The displacement due to a wave moving in the positive $x-$direction is given by $y = \frac{1}{{(1 + {x^2})}}$ at time $t = 0$ and by $y = \frac{1}{{[1 + {{(x - 1)}^2}]}}$ at $t = 2$ seconds, where $x$ and $y$ are in metres. The velocity of the wave in $m/s$ is
Diffcult
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(a) In a wave equation, $x$ and $t$ must be related in the form $(x - v\,t)$.
We rewrite the given equations $y = \frac{1}{{1 + {{(x - v\,t)}^2}}}$
For $t = 0,$ this becomes $y = \frac{1}{{(1 + {x^2})}}$, as given
For $t = 2,$ this becomes $y = \frac{1}{{[1 + {{(x - 2v)}^2}]}} = \frac{1}{{[1 + {{(x - 1)}^2}]}}$
==> $2v = 1$ or $v = 0.5m/s$.
We rewrite the given equations $y = \frac{1}{{1 + {{(x - v\,t)}^2}}}$
For $t = 0,$ this becomes $y = \frac{1}{{(1 + {x^2})}}$, as given
For $t = 2,$ this becomes $y = \frac{1}{{[1 + {{(x - 2v)}^2}]}} = \frac{1}{{[1 + {{(x - 1)}^2}]}}$
==> $2v = 1$ or $v = 0.5m/s$.
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