Waves and Oscillations

Department of Mathematics
University of Glasgow

Why look at waves?

Waves are the key to sound and colour.
Mobile phone signals, microwave ovens all use energy carried by waves.
Earthquakes and tsunamis are destructive waves of energy.

Waves affect our everyday lives in many ways.

What is a wave?

A wave is a pulse of energy. Waves carry energy away from a central transmitter. Mechanical waves, such as sound waves, need some medium of transmission. Electromagnetic waves, for example radio waves, can carry energy through a vacuum. If a wave is travelling through a medium, the particles of the medium do not move along with it. They vibrate about their equilibrium position, and the energy is transmitted through the interaction of neighbouring particles.

Here is a Maple program that plots a simple travelling sine wave. It is animated so that you can see the wave motion along the axis. The dots represent particles in the medium. If you watch the line of dots closest to the vertical axis, you will see that they oscillate up and down, but do not move with the wave.

> with(plots):

> p1:=animate(sin(x-t),x=0..4*Pi,t=0..20,frames=100,style=point,colour=magenta):

> p2:=animate(sin(x-t)-.1,x=0..4*Pi,t=0..20,frames=100,style=point,colour=magenta):

> p3:=animate(sin(x-t)-.2,x=0..4*Pi,t=0..20,frames=100,style=point,colour=magenta):

>

> display({p1,p2,p3});

Waves vs Oscillations

A wave is travelling energy: all waves -- radio, light, x-ray, sound or water waves radiate in all directions from a central source.

An oscillation is when a mass moves back and forth in a regular rhythm: a swing, the tide, a duck sitting still on a wavy pond all oscillate.

Even though oscillations and waves are different phenomena, the same mathematical functions are used to describe them and graph their motion. These functions are the sine and cosine functions.

The Wave Equation

Differential equations are used to model physical, biological and other systems. A good model explains the behaviour we see and predicts future behaviour. The wave equation is used for all linear wave motion where the speed is constant.

Here is the 3 dimensional plot of a typical solution to the wave equation.

> plot3d(sin(x-t)+sin(x+t),x=0..4*Pi,t=0..10);

Mechanical waves

Sound and water waves transmit energy through a medium. The molecules vibrate and their interaction transmits the energy. Such waves are called mechanical waves. There are two main types:
Transverse waves: the molecules vibrate at right angles to the direction of travel.

Longitudinal waves: the molecules vibrate along the direction of travel.

Wave characteristics

Imagine a long rope stretched out straight along the ground. If you vibrate one end periodically, then a transverse wave will move along it. A snapshot would look like this:

The line through BH gives the equilibrium or rest position.

The amplitude A measures the maximum displacement of a particle from equilibrium: “rest to crest”. The amplitude is related to the amount of energy the wave is carrying.

The wavelength measures the length of one complete cycle.

The period is the time a particle on the rope takes to do one cycle.

The frequency is the number of cycles a particle makes/unit time.

The mathematical functions that model periodic behaviour with a constant amplitude and wavelength are sines and cosines.

Wave interactions

A mechanical wave is a disturbance that travels through a medium. The crest moves from particle to particle in the form of a sine wave. It will continue to move in the same form until something interferes with it. This could be because it meets another wave or it reflects off a boundary.
If two single pulses meet, as they pass they interact, but as they separate, their shape is the same as before the interaction.
The picture above shows pulses that are moving in opposite directions. You can see how they cancel each other out as they cross, but once they have passed each other the energy travels on with the same wavelength and amplitude.

Now, pulses moving at different speeds in the same direction:

The one carrying more energy is moving faster. Notice that as they cross the amplitudes add. After the interaction the wave shapes are unchanged, but the faster one has overtaken the slow one.

Adding travelling waves

When travelling waves interact, they form another waveform. In general it will not be a simple sine wave, but there will be a periodic pattern and a fixed wavelength. In the picture below, the red graph is the function . The green and blue graphs give each of these functions -- which is which?

Although the wave form is not a simple sine wave it is clearly periodic and has a fixed maximum amplitude. Below is a picture showing how the waveforms move. Again, there is a fixed wavelength and amplitude.

Waves on a guitar string

A guitar string is fixed at both ends, so waves are reflected and interact in the length of the string. As both ends have to be at rest, this restricts the possible wavelengths. The possible waves are called modes of vibration.
As the length and thickness of the string is fixed, the pitch of the note is determined by its tension. The tone is made up of harmonics which are the different frequencies of the modes of vibration.

Here is a picture of the first harmonic for a guitar string. Below are animations of the first 4 modes of vibration. You can see that as the wavelength shortens the frequency increases. The higher the frequency, the higher the pitch.

Standing Waves

Because the ends of the guitar string are fixed, the waves are reflected back into the string. As the last section said, only certain wavelengths are possible as solutions of the wave equation.

As a result, if the string is made to vibrate with the frequency of one of the harmonic modes, you see a standing wave. The incident and reflected wave interact so that certain points -- the nodes -- are standing still. A standing wave looks as if the string is vibrating at right angles to its length and there is no wave transmitting energy along the string.

Written by Frances Goldman; research by Shona Maclean

© Copyright 2000, Frances Goldman, University of Glasgow