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Project 1: Stroh formalism and surface acoustic waves Bulk acoustic wave propagation in anisotropic media is governed by the Christoffel equation and the velocity and its direction dependence can be described by the slowness surface. The slowness surface consists of three sheets associated with three eigenvalues of the Christoffel equation for the wave vectors with |k|=1 and the two outer sheets may have zero curvature locally. Such zero-curvature slowness curves can also be studied by using the Stroh formalism. As a six-dimensional formalism, the Stroh eigenvalues equation produces three pairs of complex conjugated eigenvalues in the subsonic regime, and they become degenerate and real at transonic states. When four or six eigenvalues becoming degenerated and real, the transonic states are in fact associated with the slowness curve having zero curvature. It has been shown that there exist extraordinary zero-curvature transonic states in monoclinic media by claculating the curvature directly from the Christoffel equation using the perturbation method. The zero-curvature transonic state was first investigated in relation to the theory of surface waves because the property of transonic states plays an important role in the establishment of the existence and uniqueness theorem of the surface waves in anisotropic media. Generally we denote surface wave solutions in anisotropic media the generalized surface waves and they can exist in either subsonic or supersonic regimes. The following question can then be asked: When an medium admitts the extraordinary zero-curvature transonic state, will it be possible to find one-component surface waves? The discovery of the generalized surface waves triggered intensive investigations on the degeneracies occurred in the Stroh formalism and a series of existence criteria for the degeneracies were formulated. It was shown that even when the Stroh eigenvalues is extraordinarily degenerated it is possible to construct a generalized surface wave solution satisfying boundary condition of traction-free surface. However, there remains a practical problem how the generalized surface waves can be located, if exists, for a given anisotropic medium and surface configuration. Our earlier study of transversely isotropic elastic media has shown that the one-component surface wave would be located on the so-called space of degeneracy along which two or three Stroh eigenvalues are degenerated. In this subject, we reiterate the importance of the zero-curvature transonic states because they are associated with Stroh degeneracy and are very populous. We will establish first the existence conditions for various type of zero-curvature transonic states for the monoclinic elastic media, and then explore their relation to the space of degeneracy in some symmetrical surface configurations. By tracing the spaces of degeneracy, we will demonstrate how the supersonic generalized surface waves can be located and how the subsonic generalized surface waves can be revealed on the ordinary subsonic (Rayleigh) surface wave branch. Zero curvature transonic states and supersonic surface wavesExistence of the zero curvature slowness curves
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Project 2: Stroh formalism and Phonon imaging The slowness surface defined by the Christoffel equation for bulk waves propagation in anisotropic media exhibits many interesting features. Because of the folds on the surface, the wave surface which defines the group velocity will have higher degree of freedom and complex geometry. The surface geometry of the wave surface can be either illustrated by the Gauss map of the slowness surface theoretically, or by the phonon focusing patterns experimentally in form of various sets of caustic lines. It is well known that the each sheet of the slowness surface can be divided in to concave/saddle and convex regions separating be the so-called parabolic line. The caustic lines are then simply the polar projection of the parabolic lines. In cubic media, however, the parabolic lines consist, in fact, of mainly inflection points on the slowness surface. The pure parabolic region on the slowness surface can only be found at limited points with respect to some symmetry planes. Such a parabolic region can be characterized by its vanishing transverse curvature normal to the symmetry planes. Some attempts have been made to identify the zero-transverse curvature in the symmetry planes using either direct deduction or the perturbation method the Christoffel equation. Still, the normal for the parabolic points, which produces many characteristic points in the phonon focusing patterns, have not been investigated analytically. The problem can also be better illustrated in the context of the phonon focusing patterns in the cubic crystals (see linker to the right). The main challenge here is then not only to identify the parabolic points but also to find the surface normal at the points and further to formulate the direction of the surface normal explicitly. In this investigation, we will confine ourselves to parabolic points located in the symmetry planes so that the surface normal can be represented by the curve normal due to the symmetry. In this project, inistead of calculating the Gaussian curvature, we apply the Stroh formalism for two-dimensional elastodynamics to the principal symmetry planes in cubic crystals. The determination of the parabolic points in the symmetry plane becomes identifying the socalled zero-curvature transonic states. The calculation of the surface normal is reduced to finding the degenerate Stroh eigenvalues within the symmetry plane, which in turn descibes the parabolic caustic points. It is important to realize that the parabolic line defined on the slowness surface consists of both the inflection points and the parabolic points. The signature along the symmetry planes can also be dived into two types: inflection points and parabolic points. As the inflection point results in cuspidal point on the wave surface, the parabolic point brings about a focal point on the wave surface. By recognizing the connection between the parabolic points and the degeneracy in the Stroh eigenvalue problem, we are able to obtain directions for the parabolic points in terms of two simple functions. The approach requires application of the Stroh formalism twice: first to resolve the zero transverse curvature transonic state and thereafter to find the surface normal at the transonic state. Together with the previous examination of the caustic/anticaustic points, we have successfully resolved analytically all the characteristic points in the phonon focusing pattern in the cubic media. The method outlined can also be applied to crystals with other symmetries. Although these characteristic points on the slowness surface were investigated in the context of the phonon imaging, it is expected that the results can be readily applied to other related field, such as determination of elastic constants, phonon dispersion, and so on. Some PowerPoint presentations:
Some demostrations:
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