A standing wave is formed by the superposition of two waves travelling in the opposite directions. The transverse displacement is given by
`y(x, t) = 0.5 sin ((5pi)/(4) x) cos (200 pi t)`
What is the speed of the travelling wave moving in the postive X direction ?

Options

A. 160 m/s
B. 90 m/s
C. 180 m/s
D. 120 m/s

Explanation

To find the speed of the traveling wave moving in the positive x direction from the given standing wave equation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the standing wave equation**: The given equation is
\[ y(x, t) = 0.5 \sin\left(\frac{5\pi}{4} x\right) \cos(200\pi t) \]
This is in the form of a standing wave, which can be expressed as
\[ y(x, t) = 2a \sin(kx) \cos(\omega t) \]
where \( a \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

2. **Extract the wave number (k) and angular frequency (ω)**: From the equation, we can identify:
- \( k = \frac{5\pi}{4} \)
- \( \omega = 200\pi \)

3. **Use the relationship between speed (v), wave number (k), and angular frequency (ω)**: The relationship is given by:
\[ \omega = v \cdot k \]
Rearranging this gives us:
\[ v = \frac{\omega}{k} \]

4. **Substitute the values of ω and k into the equation**:
\[ v = \frac{200\pi}{\frac{5\pi}{4}} \]

5. **Simplify the expression**:
- First, simplify the fraction:
\[ v = 200\pi \cdot \frac{4}{5\pi} \]
- The \( \pi \) terms cancel out:
\[ v = 200 \cdot \frac{4}{5} \]
- Now calculate:
\[ v = 200 \cdot 0.8 = 160 \text{ m/s} \]

6. **Conclusion**: The speed of the traveling wave moving in the positive x direction is
\[ v = 160 \text{ m/s} \]