The amplitude of a wave disturbance propagating along positive X-axis is given by `y=1/(1+x^(2))` at t=0 and `y=1/[1+(x-2)^(2)]` at t=4 s where x and y are in metre. The shape of wave disturbance does not change with time. The velocity of the wave is

Options

A. 1 m/s
B. 0.5 m/s
C. 2 m/s
D. 4 m/s

Answer

To find the velocity of the wave disturbance, we can follow these steps:

### Step 1: Understand the given wave equations
At time \( t = 0 \), the wave disturbance is described by:
\[ y = \frac{1}{1 + x^2} \]
At time \( t = 4 \) seconds, the wave disturbance is given by:
\[ y = \frac{1}{1 + (x - 2)^2} \]

### Step 2: Determine the amplitude at \( t = 0 \)
To find the amplitude at \( t = 0 \), we can substitute \( x = 0 \) into the equation:
\[ y(0) = \frac{1}{1 + 0^2} = \frac{1}{1} = 1 \text{ meter} \]

### Step 3: Determine the position at \( t = 4 \) seconds
Next, we need to find the position \( x \) when the amplitude is still 1 meter at \( t = 4 \) seconds. We set the equation for \( t = 4 \) seconds equal to 1:
\[ 1 = \frac{1}{1 + (x - 2)^2} \]
Cross-multiplying gives:
\[ 1 + (x - 2)^2 = 1 \]
This simplifies to:
\[ (x - 2)^2 = 0 \]
Taking the square root, we find:
\[ x - 2 = 0 \implies x = 2 \text{ meters} \]

### Step 4: Calculate the change in position (\( \Delta x \))
Now, we can calculate the change in position:
\[ \Delta x = x_{\text{final}} - x_{\text{initial}} = 2 \text{ meters} - 0 \text{ meters} = 2 \text{ meters} \]

### Step 5: Calculate the change in time (\( \Delta t \))
The change in time is given as:
\[ \Delta t = 4 \text{ seconds} \]

### Step 6: Calculate the velocity (\( v \))
The velocity of the wave can be calculated using the formula:
\[ v = \frac{\Delta x}{\Delta t} \]
Substituting the values we found:
\[ v = \frac{2 \text{ meters}}{4 \text{ seconds}} = 0.5 \text{ meters per second} \]

### Conclusion
The velocity of the wave is:
\[ \boxed{0.5 \text{ m/s}} \]