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We investigate curved surfaces operating as geodesic lenses for elastic waves. Consistently with findings in optics, we show that wave propagation occurs along rays that correspond to the geodesics of the curved surfaces, and we establish the geometric equivalence between Gaussian curvature and refractive index. This equivalence is formulated for flexural waves in curved shells by showing that, in the short wavelength limit, the ray equation corresponds to the classical equation of geodesics. We leverage this result to identify a non-Euclidean transformation that maps the geometric profile of a isotropic curved waveguide into a spatially varying refractive index distribution for a planar waveguide. These theoretical predictions are validated first through numerical simulations, and subsequently through experiments on 3D printed curved membranes with different curvature distributions. Numerical and experimental findings confirm that focal regions and caustic networks are correctly predicted based on geodesic evaluations. Our results form the basis for the design of curved profiles that correspond to spatial distributions of the refractive index and induce focal points by forcing waves to propagate along predefined trajectories. The findings of this study also suggest curvature as an attractive alternative to strategies based on the local tailoring of material properties and geometrical patterns that have gained in popularity for gradient-index lens design. 

This work is centered on high-fidelity modeling, analysis, and rigorous experiments of vibrations and guided (Lamb) waves in a human skull in two connected tracks: (1) layered modeling of the cranial bone structure (with cortical tables and diploë) and its vibration-based elastic parameter identification (and validation); (2) transcranial leaky Lamb wave characterization experiments and radiation analyses using the identified elastic parameters in a layered semi analytical finite element framework, followed by time transient simulations that consider the inner porosity as is. In the first track, non-contact vibration experiments are conducted to extract the first handful of modal frequencies in the auditory frequency regime, along with the associated damping ratios and mode shapes, of dry cranial bone segments extracted from the parietal and frontal regions of a human skull. Numerical models of the bone segments are built with a novel image reconstruction scheme that employs microcomputed tomographic scans to build a layered bone geometry with separate homogenized domains for the cortical tables and the diploë. These numerical models and the experimental modal frequencies are then used in an iterative parameter identification scheme that yields the cortical and diploic isotropic elastic moduli of each domain, whereas the corresponding densities are estimated using the total experimental mass and layer mass ratios obtained from the scans. With the identified elastic parameters, the average error between experimental and numerical modal frequencies is less than 1.5% and the modal assurance criterion values for most modes are above 0.90. Furthermore, the extracted parameters are in the range of the results reported in the literature. In the second track, the focus is placed on the subject of leaky Lamb waves, which has received growing attention as a promising alternative to conventional ultrasound techniques for transcranial transmission, especially to access the brain periphery. Experiments are conducted on the same cranial bone segment set for leaky Lamb wave excitation and radiation characterization. The degassed skull bone segments are used in submersed experiments with an ultrasonic transducer and needle hydrophone setup for radiation pressure field scanning. Elastic parameters obtained from the first track are used in guided wave dispersion simulations, and the radiation angles are accurately predicted using the aforementioned layered model in the presence of fluid loading. The dominant radiation angles are shown to correspond to guided wave modes with low attenuation and a significant out-of-plane polarization. The experimental radiation spectra are finally compared against those obtained from time transient finite element simulations that leverage geometric models reconstructed from microcomputed tomographic scans.