M | A | T | H |

2 | 1 | B |

Mathematics Math21b Spring 2009

Linear Algebra and Differential Equations

Exhibit: Linear algebra: membrane motion

Course Head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Evolution of a membrane. We assume the amplitude u(x,y,t) satisfies the wave type equation
uIf the initial condition is an eigenvector u(x,y,0) = bto the operator T then u(x,y,t) = bwhere c _{nm} is the square root of n^{2}+m^{2}+n^{4}
(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@fas.harvard.edu

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