The shape of a wave propagating in the positive $x$ or negative $x$ - direction is given $y = \frac{1}{\sqrt{1 + x^{2}}}$ at $t = 0$ and $y = \frac{1}{\sqrt{2 - 2 x + x^{2}}}$ at $t = 1 s$ where $x$ and $y$ are in meters. The shape the wave disturbance does not change during propagation. Find the velocity of the wave.

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$1 m / s$ in positive x direction

$1 m / s$ in negative x direction

$\frac{1}{2} m / s$ in positive x direction

$\frac{1}{2} m / s$ in negative x direction

answer is A.

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Detailed Solution

$u = f (x \pm c . t)$ is the general wave equation
At $t = 0, y = f (x) \Rightarrow y = \frac{1}{\sqrt{1 + x^{2}}}$
$y = \frac{1}{\sqrt{2 - 2 x + x^{2}}} = \frac{1}{\sqrt{1 + {(x - 1)}^{2}}} = f (x - 1)$
$\Rightarrow f (x - c t) = f (x - 1) a t t = 1$
$\Rightarrow c = 1 m / s .$