21 B
Mathematics Math21b Spring 2009
Linear Algebra and Differential Equations
Exhibit: Linear algebra: membrane motion
Course Head: Oliver Knill
Office: SciCtr 434

Evolution of a membrane. We assume the amplitude u(x,y,t) satisfies the wave type equation
utt = uxx + uyy - uxxxx = T(u) .
If the initial condition is an eigenvector
u(x,y,0) = bn m  sin(n x) sin(m y)
to the operator T then
u(x,y,t) = bn m sin(n x) sin(m y) cos(cnm t) 
where cnm is the square root of n2+m2+n4 (which is the negative of the eigenvalue of T).

In general (as in the simulation to the right), the initial condition is a linear combination of eigenfunctions. The evolution is the sum of the evolutions of the eigenfunctions.

Wave simulation with Mathematica. 2000 frames were computed. The background music is from "Lambada" by Kaoma.

Please send questions and comments to
Math21b (Exam Group 1)| Oliver Knill | Spring 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University