Waves and simple harmonic motion

Is it possible to always assume that a wave is generated from the medium components oscillating in simple harmonic motion with same amplitude but just out of phase with each other? For example, I’ve come across this picture many times:

Also, the displacement from the particle’s mean position is given as:

$ y = A cos(omega t – phi) $ where $omega$ is the frequency of the oscillating particle. We construct the wave equation by defining $ phi = kx$ where $x$ is path difference of the particle and $k$, the wave number. And substituting, we get the wave equation:

$$y = A cos(omega t – kx) $$

And we see that the wave has the same frequency as it’s oscillating particle. Is this always true? We know that wave velocity and particle velocity is different. What will be the total energy of the wave? (Total Energy of the oscillating particle will be $ frac{1}{2}momega^2A^2$ where A is the amplitude.