A wave travels on a string in the positive x -direction at a...

The general equation of a wave traveling in the positive $x$-direction with velocity $v$ is given by $f(x, t) = h(x - vt)$.

At $t = t_0$, the wave equation is given as $f(x, t_0) = h(x - vt_0) = A \sin \left(\dfrac{x}{a}\right).<br><br>Let$z = x - vt_0$, which implies$x = z + vt_0$.<br><br>Substituting this into the function gives$h(z) = A \sin \left(\dfrac{z + vt_0}{a}\right)$.<br><br>For a general time$t$, replace$z$with$x - vt$:<br><br>$f(x, t) = h(x - vt) = A \sin \left(\dfrac{x - vt + vt_0}{a}\right)$<br><br>$f(x, t) = A \sin \left(\dfrac{x - v(t - t_0)}{a}\right)