Which of the following function represent a travelling wave? Here a,b and c are constant.

Options

A. `y=a cos (bx) sin (ct)`
B. `y=a sin (bx+ct)`
C. `y=a sin (bx+ct)+a sin (bx-ct)`
D. `y=a sin (bx-ct)`

Explanation

To determine which of the given functions represents a traveling wave, we need to analyze each option based on the standard forms of traveling wave equations. The general forms of traveling waves are:

1. \( y = f(x - vt) \) for waves traveling in the positive x-direction.
2. \( y = f(x + vt) \) for waves traveling in the negative x-direction.

Let's evaluate the given options one by one.

### Step-by-Step Solution:

**Option 1: \( y = a \cos(bx) \cdot \sin(ct) \)**
- This function is a product of a cosine function in terms of \( x \) and a sine function in terms of \( t \). It does not fit the standard form of a traveling wave. Therefore, this option does not represent a traveling wave.

**Option 2: \( y = a \sin(bx + ct) \)**
- This function can be rewritten as \( y = f(x + vt) \) where \( v = \frac{c}{b} \). This indicates that the wave is traveling in the negative x-direction. Thus, this option represents a traveling wave.

**Option 3: \( y = a \sin(bx + ct) + a \sin(bx - ct) \)**
- This function is a superposition of two waves: one traveling in the negative x-direction and the other in the positive x-direction. The superposition of waves does not represent a single traveling wave but rather a standing wave. Therefore, this option does not represent a traveling wave.

**Option 4: \( y = a \sin(bx - ct) \)**
- This function can be rewritten as \( y = f(x - vt) \) where \( v = \frac{c}{b} \). This indicates that the wave is traveling in the positive x-direction. Thus, this option represents a traveling wave.

### Conclusion:
The functions that represent traveling waves are:
- Option 2: \( y = a \sin(bx + ct) \) (traveling in the negative x-direction)
- Option 4: \( y = a \sin(bx - ct) \) (traveling in the positive x-direction)

### Final Answer:
Options 2 and 4 represent traveling waves.

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