The equation of a wave pulse moving with a speed 1 m/sec at time t = 0 is given as `y=f(x)=1/(1+x^(2))`. Its equation at time t = 1 second can be given as

Options

A. `y=1/([1+(1+x)]^(2))`
B. `y=1/(1+(1-x)^(2))`
C. `y=1/(1+(x^(2)+1))`
D. `y=1/(1+(1/(1+x^(2))))`

Answer

To find the equation of the wave pulse at time \( t = 1 \) second, we start with the initial wave equation given at \( t = 0 \):

\[ y = f(x) = \frac{1}{1 + x^2} \]

The wave is moving at a speed of \( v = 1 \) m/s. We need to determine how far the wave has traveled after \( t = 1 \) second.

### Step 1: Calculate the distance traveled by the wave
The distance traveled by the wave in time \( t \) is given by the formula:

\[ \text{Distance} = v \times t \]

Substituting the values:

\[ \text{Distance} = 1 \, \text{m/s} \times 1 \, \text{s} = 1 \, \text{m} \]

### Step 2: Modify the wave equation for the new position
Since the wave has moved 1 meter to the right, we need to adjust the original equation by replacing \( x \) with \( x - 1 \) (since it moves in the positive x-direction).

The new equation at \( t = 1 \) second becomes:

\[ y = f(x - 1) = \frac{1}{1 + (x - 1)^2} \]

### Step 3: Simplify the equation
Now, we can simplify the equation:

\[ y = \frac{1}{1 + (x - 1)^2} \]

Expanding \( (x - 1)^2 \):

\[ (x - 1)^2 = x^2 - 2x + 1 \]

Thus, the equation becomes:

\[ y = \frac{1}{1 + x^2 - 2x + 1} = \frac{1}{2 + x^2 - 2x} \]

### Final Equation
So the equation of the wave at \( t = 1 \) second is:

\[ y = \frac{1}{2 + x^2 - 2x} \]