# Is the propagation of a wave simple harmonic motion?

1. Aug 21, 2014

### needingtoknow

Is the propagation of a wave simple harmonic motion?

Simple harmonic motion is defined when the restoring force is proportional to the displacement. Hooke's Law F = -kx is an example. However at my level of understanding I have not yet read about the relationship between forces and waves and was just curious if the restoring force is proportional to the displacement when it comes to the propagation of a wave?

2. Aug 21, 2014

### Orodruin

Staff Emeritus
Each frequency in a wave is essentially a simple harmonic oscillator. However, a general wave is typically built up out of several different frequencies. If you have a finite string (for example), it will have a discrete (but infinite) set of eigenmodes, each with its own frequency and which can be considered to be a harmonic oscillator. An infinite string will support a continuum of frequencies. The general wave is a linear combination of these and will therefore not be a simple harmonic oscillator. The mathematics for this is handled mainly through Fourier series and transforms.

3. Aug 21, 2014

### Kyle.Nemeth

The propagation of a wave is modeled by SHM (Simple Harmonic Motion)

The thumbnail I have attached represents the configuration of a wave propagating on a string. We take a tiny length element (or particle) of the string and model it's vertical motion as that of SHM. When we do this, we end up with

$y(x,t) = Asin(kx - ωt)$

Here, y (the vertical position of a particle on the string) is determined by x and t. The amplitude, A, of the waves is usually fixed for this elementary case. The wave speed is determined by

$v = k/ω$

where k is the wave number and ω is the angular frequency. The wave number, k, is defined as

$k = 2π/λ$

and the angular frequency, ω, is defined as

$ω = 2πf$

Thus,

$v = λf$

where λ is the wavelength between waves on the string and f is the frequency of the wave motion.

Let's consider a "snapshot" of the string (sort of like the picture I have attached) such that t = 0. Then,

$y(x,0) = Asin(kx)$

Here, we can see that as we increase or decrease our x value, the vertical position of particles on the string varies periodically such as that of SHM.

So, in general, we can model the vertical positions of particles on a string as a function of x and t (that is, space and time). This is a very simplified elementary case though and it only accounts for wave propagation in one spatial dimension. I have also demonstrated a case for which ø = 0 (phase constant) for the sake of simplicity. The phase constant is something you should take more seriously once you start reading more about wave propagation, but hopefully I have helped relate SHM and wave propagation for you on an elementary basis.

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4. Aug 21, 2014

### olivermsun

Also, keep in mind that there are other waves out there which behave quite differently from SHMs and the 1-d wave on a string.