A wave pulse is propagating on a long string along the length taken as +x axis. Shape of the string

at $$t=0$$ is given by $$y=\frac{1}{x^2+8x+19}$$ and at $$t=2\ \sec$$ the shape is $$y=\frac{1}{x^2+3}$$. find the speed of wave on string

At $$t=0$$, the shape of the pulse is

$$y=\frac{1}{x^2+8x+19}=\frac{1}{(x+4)^2+3}$$

At $$t=2\,\text{s}$$, the shape is

$$y=\frac{1}{x^2+3}$$

Thus, the pulse has shifted from $$x=-4$$ to $$x=0$$, i.e. it has moved $$4\,\text{m}$$ in the $$+x$$ direction.

Hence, the speed of the wave is

$$v=\frac{\text{distance travelled}}{\text{time taken}}=\frac{4}{2}=2\,\text{m s}^{-1}$$

Therefore,  $${v=2\,\text{m s}^{-1}}$$

Hence, the correct option is C.