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 $\left(\frac{x}{a}\right)$ Write the wave equation for a general time t.

• A. $Asin[\frac{x}{a}-v(t-t_0\left)\right]$
• B. $Asin\left[\frac{x-v(t-t_0)}{a}\right]$
• C. $Asin\left[\frac{x+v(t-t_0)}{a}\right]$
• D. None of these

Answer 1

The correct option is B

$Asin\left[\frac{x-v(t-t_0)}{a}\right]$

Here given is $g(x,t_0)=Asin\left(\frac{x}{a}\right)$ at $t=t_0$

Comparing with the general equation g(x,t)=$Asin(kx-\omega t+\varphi )$

We get at $t=t_0$

$k=\frac{1}{a}$

also$-\omega t_0+\varphi =0$

$\Rightarrow \varphi =\omega t_0$

We also known that $k=\frac{\omega }{v}$

$\Rightarrow \omega =\frac{v}{a}$

so we get

$g(x,t)=Asin(\frac{x}{a}-\frac{vt}{a}+\frac{v{t}_0}{a})$

$g(x,t)=Asin[\frac{x}{a}-\frac{vt}{a}+\frac{v{t}_0}{a}]$

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