The key is to see if they can be represented as a product of a function of position (like \( \sin(kx) \) or \( \cos(kx) \)) and a function of time (like \( \sin(\omega t) \) or \( \cos(\omega t) \)).

2. **Examine Each Combination**:
For each of the four combinations provided in the question, we will check if they can be rewritten in the standard form of a standing wave.

- **Combination 1**: Check if it can be expressed as \( A \sin(kx) \cos(\omega t) \) or \( A \cos(kx) \sin(\omega t) \).
- **Combination 2**: Similarly, check if it fits the standing wave form.
- **Combination 3**: Repeat the process for this combination.
- **Combination 4**: Finally, check this combination as well.

3. **Conclusion**:
If all four combinations can be expressed in the form of \( 2A \sin(kx) \cos(\omega t) \) or \( 2A \cos(kx) \sin(\omega t) \), then they can all produce standing waves.

### Final Answer:
All four combinations can give standing waves as they can be expressed in the standard form of standing wave equations.

---