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Mathematics Resources by Topic - Algebra
Periodic functions

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Modelling with periodic functions - the data sets page includes the following links to data sets for which a periodic model may be appropriate.

Where is the dot? - cosine and sine introduction - NRICH

definitions and graphs of sine, cosine and tangent

geometric representation of sine, cosine, tangent, secant, cosecant, cotangent

Ferris wheel demonstration
Observe the Ferris wheel from different viewpoints. Focus on the vertical motion.

sine rule demonstration
Find the height of a triangle in two different ways.

cosine rule demonstration

periodic drum beats
Create your own repeating patterns.

y = a sin b(x-c)
Transformations of the graph y=sin(x) show the effects of the values a,b,c - works best in Internet Explorer.

y = a sin b(x-hp) + k
Select the values of a,b,h,k and see the effect on the graph of y=sin(x) (also cosine function).

y = A sin (x+B)
Given the graph, find the values of A and B.

matching functions and graphs - Maths Online
two sets of matching puzzles

y = a sin(x) + b cos(x)
Geometric interpretation that allows you to investigate the effect on the amplitude and phase shift of different values of a and b.

pendulum simulations with periodic graphs - Walter Vendt

Mathematics for Technology software
Freeware includes rotating vectors and sinusoidal waveforms.

The Soundry
The site includes sections on the human ear, the physics of sound, applications of sound and the history of sound. The interactive lab contains excellent java applets that allow you to:

make beats
listen to the Doppler effect
create instruments by adjusting harmonics and play tunes
investigate the effect of the shape of a wave on the sound
mix sound effects.

Sound Waves and Music - The Physics Classroom
The nature of sound as a longitudinal, mechanical pressure wave is explained and the properties of sound are discussed. Wave principles are used to analyse musical instruments.

Acoustics Animations - Dr. Dan Russell
illustrations of longitudinal waves and transverse waves showing the motion of individual particles

interesting sounds

Fourier Synthesis - make sounds as the sum of sine and cosine functions.

Fourier series
This applet lets you enter a function and represent it as a Fourier series.

Math and Music - Electronic Bookshelf - Dartmouth College
Topics include the MIDI System, waves, synthesizers, coding and computer music applications. The section on waves is comprehensive and easy to follow.

Seeing Sound - Queensland Sciencentre
explanations, waveforms and sound files

addition of two waves - Jerzy Kocik

Enter the frequencies f₁ and f₂.
See the curves y = sin (2 p f₁x), y = sin (2 p f₂x) and y = sin (2 p f₁x) + sin (2 p f₂x).
Investigate beats when the frequencies are close.

addition of two waves - ExploreLearning
Click on two speakers and hear the beats.

demonstration of wave forms for beats - Walter Fendt
no sound but good graphics

Listen to music

sound effects

play a piano
Play a piano and see the wave forms. Investigate the effect of overtones.

Hammond Drawbar Organ simulation
Create waveforms for different sounds.

Making Windchimes - Eric Reiswig
all you need to know about making windchimes (xylophone / tubular bells)

How the Radio Spectrum Works - HowStuffWorks
What is VHF, AM, ...?

Damping Functions - Coolmath
logarithmic fade in music

bungee jumping
Simulate a bungee jump and produce graphs of momentum and height against time (damped trigonometric functions).

Mathematics of Tide Prediction - Rex Boggs and Kiddy Bolger
In this PowerPoint presentation, addition of periodic functions is used to model the tides on 1 January 2002 at Mooloolaba in Queensland.

tide predictions for Australia and the Pacific
Find high and low tide heights and times.

tide predictions for over 7000 places around the world - XTide
provides excellent graphs as well as high and low tide heights and times

tidal prediction with harmonic analysis
Tides can be predicted by harmonic analysis i.e the addition of periodic functions. This java applet provides data for Breakwater Harbour in Delaware (up to 18 components). Other places can be investigated by changing the values for amplitude and phase.

sin(A+B) and cos(A+B)
demonstration of expressions for sin(A+B) and cos(A+B)

Digital Oscilloscope software
The software uses the computer's sound card as an Analog-to-Digital Converter to digitize any input waveform (speech, music, electric signal, etc.) and to present it on the monitor in real time. Data can be saved and pasted into a spreadsheet.

NCH Tone/Waveform Generator software
Create sounds with a variety of waveforms at selected frequencies between 1 to 20000Hz.

Tuning Fork software
Create notes A, A#/Bb, B, C, C#/Db, D, ... and view their frequencies. Select from sine, triangular, sawtooth and square waveforms. Move up/down the scale by a single note or by an octave.

Sound Analyser and Tuner software
As well as the features of the Tuning Fork, you can analyse sounds by frequency, note/octave, waveform and frequency spectrum analysis.

Frequency Analyser software
Split sounds into component frequencies. Watch patterns of speech, the harmonics of vowels and the noise of sibilants.

Finale Notepad software
Compose, print and play music. Use the program to play music downloaded from Project Gutenberg Music.

NLREG software
NLREG is a powerful statistical analysis program that performs linear and nonlinear regression analysis and curve fitting. It can handle linear, polynomial, exponential, logistic, periodic and any other function that can be described algebraically.

learning resources and sources of problems

resources page - algebra