Question

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

Answer 1

At $t=t_0,g(x,t_0)=A\sin \left(x/a\right)$ ___(1)

For a wave traveling in the positive x-direction, the general equation is given by

$y=f(\frac{x}{a}-\frac{t}{T})$

Putting $t=-t_0$ and comparing with equation (1), we get

$\Rightarrow g(x,0)=A\sin \left\{\right(x/a)+(t_0/T\left)\right\}$

$\Rightarrow g(x,t)=A\sin \left\{\right(x/a)+(t_0/T)-(t/T\left)\right\}$

As $T=a/v$ (a = wave length, v = speed of the wave)

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

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

$\Rightarrow y=A\sin \left[\frac{x-v(t-t_0)}{a}\right]$