Which of the following expressions is that of a simpleharmonic progressive wave ?

Options

A. `A sin omegat`
B. `A sin omegat cos kx`
C. `A sin (omegat-kx)`
D. `A cos kx`

Answer

To determine which of the given expressions represents a simple harmonic progressive wave, we need to analyze each option based on the characteristics of such waves.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- A simple harmonic progressive wave is characterized by a sinusoidal function that varies with both time and position. The general form of a simple harmonic wave traveling in the positive x-direction can be expressed as:
\[ y(x, t) = A \sin(\omega t - kx) \]
- Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

2. **Analyzing the Options**:
- **Option 1**: \( A \sin(\omega t) \)
- This expression only depends on time and does not include the spatial variable \( x \). Therefore, it represents simple harmonic motion (SHM) but not a progressive wave. **(Incorrect)**

- **Option 2**: \( A \sin(\omega t) \cos(kx) \)
- This expression represents a standing wave, which occurs when two waves of the same frequency and amplitude travel in opposite directions. It is not a progressive wave. **(Incorrect)**

- **Option 3**: \( A \sin(\omega t - kx) \)
- This expression matches the general form of a simple harmonic progressive wave. It indicates a wave traveling in the positive x-direction. **(Correct)**

- **Option 4**: \( A \cos(kx) \)
- This expression only varies with position \( x \) and does not include time \( t \). Thus, it does not represent a progressive wave. **(Incorrect)**

3. **Conclusion**:
- Among the four options analyzed, the only expression that represents a simple harmonic progressive wave is **Option 3: \( A \sin(\omega t - kx) \)**.

### Final Answer:
**Option 3** is the correct answer.

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