Lecture 26. january 1999 We mentioned what role do the eigenmodes have. How to excite an eigenmode. (linear equations -> superposition) We mentioned the 'resonances' - they are excitations of eigenmodes. The string with a basis. Derivation of the equations. ..... In the lecture a mistake - 'a' was supposed to be the distance between the atoms m1 and m2. In the standard derivation 'a' is the length of the cell, i.e. from m1 to m1. Acoustical branch and optical branch. Origin of the names. As omega goes to zero: Density of modes in one-dimensional space (constant) Density of modes in two-dimensional space (prop. to omega) Density of modes in three-dimensional space (prop. to omega*omega) Solutions in two and three dim. space. vector displacement vector wave number the modes have 'polarization vectors' The equations become only more complicated, in principle the same method.