The displacement function of a wave traveling along positive x-direction is `y=(1)/(2+2x^(2))` at `t=0 and ` by `y=(1)/(2+2(x-2)^(2))` at t=2s, where y and x are in metre. The velocity of the wave is:

Options

A. `2m//s^(-1)`
B. `0.5m//s`
C. `1m//s`
D. `3m//s`

Explanation

To find the velocity of the wave given the displacement functions at two different times, we can follow these steps:

### Step 1: Write down the displacement functions
At \( t = 0 \):
\[ y = \frac{1}{2 + 2x^2} \]
At \( t = 2 \):
\[ y = \frac{1}{2 + 2(x - 2)^2} \]

### Step 2: Identify the form of the wave equation
The general form of a wave traveling along the positive x-direction can be expressed as:
\[ y(x, t) = f(x - vt) \]
where \( v \) is the velocity of the wave.

### Step 3: Rewrite the second displacement function
To compare the two functions, we can rewrite the second displacement function:
\[ y = \frac{1}{2 + 2((x - 2)^2)} = \frac{1}{2 + 2(x^2 - 4x + 4)} = \frac{1}{2 + 2x^2 - 8x + 8} = \frac{1}{2x^2 - 8x + 10} \]

### Step 4: Compare the two equations
We have:
1. At \( t = 0 \):
\[ y = \frac{1}{2 + 2x^2} \]
2. At \( t = 2 \):
\[ y = \frac{1}{2x^2 - 8x + 10} \]

### Step 5: Set the arguments equal to find the relationship
Since both equations represent the same wave at different times, we can set the arguments equal to each other:
\[ 2 + 2x^2 = 2 + 2(x - vt)^2 \]
Substituting \( t = 2 \):
\[ 2 + 2x^2 = 2 + 2(x - 2v)^2 \]

### Step 6: Simplify the equation
This simplifies to:
\[ 2x^2 = 2(x - 2v)^2 \]
Dividing both sides by 2:
\[ x^2 = (x - 2v)^2 \]

### Step 7: Expand and rearrange
Expanding the right side:
\[ x^2 = x^2 - 4vx + 4v^2 \]
Cancelling \( x^2 \) from both sides gives:
\[ 0 = -4vx + 4v^2 \]
Factoring out \( 4v \):
\[ 0 = 4v(v - x) \]

### Step 8: Solve for the velocity
Since \( v \) cannot be zero (as it represents wave velocity), we have:
\[ v = x \]
Given that the wave travels a distance of 2 meters in 2 seconds, we can determine the velocity:
\[ v = \frac{\text{distance}}{\text{time}} = \frac{2 \text{ m}}{2 \text{ s}} = 1 \text{ m/s} \]

### Final Answer
The velocity of the wave is:
\[ \boxed{1 \text{ m/s}} \]