Question

A wave propagates on a string in the positive x-direction at a velocity $\nu$. The shape of the string at $t=t_0$ is given by $g\left(x,t_0\right)=A\sin \left(x/a\right)$. Write the wave equation for a general time t.

Answer 1

Given,
Wave velocity = $\nu$
Shape of the string at $t=t_0$ = $g\left(x,t_0\right)=A\sin \left(x/a\right)$ ...(i)
For a wave travelling in the positive x-direction, the general equation is given by
$y=A\sin \left(\frac{x}{a}-\frac{t}{T}\right)$
Putting t = − t and comparing with equation (i), we get:
$$\begin{gathered} g\left(x,0\right)=A\sin \left\{\left(\frac{x}{a}\right)+\left(\frac{t_0}{T}\right)\right\} \\ \Rightarrow g\left(x,t\right)=A\sin \left[\left\{\left(\frac{x}{a}\right)+\frac{t_0}{T}\right\}-\left(\frac{t}{T}\right)\right] \\ \mathrm{Now}, \\ T=\frac{a}{\nu } \\ \mathrm{Here}, \\ a=\mathrm{Wave}\mathrm{length} \\ \nu =\mathrm{Velocity}\mathrm{of}\mathrm{the}\mathrm{wave} \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ y=A\sin \left[\left(\frac{x}{a}\right)+\frac{t_0}{\left({\displaystyle \frac{a}{\nu }}\right)}-\frac{t}{\left({\displaystyle \frac{a}{\nu }}\right)}\right] \end{gathered}$$

$\Rightarrow y=A\sin \frac{x+\nu \left(t_0-t\right)}{a}$