In shm F= -m^2x = - kx

k = elastic const. i.e when displ is towards right when force points towards left.This sort of a force acts when an elastic body such as a spring is deformed.Now F = - dU/dx

or dU/dx= kx, now on integrating within limits of 0 to U, we get U = 1/2 kx^2  = negative of work done by this force.

total mechanical energy is a const qty and since the force is conservative ,

E = 1/2 k A^2.Pot energy in any kind of osscilation,increeases at the expense of K.E. while moving away from the mean position.

Now when we come to waves,

let the eqn of a wave be y = f(ax+- bt) , speed of the wave is dy/dt x is considerd to be const.

Now pot energy here is the amt of work done in stretching  the string and it depends on the tension and the slope .

So pot energy per unit vol = 1/2 (wave speed)^2 * (dy/dx)^2*mass ,it can be concluded that

for a mass attached to a spring and osscilating simple harmonic wave the results come out to be the same