# Can a wave undergo SHM?

I am recently studying about Simple Harmonic motion and during that I thought,

Do a wave undergo simple harmonic motions ?

Do they ?

The equation for SHM is given by

$$\frac{\partial^2y}{\partial t^2}+ \omega_0^2y = 0\, .$$

Where $$y$$ is the displacement of the body with respect to a defined origin, and $$\omega_0$$ is the characteristic frequency of the system.

I believe when you are asking whether a wave undergoes SHM, you are effectively asking whether the points on a wave follow a SHM motion about the equilibrium point. Now the wave equation is given by

$$\frac{\partial^2{y}}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2{y}}{\partial t^2}\, ,$$

where $$v$$ is the propagation speed of the wave.

So in effect you would require both equations to hold.To find the condition for such a system, we substitute $$\partial^2y/\partial t^2$$, and we get the equation

$$\frac{\partial^2{y}}{\partial x^2} = -\frac{\omega_0^2}{v^2}y\, .$$

Does this form look similar? Yes it does, it is a SHM in the spatial dimension. As such, a general solution for a wave whose particles riding on the crest undergo SHM about their equilibrium point, would require a SHM in the $$x$$ dimension, i.e. a linear combination of sinosoidal waves with respective amplitudes and phase displacements. Note the new characteristic frequency $$\omega_{0, new} \equiv \omega_0/v$$

They can do, in a sinusoidal wave each part of the wave is undergoing simple harmonic motion about the point of zero amplitude as long as it has a linear time dependence. As in this image from Hyperphysics: • What happen if it doesn't have linear time dependence ? – Annamalai Sriram Mar 26 at 11:28
• Someone might correct me with an counterexample, but google "plot sin(x^2)" and look at its behaviour around the origin, that's not SHM in the same way. Although it might still fit the definition. – Charlie Mar 26 at 11:30

If the wave can be described by a single frequency (i.e. a sinusoidal wave of infinite length) then yes, by definition the medium is undergoing simple harmonic motion.

If we drop the requirement of a single frequency, the whole wave does not strictly have simple harmonic behavior. However if we still consider the wave to be periodic with infinite extent, then by Fourier's theorem this wave can be described as a combination of many single-frequency waves, each of which undergoes simple harmonic motion.

Of course, in the real world most waves undergo damping, which means they do not go on forever and thus are not strictly periodic. For example, waves in water reduce in amplitude over time due to friction (and/or due to spreading out). Similarly, light (electromagnetic waves) is attenuated when it travels through an absorbing material.

Thus, for most real-world examples of waves simple harmonic motion does not strictly apply, but it can be an excellent approximation if the damping and finite-time effects are small.