Periodic Wave

A periodic phenomenon is one that occurs at regular intervals. Periodic phenomena are all around us, from the rising of the sun every morning to the repeating structure of the DNA in our cells, it's clear that nature loves to repeat itself!

Analyzing these phenomena scientifically is particularly elegant as we only have to understand a few characteristic features to get the full picture. In this article, we will apply this analysis to one of the fundamental phenomena in physics, periodic waves. By doing so we can find a full mathematical description of periodic waves from just a couple of key quantities, the wavelength, and the frequency of a wave.

Definition of Periodic Waves

In physics, waves are a type of energy transfer caused by an initial disturbance that is then propagated through space and time. The size of the disturbance defines the amplitude of the wave. If a wave has a continually repeating pattern, made up of cycles, such as particles oscillating around an equilibrium point, we call it a Periodic Wave. Periodic waves have several key quantities by which they can be characterized, depending on the time and distance between cycles. The characteristic time taken, in seconds \(\mathrm{s}\), for each cycle is known as the Time Period \(T\) of the wave. The number of cycles that occur within one second is known as the Frequency \(f\) of the wave, and is measured in Hertz \(\mathrm{Hz}\) equivalent to \(\mathrm{s}^{-1}\). By definition, the frequency and time period of a wave are inverses.

\[f=\frac{1}{T}\]

Fig. 1 - A graph of a periodic wave, demonstrating how displacement from equilibrium varies over time. From the graph, we can see the amplitude \(A\) and time period \(T\) of the wave.

Similarly, the characteristic distance taken for a cycle is known as the Wave Length \(\lambda\). The wavelength is related to the frequency of the wave by the velocity \(v\) with which the wave propagates through space.

\[v=f\lambda=\frac{\lambda}{T}\]

Fig. 2 - The wavelength of a periodic wave is defined as the distance between two equivalent points of the wave.

Transverse and Longitudinal Periodic Waves

Periodic waves can be split into two main types of waves, depending on the direction of the displacement the wave causes. Longitudinal waves cause oscillations that are parallel to the direction of energy transfer. Some key examples of longitudinal are sound waves and stress waves within materials.

On the other hand, many periodic waves are such that the direction of oscillation is perpendicular to the direction of energy transfer. For example, light is a type of electromagnetic radiation caused by electric and magnetic fields which oscillate at right angles to the direction the light travels in.

Fig. 4 - Electromagnetic waves are a type of transverse wave. The electric \(E\) and magnetic \(B\) fields oscillate perpendicularly to the direction the wave travels in \(z\).

Pulse vs Periodic Wave

Whilst the focus of this article is on periodic waves, its worth briefly looking at a-periodic waves known as pulses to emphasize the defining features of a periodic wave. Pulses are a very common type of energy transfer in physics, caused by sudden short disturbances which propagate as a short burst of energy. Whilst a pulse may have cycles like a periodic wave, pulses usually only contain one or two cycles such that we cannot properly define a wavelength or frequency of a pulse.

Fig. 5 - We can model pulses as moving sharp peaks, note that a pulse contains no oscillations.

For example, consider dropping a pebble into a pool. The disturbance caused by the pebble would lead to a ripple of a few water waves traveling outwards, however shortly after the pebble was dropped the water would return to equilibrium and no further waves would be produced. We say that the pebble produced pulses in the water, if instead the water was continuously disturbed, like with a wave machine, then the water waves would be periodic.

The formula for Periodic Waves

Given what we know about periodic waves so far, let's consider how we could best represent periodic waves using mathematical functions. As we have seen, the period of a wave is defined by its characteristic wavelength or time period. Consider a periodic wave, with an amplitude A, and a wavelength of \(\lambda\). This means we are looking for a function that satisfies the following conditions.

\[\begin{align}f(x)&=f(x+\lambda)\\\max |f(x)|&=A\end{align}\]

As you might be able to guess, the trigonometric sine and cosine functions are the functions we are looking for, given that they also satisfy similar periodicity conditions.

\[\sin(x)=\sin(x+2n\pi),\,\cos(x)=\sin(x+2n\pi)\]

As the sine and cosine functions can be made equivalent by adding a phase of \(\frac{\pi}{2}\), we simply choose which function we want, depending on the initial conditions of the wave. As we usually think of waves as starting with maximum displacement, we'll consider the cosine function.

By smartly choosing scaling factors for the cosine function, we can satisfy the conditions needed

\[\begin{align}f(x)&=A\cos\left(\frac{2\pi}{\lambda}x\right)\\\max|f(x)|&=A\\f(x+\lambda)&=A\cos\left(\frac{2\pi}{\lambda}\left(x+\lambda\right)\right)\\&=A\cos\left(\frac{2\pi}{\lambda}x+2\pi\right)\\&=A\cos\left(\frac{2\pi}{\lambda}x\right)\\&=f(x)\end{align}\]

So a periodic wave, with amplitude \(A\) and wavelength \(\lambda\), at one fixed instant of time \(t\), is described by the function

\[f(x)=A\cos\left(\frac{2\pi}{\lambda}x\right)\]

In physics the quantity \(\frac{2\pi}{\lambda}\) is called the 'angular wavenumber' usually denoted \(k\).

Applying the same thinking, we can find the function describing the oscillation of a single point on the wave over time. If the wave has a time period of \(T\) and amplitude \(A\), then the oscillations as a function of time are given by

\[\begin{align}h(t)&=A\cos\left(\frac{2\pi}{T}t\right)\\&=A\cos\left(2\pi f t\right)\\&=A\cos\left(\omega t\right)\end{align}\]

The quantity \(\omega=2\pi f\) is known as the angular frequency of the wave.

Periodic Waves Examples