MATLAB script, named
waveeqn.m,
which creates an animation illustrating the behaviour of solutions to the wave equation. The initial position is a single spike. The initial velocity is zero.
MATLAB script, named
heateqn.m,
which creates an animation illustrating the behaviour of solutions to the heat equation. The initial position is a single spike.
Derivation of the Heat Equation for One Space Dimension
[
pdf
]
Fourier Series.
This demo shows a Fourier series adding up to the expected answer.
The Wave Equation.
This demonstration illustrates the behaviour of solutions of the
wave equation. The demonstration plots the solution
given by separation of variables that you have found in class.
Separation of variables expresses the solution as a sum
b_1(t) sin(pi x/l) + b_2(t) sin(2 pi x/l) + ...
of modes. When the demonstration starts, the initial amplitude is plotted.
By clicking the "Advance time" button, you instruct the computer
to increase the time by an amount specified in the "time interval window".
The Wave Equation (animated) [
one bump |
two bumps ].
These two demonstrations also illustrate the behaviour of solutions of the
wave equation. They also plot the solution
given by separation of variables that you have found in class.
The demonstrations animate the solution by successively plotting
u(x,0), followed by u(x,dt), followed by u(x,2dt) and so on.
The two demonstrations use different initial conditions.
The Telegraph Equation.
This program gives an animated demonstration of the solution to the
telegraph equation. The viewer
may adjust the equation parameters to give signal transmission with and without
distortion due to dispersion.
Fourier Synthesis:
Real-time Java applet lets you try out Fourier synthesis and hear the sounds the waveforms produce. (Link added 30 Dec 2004, thanks to P. Loewen.)
History of Music Site:
An annotated reduction of Also Sprach Zarathustra (aka the theme from the movie 2001) shows some of the theory of modes and overtones in action, with amusing MIDI play-along.
(Link added 30 Dec 2004, thanks to P. Loewen.)
UNSW Acoustics Lab:
Physicists at the University of New South Wales have put together a wonderfully illustrated and informative collection of pages on the science of various instruments.
Check out the
Chladni patterns for a handmade violin.
(Links added 30 Dec 2004, thanks to P. Loewen.)