Supplements for Chapter 3:
Sound and sinusoids
The atom of sound
As we said in Chapter 3, sinusoids are the atoms of sound; alone they give a pure but barren sound. More complex tones like from an oboe playing one note with no vibrato are combinations of many such "atoms": a molecule of sound. Many of the software programs are easily coaxed into producing pure sine tones, including Fourier Series, MAX Partials, and Loaded String. If you like using CDF Player codes, Steve Wolfram has written Pure Tones, http://demonstrations.wolfram.com/PureTones/, a very simple exploration tool for pure suinusoids with a slider for varying their frequency.
Paul Falstad's Fourier
The "atom of sound" principle of the sinusoidal functions is strongly suggested by Falstad's Fourier, http://www.falstad.com/fourier/, by starting with a single "stalk" (a single sinusoid) and then raising other stalks, noting how the function and its sound changes. You can use amplitude (stalk height), phase, and choice of different stalks to create an infinite variety of functions. Then, draw any (automatically periodic in this case) waveform you like in the upper panel and note that the Fourier analysis always finds a combination of amplitudes and phases (or, equivalently, a sum of sine and a cosine functions - the applet can display either) to reproduce the function you drew in terms of a Fourier series.
Jean-Francois Charles' MAX Partials
Jean-Francois Charles has written this research-level (but still easy to use) sine-tone generator with up to 50 Partials under complete amplitude, frequency, and phase control, plus the ability to record. The autocorrelation and spectrum is also displayed. See MAX patches.
PhET Fourier has overlap with Falstad's Fourier and MAX partials (with the latter being the most flexible) but it does offer some unique and well organized visual (and limited auditory) feedback, especially under the tab "Discrete to Continuous", where the result of a phased sum of partials as frequency spacing and frequency span of the partials is varied (there are Uncertainty Principle lessons here).
The sum of two sinusoids
A simple interactive graphical representation of the addition of two sinusoidal functions with different frequency, amplitude, and phase can be found at http://demonstrations.wolfram.com/SumOfSines/, to be run on the free Wolfram CDF player (http://www.wolfram.com/cdf-player/). Note especially how different the sum can be if only the relative phase is adjusted. Often we cannot hear the difference, but the exceptions play a large role in controversies over the perception of tone and pitch, as will become clear in succeeding chapters. If you have installed the CDF player, the application should appear below.
Derivatives (slopes) of sinusoidal functions
Sound of waveforms
The following Mathematica Demonstration Project is a good quick way to see and hear waveforms, and examine their power spectra. More powerful applets like Falstad Fourier and MAX Partials can also do this but perhaps not so quickly.
A Lissajous plot is generated by choosing two sinusoids, with arbitrary amplitude, phase, and frequency. One sinusoid becomes the value of x(t), and the other becomes y(t); the resulting Lissajous curve is the result of plotting the (x,y) points for a continuous range of times. Lissajous curve arise in some visualizations of sound, as when mirrors are attached to two different tuning forks and a light beam is bounced off both mirrors and onto a wall
A good Lissajous figure generator can be found at the Wolfram Demonstrations website, http://demonstrations.wolfram.com/LissajousFigures/ . You first need to download the free Wolfram CDF player at http://www.wolfram.com/cdf-player/. This simple but effective applet was written by Stephan Wolfram himself. On the Wolfram Demonstrations you will find many fancier Lissiajous CDF applications, that includes things we don't need or haven't introduced, like friction, but you might want to explore them if you are adventurous. Changing the amplitudes is really not as interesting so LissajousFigures doesn't bother with it. The figure below shows LissajousFigures in action for one choice of phase and frequency parameters: