Displacement of string and simple harmonic motion

Question

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:

$$
vec{F} = -kvec{x}
$$

where $vec{x}$ is the displacement. I am not sure how to tackle this exercise.

Hooke's law in differential form is

$$
mcdot frac{partial^2 y(x,t)}{partial t^2} = -k x
$$

$$
frac{partial^2 y(x,t)}{partial t^2} = omega^2 A sin (kx -omega t + phi _0)
$$

I'm not sure how this shows that $y(x,t)$ is a solution to the differential Hooke's law

$$
-kx = m omega^2 A sin (kx -omega t + phi _0)
$$

Lelouch · Answer

Consider the following proof:
The wave equation is given by

$$
frac{partial ^2 y}{partial x^2} = c^2 nabla ^2 y.
$$

Suppose, i consider a rope with on end attached to a wall and the other end free to move. Clearly, we can find the boundary condition for this system at any time t : y(L,t) = 0 and y(0,t) = $Asin(phi_0 - omega t)$.  (note this is already stated in the fact that the rope undergoes shm). Also at any given x, the laplacian is defined. Now,we want to find out how  the source moves. It is not difficult to guess such a scenario, where the rope and source both move together in shm. Now for the fun part. Uniqueness theorem gurantees that this is the only solution to the system given the boundary conditions for any x. Hence the source must be undergoing shm too. You can read up on uniqueness theorem online. I may be wrong in the proof, so any doubts are welcome.

Displacement of string and simple harmonic motion

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