At time `t=0`,`y(x)` equation of a wave pulse is
`y=(10)/(2+(x-4)^(2))`
and at `t=2s`, `y(x)` equation of the same wave pulse is
`y=(10)/(2+(x+4)^(2))`
Here, `y` is in mm and `x` in metres. Find the wave velocity.

Explanation

To find the wave velocity based on the given wave pulse equations at two different times, we can follow these steps:

### Step 1: Understand the wave equations
The wave pulse at time \( t = 0 \) is given by:
\[ y(x) = \frac{10}{2 + (x - 4)^2} \]
At time \( t = 2 \) seconds, the wave pulse is given by:
\[ y(x) = \frac{10}{2 + (x + 4)^2} \]

### Step 2: Identify the general wave equation
The general form of a wave pulse can be expressed as:
\[ y(x, t) = \frac{A}{B + (x - vt)^2} \]
where \( A \) is the amplitude, \( B \) is a constant, \( v \) is the wave velocity, and \( t \) is time.

### Step 3: Compare the equations
For \( t = 0 \):
\[ y(x, 0) = \frac{10}{2 + (x - 4)^2} \]
This can be rewritten in the form:
\[ y(x, 0) = \frac{10}{2 + (x - 4)^2} \]
This indicates that the wave is centered at \( x = 4 \).

For \( t = 2 \):
\[ y(x, 2) = \frac{10}{2 + (x + 4)^2} \]
This indicates that the wave is centered at \( x = -4 \).

### Step 4: Determine the shift in position
The wave pulse moves from \( x = 4 \) at \( t = 0 \) to \( x = -4 \) at \( t = 2 \). The distance moved by the wave pulse is:
\[ \Delta x = -4 - 4 = -8 \text{ meters} \]

### Step 5: Calculate the wave velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = \frac{\Delta x}{\Delta t} \]
where \( \Delta t = 2 \) seconds. Thus,
\[ v = \frac{-8 \text{ m}}{2 \text{ s}} = -4 \text{ m/s} \]
Since we are interested in the magnitude of the wave velocity, we take:
\[ v = 4 \text{ m/s} \]

### Final Answer
The wave velocity is \( 4 \text{ m/s} \).

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