The cochlea is a spiral-shaped, liquid-filled organ in the inner ear that converts sound with high frequency selectivity over a wide pressure range to neurological signals that are eventually interpreted by the brain. The cochlear partition, consisting of the organ of Corti supported below by the basilar membrane and attached above to the tectorial membrane, plays a major role in the frequency analysis. In early fluid-structure interaction models of the cochlea, the mechanics of the cochlear partition were approximated by a series of single-degree-of-freedom systems representing the distributed stiffness and mass of the basilar membrane. Recent experiments suggest that the mechanical properties of the tectorial membrane may also be important for the cochlea frequency response and that separate waves may propagate along the basilar and tectorial membranes. Therefore, a two-dimensional two-compartment finite difference model of the cochlea was developed to investigate the independent coupling of the basilar and tectorial membranes to the surrounding liquid. Responses are presented for models using two- or three-degree-of-freedom stiffness, damping, and mass parameters derived from a physiologically based finite element model of the cochlear partition. Effects of changes in membrane and organ of Corti stiffnesses on the individual membrane responses are investigated.
Nonlinear wave equations are obtained for the two plane shear wave modes in a transversely isotropic soft solid. The material is modeled using a general expansion of the strain energy density up to fourth order in strain. Whereas, in an isotropic soft solid, leading order nonlinearity for plane wave propagation appears at cubic order in strain, elastic anisotropy in a transversely isotropic material introduces nonlinear effects at quadratic order, including interaction between the modes of a wave with two displacement components. Expressions for second harmonic generation in an elliptically polarized wave field illustrate the low efficiency of nonlinear interactions between the two displacement components, which results from the disparity between propagation speeds of the two shear wave modes. Coupled wave equations with up to cubic nonlinearity are presented and then approximated to describe linearly polarized waves by neglecting interaction between modes. Evolution equations are obtained for linearly polarized progressive waves, and explicit expressions are given in terms of elastic moduli and propagation direction for the coefficients of leading order nonlinearity. Expressions are presented for up to third harmonic generation from a time-harmonic source.
Model equations with cubic nonlinearity are developed for a plane shear wave of finite amplitude in a relaxing medium. The evolution equation for progressive waves is solved analytically for a jump in stress that propagates into an undisturbed medium. Weak-shock theory is used to determine the amplitude and location of the shock when the solution predicts a multivalued waveform. The solution is similar to that obtained by Polyakova, Soluyan, and Khokhlov [Sov. Phys. Acoust. 8, 78-82 (1962)] for a compressional wave with quadratic nonlinearity in a relaxing fluid. Numerical simulations illustrate the effect of relaxation on shock formation in an initially sinusoidal shear wave. The minimum source amplitude required for an initially sinusoidal waveform to develop shocks in a relaxing medium is determined as a function of the dispersion and relaxation time. Limiting forms of the evolution equation are considered, and analytical solutions incorporating weak-shock theory are presented in the high-frequency limit. A Duffing-type model for a nonlinear shear-wave resonator is developed and investigated.
Due to very low shear moduli for soft tissue or tissue-like media, shear waves propagate very slowly, on the order of meters per second, making it relatively easy to produce shear waves exhibiting waveform distortion and even shock formation. Finite amplitude effects in plane shear waves result from cubic nonlinearity, compared with quadratic nonlinearity in compressional waves. Both attenuation and dispersion also significantly affect propagation of shear waves in tissue. Here we account for these complex viscoelastic effects by considering a medium with one relaxation mechanism. An analytical solution similar to that of Polyakova, Soluyan, and Khokhlov [Sov. Phys. Acoust. 8, 78 (1962)] for a compressional wave with a step shock in a relaxing medium is obtained for a shear wave with a step shock in a relaxing medium. The wave profile with cubic nonlinearity closely resembles that with quadratic nonlinearity. For weak nonlinearity the solution reduces to an expression obtained by Crighton [J. Fluid Mech. 173, 625 (1986)] for a Taylor shock in a viscous medium with cubic nonlinearity. Numerical simulations are presented comparing shock formation with quadratic and cubic nonlinearity for other wave profiles in relaxing media. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.