MAT 461 Animations and Figures

A Mathematica notebook with a demonstration of problem 2.3.2g, and a possible project.

Problem 2.3.3(d)
First of all, here are still pictures (about 2K) of the approximation to the initial condition using a finite Fourier series with 50 terms and 100 terms.
Now, here are some animations (about 200K) of this problem, with time advancing slowly and quickly.

Problem 2.4.1(a)
Here are animations (about 200K) with time advancing slowly and quickly.

Problem 2.4.1(c) Here are animations (about 200K) with time advancing slowly and quickly.

The 2D Heat Equation
Here is a DPGraph of the solution to the heat equation on the square with fixed temperature u=0 on the boundary, and initial condition u(x,y,0) = 1.

Unusual Coordinate Systems
There are 5 coordinate systems in the plane in which the Laplacian is separable. Aside from rectangular and polar coordinates, there are three more shown here.

Problem 2.5.1(g)
Laplace's equation on the rectangle 0 < x < L, 0 < y < W, with the sides and top insulating boundaries, and u(x, 0) = 0 if x > L/2 and u(x, 0) = 1 if x < L/2:
Here are contour maps for L/W =1, L/W =2,and L/W =3. Here are contour maps showing just the contour that hits the corner and two nearby contours, for L/W =1,and L/W =3.

Laplace's Equation in the disk
The contour map of the solution to Laplace's equation inside the disk of radius 1 with boundary conditions

Fourier Series
Here are some Mathematica notebooks, showing the Gibbs Phenomenon, a FSS that diverges, and a FSS that appears to converge even though f(x) is not piecewise smooth.
Here are simple Mathematica notebooks for computing the Fourier Series, Fourier Sine Series, and Fourier Cosine Series.

d'Alembert Solutions to the 1-D Wave Equation (using DPGraph)
Fig. 1 A standing wave is the sum of two traveling waves with the same amplitude.
Fig. 2 Two traveling waves with different amplitudes.
Fig. 3 Initial position is a gaussian, initial velocity is zero.
Fig. 4 Initial position is zero, initial velocity is the derivative of a gaussian.
Fig. 5 Collision of two pulses.
Fig. 6 Collision of a pulse and an "antipulse."
Guitar String d'Alembert solution for a plucked guitar string.

d'Alembert Solutions to the 1-D Wave Equation (using Mathematica)
The initial conditions, u(x,0) = f(x) and u_t(x,0) = g(x), can be changed in the notebook and re-run. A few ICs are saved:
f(x) = exp(-x^2), g(x) = 0 (Collision of two pulses.)
f(x) = 0, g(x) = exp(-x^2) (Collision of two steps.)
f(x) = arctan(x), g(x) = -c/(1+x^2) (A traveling step.)
guitarString.nb. (d'Alembert solution for a plucked guitar string.)

Normal Modes of a Square Drum
Mode 1,1
Mode 2,1   Modes 1,2 and 2,1 in phase   Modes 1,2 and 2,1 combined in a rotating wave
Mode 2,2
linear combination of modes

Normal Modes of a Rectangular Drum
Modes 2,1 and 1,2 on [0, 1.1] x [0, 1]. The frequencies of the normal modes are slightly different.
Modes 1,3 and 3,2 on [0, 4] x [0, 3]. The frequencies of the normal modes differ by about 6%.
Modes 2,2 and 3,1 on [0, 4] x [0, 3]. The frequencies of the normal modes differ by about 3%.

Page on Vibrating Drumheads.

Normal Modes of a Fractal Drumhead
Nandor Sieben's page.
Jim Swift's page.
See also the poster display in the halls of the math building.

Mathematica notebooks for Lab day
Laplace Equation in a wedge
Schroedinger's Equation