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## Homework Statement

The displacement due to a wave moving in the positive x-direction is given by ##y=\frac{1}{1+x^2}## at time t=0 s and by ##y=\frac{1}{1+(x-1)^2}## at t=2 s, where x and y are in meters. Find the velocity of the wave in m/s.

## Homework Equations

## \frac{∂^2y}{(∂x)^2}=\frac{1}{v^2}.\frac{∂^2y}{(∂t)^2}##

## The Attempt at a Solution

Since we have to deal with partial derivative wrt x on L.H.S., we can simply take the derivative of given x-y relations.

For case, t=0 we have:

##y=\frac{1}{1+x^2}##

or ##y.(1+x^2)=1##

differentiating on both sides wrt x, we get

##(1+x^2).\frac{∂y}{∂x}=-2.x.y##

differentiating again and substituting value of y, we get

## \frac{∂^2y}{(∂x)^2}=\frac{2.(3.x^2-1)}{(1+x^2)^3}##

Similarly for case t=2, we get:

## \frac{∂^2y}{(∂x)^2}=\frac{2.(3.(x-1)^2-1)}{(1+(x-1)^2)^3}##

The only common factor they seem to have is 2 so that must be equal to ##\frac{1}{v^2}##.

##\frac{1}{v^2}=2##

that means ##v= \sqrt{\frac{1}{2}} m/s≈0.71 m/s##

But my book says the answer is ##v=0.5 m/s##, I haven't got a clue what went wrong or how to approach it with some other method.

Any help would be greatly appreciated, thanks.

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