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

In a wave equation, x and t must be related in the form $(\mathrm{x}-\mathrm{v}\mathrm{t})$.

We rewrite the given equations $\mathrm{y}=\frac{1}{1+{(\mathrm{x}-\mathrm{v}\mathrm{t})}^2}$
For t = 0, this becomes $\mathrm{y}=\frac{1}{(1+{\mathrm{x}}^2)}$, as given
For t = 2, this becomes $\mathrm{y}=\frac{1}{[1+{(\mathrm{x}-2\mathrm{v})}^2]}=\frac{1}{[1+{(\mathrm{x}-1)}^2]}$
$\Rightarrow 2\mathrm{v}=1 \mathrm{or} \mathrm{v}=0.5\mathrm{m}/\mathrm{s}$.