Homogenization of the wave equation with non-uniformly oscillating coefficients

The focus of our work is a dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g., functionally graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in [Formula: see text], [Formula: see text], within the framework of multiple scales expansion. When either [Formula: see text] or [Formula: see text], this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

Age-related changes in the viscoelasticity of rabbit lens characterised by surface wave dispersion analysis

The viscoelastic properties of the young and mature rabbit lenses in situ are evaluated using wave-based optical coherence elastography (OCE). Surface waves in the crystalline lens are generated using acoustic radiation force (ARF) focused inside the eyeball. Surface-wave dispersion is measured with a phase-stabilised optical coherence tomography (OCT) system. The Young's modulus and shear viscosity coefficient are quantified based on a Scholte wave model. The results show that both elasticity and viscosity are significantly different between the young and mature lenses. The Young's modulus of the lenses increased with age from 7.74 ± 1.56 kPa (young) to 15.15 ± 4.52 kPa (mature), and the shear viscosity coefficient increased from 0.55 ± 0.04 Pa s (young) and 0.86 ± 0.13 Pa s (mature). It is shown that the combination of ARF excitation, OCE imaging, and dispersion analysis enables nondestructive quantification of lenticular viscoelasticity in situ and shows promise for in vivo applications.

Wave dispersion and dissipation in landfast ice: comparison of observations against models

Observations of wave dissipation and dispersion in sea ice are a necessity for the development and validation of wave–ice interaction models. As the composition of the ice layer can be extremely complex, most models treat the ice layer as a continuum with effective, rather than independently measurable, properties. While this provides opportunities to fit the model to observations, it also obscures our understanding of the wave–ice interactive processes; in particular, it hinders our ability to identify under which environmental conditions these processes are of significance. Here, we aimed to reduce the number of free variables available by studying wave dissipation in landfast ice. That is, in continuous sea ice, such as landfast ice, the effective properties of the continuum ice layer should revert to the material properties of the ice. We present observations of wave dispersion and dissipation from a field experiment on landfast ice in the Arctic and Antarctic. Independent laboratory measurements were performed on sea ice cores from a neighboring fjord in the Arctic to estimate the ice viscosity. Results show that the dispersion of waves in landfast ice is well described by theory of a thin elastic plate, and such observations could provide an estimate of the elastic modulus of the ice. Observations of wave dissipation in landfast ice are about an order of magnitude larger than in ice floes and broken ice. Comparison of our observations against models suggests that wave dissipation is attributed to the viscous dissipation within the ice layer for short waves only, whereas turbulence generated through the interactions between the ice and waves is the most likely process for the dissipation of wave energy for long periods. The separation between short and long waves in this context is expected to be determined by the ice thickness through its influence on the lengthening of short waves. Through the comparison of the estimated wave attenuation rates with distance from the landfast ice edge, our results suggest that the attenuation of long waves is weaker in comparison to short waves, but their dependence on wave energy is stronger. Further studies are required to measure the spatial variability of wave attenuation and measure turbulence underneath the ice independently of observations of wave attenuation to confirm our interpretation of the results.