Conformal Fourier wavenumber decompositions on continuous differentiable surfaces.

In this manuscript, a method is introduced for the evaluation of Fourier wavenumber decompositions on C(1) vibrating surfaces for spatial-spectral analysis. Whereas typical Fourier analysis is restricted to geometries that are separable for meaningful interpretations of the corresponding wave motion, this approach allows for conformal spectral analysis along curved surfaces. This is accomplished by restricting the wavevectors to lie within the local tangent to the surface and to be spatially dependent. The theoretical development is presented and it is demonstrated that commonly utilized kernels appropriate for some simple geometries can be recovered. Additionally, this approach is applied in the analysis of the vibration and radiation of a point driven, fluid loaded cone, where the displacements and pressures have been obtained using the finite element method.

Determination and analysis of guided wave propagation using magnetic resonance elastography.

We present a novel extension of standard magnetic resonance elastography (MRE) measurement and analysis methods, which is applicable in cases where the medium is characterized by waveguides or fiber bundles (i.e., muscle) leading to constrained propagation of elastic waves. As a demonstration of this new method, MRI is utilized to identify the pathways of the individual fibers of a stalk of celery, and 3D MRE is then performed throughout the volume containing the celery fibers for a measurement of the displacements. A Helmholtz decomposition is performed permitting a separation of the displacements into longitudinal and transverse components, and an application of a hybrid Radon transform permits a spectral decomposition of wave propagation along the fibers. Dot product projections between these elastic displacements measured in the global coordinate system and three Frenet vectors representing the tangent and two corresponding orthogonal vectors along any particular fiber orientation yield the displacement contributions to wave propagation along the fiber as if it were a waveguide. A sliding window spatial Fourier transform is then performed along the length of each fiber to obtain dispersion images that portray space-wavenumber profiles. Therefore, this method can permit localized tracking and characterization of wave types, velocities, and coupling along arbitrarily oriented fibers.