Of the three traveling waves listed below, which one(s) is(are) traveling in the + x direction?
(1) y= +3.2 sin (4.1t +2.3x]
(2) y= -6.8 sin [-3.0t +1.5x]
(3) y=+4.9 sin (12.0t + 18x]

Options

A. 1 only
B. 2 only
C. 3 only
D. 1,2, and 3

Answer

To determine which of the three traveling waves is traveling in the +x direction, we need to analyze the wave equations given. The general form of a traveling wave can be expressed as:

\[ y = A \sin(\omega t \pm kx) \]

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

The sign in front of \( kx \) indicates the direction of wave propagation:
- A positive sign (\(+\)) indicates that the wave is traveling in the -x direction (to the left).
- A negative sign (\(-\)) indicates that the wave is traveling in the +x direction (to the right).

Now, let's analyze each wave equation one by one:

1. **Wave 1: \( y = 3.2 \sin(4.1t + 2.3x) \)**
- Here, we have \( +2.3x \).
- Since the sign before \( kx \) is positive, this wave is traveling in the -x direction.

2. **Wave 2: \( y = -6.8 \sin(-3.0t + 1.5x) \)**
- We can factor out the negative sign:
- \( y = -6.8 \sin(-3.0t + 1.5x) = 6.8 \sin(3.0t - 1.5x) \)
- Now, we see \( -1.5x \) which indicates that this wave is traveling in the +x direction.

3. **Wave 3: \( y = 4.9 \sin(12.0t + 18x) \)**
- Here, we have \( +18x \).
- Since the sign before \( kx \) is positive, this wave is also traveling in the -x direction.

### Summary of Directions:
- Wave 1: Travels in -x direction
- Wave 2: Travels in +x direction
- Wave 3: Travels in -x direction

Thus, the only wave that is traveling in the +x direction is **Wave 2**.

### Final Answer:
Wave 2 is traveling in the +x direction.

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