Question

The disturbance $y (x, t)$ of a wave propagating in the positive x-direction is given by $y = \frac{1}{1+x^{2}}$ at time $t = 0$ and by $y = \frac{1}{\left[1+\left(x-1^{2}\right)\right]}$ at $t = 2\,s,$ where $x$ and $y$ are in meters. The shape of the wave disturbance does not change during the propagation. The velocity of wave in $m/s$ is

• A. 2.0
• B. 4.0
• C. 0.5
• D. 1.0

Answer 1

Answer: (C)

The equation of wave at any time is obtained by putting $X = x - vt$
$y = \frac{1}{1+x^{2}} \frac{1}{1+\left(x-vt\right)^{2}} \quad\quad\ldots\left(i\right)$
We know at $t = 2 \,sec$,
$y = \frac{1}{1+\left(x-1\right)^{2}}\quad \quad \ldots \left(ii\right)$
On comparing $\left(i\right)$ and $\left(ii\right)$ we get
$vt = 1$
$V = \frac{1}{t}$
As $t = 2 \,sec$
$\therefore V = \frac{1}{2} = 0.5 \,m/s.$