ESM 5344 Wave Propagation in Solids: Wave Surface "Cross-Overs"

Wave surfaces intersect when there exists a eigenvalue degeneracy in the Christoffel's equations. At these intersections the eigenvalues (wave vibration directions, 0-90 degrees shown as color: purple-0 (longitudinal) and red-90(transverse)) remain associated with the same surface and appear to "jump" or "hop" to an adjoining surface when viewed in three-dimensions (3D) as shown below. We refer to these intersections as wave surface "cross-overs". A closer examination reveals that these cross-overs are the basis for new types of wave surface geometries that can be used for crystal classification.

For example wave surface cross-overs in a Tetragonal symmetry crystal called Tellurium Dioxide (TeO2) are shown below in 3D.

To view larger images select: Top View / Rotated View / Cross-Over Close-up

To view these wave surfaces in three dimensions, two different VRML file formats are provided here that can be viewed with a VRML file web browser plug-in (e.g. google: cortona, cosmo, FreeWRL): VRML1 velocity surfaces, VRML2 velocity surfaces

When all of these wave surface cross-overs are considered the Christoffel's equations predicted three seperate wave surfaces as expected and is shown in the isometric figure shown below.

However for an orthorhombic symmetric crystal called Calcium Formate (HCOO2) wave surface cross-overs results in a fully connected "single-surface" geometry.

First construct three wave surfaces using Christoffel's equation. The figure below illustrates how the fourth order stiffness tensor is related to the nano-structure of a unit cell.

Consider three wave surfaces intersecting three orthogonal planes in two-dimensions (2D): x1-x2, x1-x3, and x2-x3.

Now construct these three 2D images into a 3D isometric view and note that there is only one connected surface, because of four wave surface cross-overs at points a,b,c, and d. To view these wave surfaces in three dimensions, two different VRML file formats are provided here that can be viewed with a VRML file web browser plug-in (e.g. google: cortona, cosmo, FreeWRL): VRML1 velocity surfaces, VRML2 velocity surfaces

To better interpret all cross-overs that result into a single connected surface we view these 3D wave surfaces can be viewed and manipulated interactively: 1) as VRML files on web browwers, or 2) in an immersive environment. The immersive images of Calcium format are shown below. Both VRML and files constructed to be viewed in an immersive environment can be loaded and viewed directly from ESM5344 Web pages. The figure below illustrates how these images are viewed in immersive environment.

In fact for orthorhombic symmetry there are four unique types of geometries that can result from wave surface cross-overs.

Engineering Science and Mechanics: ESM5344 Spring 2007