If the displacement of a string follows
$$ y(x,t) = A \sin(kx - \omega t + \phi_0) $$
how can I show that the "hand" generating the wave must be moving vertically in a simple harmonic motion?
Simple harmonic motion means that the restoring force is proportional to the displacement, which basically is Hooke's law:
$$ \vec{F} = -k\vec{x} $$
where $\vec{x}$ is the displacement. I am not sure how to tackle this exercise.
Hooke's law in differential form is
$$ m\cdot \frac{\partial^2 y(x,t)}{\partial t^2} = -k x $$
$$ \frac{\partial^2 y(x,t)}{\partial t^2} = \omega^2 A \sin (kx -\omega t + \phi _0) $$
I'm not sure how this shows that $y(x,t)$ is a solution to the differential Hooke's law
$$ -kx = m \omega^2 A \sin (kx -\omega t + \phi _0) $$