Question

A pulse travelling on a string is represented by the function

$y=\frac{a^2}{{\left(x-\nu t\right)}^2+a^2},$

where a = 5 mm and $\nu =20{\mathrm{cm}}^{-1}$. Sketch the shape of the string at t = 0, 1 s and 2 s. Take x = 0 in the middle of the string.

Answer 1

Given,
Pulse travelling on a string, $y=\left[\frac{{\left(a\right)}^3}{{\left(x-\nu t\right)}^2+a^2}\right]$
$$\begin{gathered} a=5\mathrm{mm}=0.5\mathrm{cm} \\ \mathrm{Wave}\mathrm{speed},\nu =20\mathrm{cm}/\mathrm{s} \end{gathered}$$
So, at $t=0\mathrm{s},y=\frac{a^3}{\left(x^2+a^2\right)}$.
Similarly, at t = 1 s,
$$\begin{gathered} y=\frac{a^3}{{\left(x-\nu \right)}^2+a^2} \\ \mathrm{And}, \\ \mathrm{At}t=2\mathrm{s}, \\ y=\frac{a^3}{{\left(x-2\nu \right)}^2+a^2} \end{gathered}$$

To sketch the shape of the string, we have to plot a graph between y and x at different values of t.