The figures shown below are general geometric surface representations of stress waves propagating through an anisotropic crystal. There are actually three surfaces corresponding to the three eigenvalue solutions to the Christoffel's equation but in this one special case (orthorhomic symmetry) all three surfaces form a continuous connected surface. For simiplicity we also show each of the three surfaces (eigenvalues solutions) disconnected from the continuously connected surface.
The directions of particle vibrations, [0 degrees (purple)] to 90 degrees (red)], are the eigenvector solutions to the same equation. In the three-dimensional surfaces we show a one-to-one mapping of eigenvectors (COLOR) onto their corresponding eigenvalue surfaces (SHAPE).
The objective here is to observe the complete solution over the entire three-dimensional region. SHAPE and COLOR are defined as solutions to Christoffel's equation below. Geometric definitions for each term in the Christoffel's equation and geometry of results are defined below.