A standing wave is formed by the superposition of two waves travelling in 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 position x direction?

Options

A. `160m//s`
B. `90m//s`
C. `180m//s`
D. `120m//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 1: Identify the given equation
The transverse displacement of the standing wave is given by:
\[ y(x,t) = 0.5 \sin\left(\frac{5\pi}{4} x\right) \cos(200\pi t) \]

### Step 2: Compare with the standard form
The standard form of a standing wave is:
\[ y(x,t) = 2a \sin(kx) \cos(\omega t) \]
From this, we can identify:
- \( k = \frac{5\pi}{4} \) (the wave number)
- \( \omega = 200\pi \) (the angular frequency)

### Step 3: Use the relationship between speed, wave number, and angular frequency
The speed \( v \) of the wave can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]

### Step 4: Substitute the values of \( \omega \) and \( k \)
Substituting the values we identified:
\[ v = \frac{200\pi}{\frac{5\pi}{4}} \]

### Step 5: Simplify the expression
To simplify:
1. The \( \pi \) in the numerator and denominator cancels out:
\[ v = \frac{200}{\frac{5}{4}} \]
2. Dividing by a fraction is the same as multiplying by its reciprocal:
\[ v = 200 \times \frac{4}{5} \]
3. Calculate:
\[ v = \frac{800}{5} = 160 \text{ m/s} \]

### Final Answer
The speed of the traveling wave moving in the positive x-direction is:
\[ v = 160 \text{ m/s} \]

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