A wave travelling on a string has the equation y = (0.10 mm) [(31.4...

The given equation of the wave is $y = (0.10 \text{ mm}) \sin[(31.4 \text{ m}^{-1})x + (314 \text{ s}^{-1})t]$.

Comparing this with the standard wave equation $y = A \sin(kx + \omega t)$, we observe that the argument of the sine function has a positive sign between the spatial and temporal terms.

For a point of constant phase, $kx + \omega t = \text{constant}$.

Differentiating with respect to time $t$, we get $k \dfrac{dx}{dt} + \omega = 0$, which gives the wave velocity $v = \dfrac{dx}{dt} = -\dfrac{\omega}{k}$.

Since the velocity is negative, the wave travels in the negative $x$-direction.