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A wave pulse is travelling along +x dire...

A wave pulse is travelling along +x direction on a string at `2 m//s`. Displacement y (in cm ) of the parrticle at x=0 at any time t is given by `2//(t^(2)+1)`. Find
(i) expression of the function y=(x,t),i.e., displacement of a particle at position x and time t.
Draw the shape of the pulse at t=0 and t=1 s.

Text Solution

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`(i)` By replacing `t` by `(1 - (x)/(v))`, we can get the desired wave funcation i.e.
`y = (2)/(t - (x)/(2))^(2) + 1`
`(ii)` We can use wave funcation at a particular instant, say `t = 0`, to find shape of the wave pulse using different values of `x`.
at `t = 0 , y = (2)/((x^(2))/(4) + 1)`
at `x = 0 , y = 2`
`x = 2 , y = 1`
`x = -2 , y = 1`
`x = 4 , y = 0.4`
`x = -4 , y = 0.4`

Using these value, shape is drawn.
Similarly for `1 = 1s`, shape can be drawn. What do you conclude about direction of motion of the wave from the graphs ? Also check how much the pulse has moved in `1s` time interval. This is equal to wave speed. Here is the procedure :
`y = (2)/((1 - (x)/(2))^(2) + 1)`
at `t = 1s`
at `x = 2 , y = 2` (maximum value)
at `x = 0 , y = 1`
at `x = 4 , y = 1`

The pulse has moved ot the right by `2` units in `1s` interval.
Also as `t - (x)/(2) =` constt.
Differentiating `w.r.t.` time
`1 - (1)/(2), (dx)/(dt) = 0 rArr (dx)/(dt) = 2`.
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