ABSTRACT
A travelling wave, such as the propagation of a small elastic deformation in a solid, can be represented by a trigonometric cosine function:
For mathematical convenience, a wave can also be expressed in complex exponential form: kxikxti AeAetxu , We can set t = 0 when we are not interested in time-dependent effects. This expression represents one possible solution to the general wave equation. Boundary conditions determine the precise nature of the solution. For example, imposition of the periodic boundary conditions means that the wave number k can only take on certain values:
n = 0, 1, 2, 3, 4… L
2
h L i h h i i l h hi h h b d di i
Now,
k
+k
w ere s t e c aracter st c engt over w c t e oun ary con t ons apply. In one-dimensional k-space, we can represent the allowed values of k as:
Each point in k-space represents a particular mode of vibration in the solid.
