John M. Cormack

Two-compartment passive frequency domain cochlea model allowing independent fluid coupling to the tectorial and basilar membranes

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.

Effect of basilar and tectorial membrane properties and gradients on cochlear response

The cochlea is a spiral-shaped, fluid-filled organ in the inner ear that converts sound with high resolution over a large frequency range to neurological signals that can then be interpreted by the brain. The organ of Corti, supported below by the basilar membrane and attached above to the tectorial membrane, plays a major role in the amplification of small signals. In early fluid-structure interaction models of the cochlea, the mechanical properties of the organ of Corti were neglected and only the basilar membrane was considered, approximated by a series of springs. Recent experiments suggest that the mechanical properties and property gradients of the tectorial membrane may also be important for frequency response of the organ of Corti and that separate waves may propagate along the basilar and tectorial membranes. Therefore, a two-dimensional two-chamber finite difference model of the cochlea was developed to investigate the independent responses of the basilar and tectorial membranes. Responses are c...

Nonlinear response of a relaxing shear wave resonator to elliptical driving motion

Soft elastic media such as rubber and soft tissue possess small shear moduli, facilitating the generation of large shear deformations. These materials may also exhibit viscoelasticity such as stress relaxation over the frequency range of interest. An augmented form of the Duffing equation was recently developed to model the response near the lowest resonance of a shear wave resonator formed with a nonlinear relaxing material that is shaken at one end and free at the other [J. Acoust. Soc. Am. 143, 1035 (2018)]. The augmented Duffing model was found to accurately describe the response of the resonator when the driving motion is linearly polarized. Here the model is extended to account for elliptical driving motion at frequencies near the lowest resonance. The augmented Duffing model in this case consists of two coupled ordinary differential equations for the two displacement components. Amplitude-dependent phenomena such as amplification of the minor displacement component and an induced phase shift betwee...

Laboratory study of linear and nonlinear elastic pulse propagation in a sandstone bar

Elastic wave propagation experiments were performed on a Texas moss sandstone sample, saw cut into a rectangular bar, some 8 cm square by 1.75 m long, in order to study linear and nonlinear effects in both the time and frequency domains. Prior studies, mostly in the frequency domain, have been a guide to the present work. Several means of exciting sound propagation in the sample were studied, including impact excitation with a pendulum hammer. Signal reception was performed with a laser Doppler vibrometer. As determined by arrival times of reflected pulses, conditioning of the bar by an initial pulse of finite amplitude was observed to reduce the effective bulk modulus by an order of magnitude more than the corresponding reduction inferred from reported measurements of bar resonance in the presence of a pump wave. Also considered is pulse attenuation as a function of initial amplitude.Elastic wave propagation experiments were performed on a Texas moss sandstone sample, saw cut into a rectangular bar, some 8 cm square by 1.75 m long, in order to study linear and nonlinear effects in both the time and frequency domains. Prior studies, mostly in the frequency domain, have been a guide to the present work. Several means of exciting sound propagation in the sample were studied, including impact excitation with a pendulum hammer. Signal reception was performed with a laser Doppler vibrometer. As determined by arrival times of reflected pulses, conditioning of the bar by an initial pulse of finite amplitude was observed to reduce the effective bulk modulus by an order of magnitude more than the corresponding reduction inferred from reported measurements of bar resonance in the presence of a pump wave. Also considered is pulse attenuation as a function of initial amplitude.

Subharmonic generation by shear waves in a 1D resonator formed with a relaxing material

The low shear moduli of soft elastic media such as rubber and tissue facilitate the excitation of shear waves that exhibit significant finite-amplitude effects. Measurements performed by Andreev et al. [Acoust. Phys. 57, 779 (2011)] of the amplitude-dependent response of a 1D shear-wave resonator formed with a rubber-like material revealed reasonable agreement with numerical simulations based on a monorelaxing material model. At the previous ASA meeting the present authors developed an augmented Duffing equation as a model for plane shear waves in a resonator formed with a monorelaxing material that is shaken at one end and free at the other [J. Acoust. Soc. Am. 142, 2723 (2017)]. The focus of the previous work was on the response at drive frequencies near the lowest resonance. Here the augmented Duffing model is used to investigate subharmonic generation associated with a drive frequency of approximately three times that of the lowest mode. Conditions on the amplitude and frequency of excitation for whic...

Transient solution for the directional response at the focus of a paraboloidal reflector

A transient solution is presented for the directional response at the focus of a paraboloidal reflector due to an incident plane wave. The validity of the solution is associated with the Kirchhoff approximation used to determine the boundary condition on the surface of the reflector. This approximation requires the wavelengths to be short compared with the minimum radius of curvature, and the radius of the aperture, of the reflector. The solution in the far field due to a point source at the focus is related by reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of a function obtained for the unit step response with the time derivative of the waveform incident on the reflector. For a point source far removed from a shallow paraboloidal reflector, it is shown that the pressure in the far field reduces to the expression obtained by Morse (Vibration and Sound) for transient radiation from a baffled circular piston. Results are presented illustrating the angular dependence of the reflected pressure waveforms at the focus for incident plane waves that include an N wave and a tone burst.A transient solution is presented for the directional response at the focus of a paraboloidal reflector due to an incident plane wave. The validity of the solution is associated with the Kirchhoff approximation used to determine the boundary condition on the surface of the reflector. This approximation requires the wavelengths to be short compared with the minimum radius of curvature, and the radius of the aperture, of the reflector. The solution in the far field due to a point source at the focus is related by reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of a function obtained for the unit step response with the time derivative of the waveform incident on the reflector. For a point source far removed from a shallow paraboloidal reflector, it is shown that the pressure in the far field reduces to the expression obtained by Morse (Vibration and Sound) for transient radiation from a baffled circular piston. Results are presented illu...

Cylindrically converging nonlinear shear waves

The low shear moduli of soft elastic media permit the generation of shear waves with large acoustic Mach numbers that can exhibit waveform distortion and even shock formation over short distances. Waves that converge onto a cylindrical focus experience significant dispersion, causing waveforms at the focus and in the post-focal region to differ significantly from the source waveform even in the absence of nonlinear distortion. A full-wave model for nonlinear shear waves in cylindrical coordinates that accounts for both quadratic and cubic nonlinearity is developed from first principles. For the special case of an infinite cylindrical source with particle motion parallel to the axis, for which nonlinearity is purely cubic, the nonlinear wave equation is solved numerically with a finite-difference scheme. The full-wave model is compared with a piecewise model based on a generalized Burgers equation for cylindrically converging waves outside of the focal region and linear diffraction theory in the focal regi...

Nonlinear shear wave resonator consisting of a relaxing material

Soft materials such as rubbers, polymers, and tissue exhibit low shear wave speeds, facilitating the generation of shear waves with large acoustic Mach numbers. In addition to finite-amplitude effects that result from cubic nonlinearity, plane shear wave propagation in these materials is subject to frequency-dependent attenuation and dispersion that result from viscoelastic effects. A wave equation for plane shear waves in a relaxing material is obtained from a nonlinear Zener constitutive model that accounts for cubic nonlinearity as well as the attenuation and dispersion associated with relaxation. The wave equation is used to analyze a one-dimensional shear wave resonator comprised of a nonlinear relaxing material that is shaken at one end and free at the other. For excitation of the lowest mode the wave equation is approximated by an augmented Duffing equation, and the resulting frequency-response equation is compared with numerical finite-difference solutions of the original wave equation. Frequency ...

Ultrasonic characterization of the complex Young's modulus of polymers produced with micro-stereolithography

Micro-stereolithography is capable of producing polymer parts on the scale of millimeters that contain micron-scale features. Accurate prediction of the dynamic performance of components produced using micro-sterolithography requires knowledge of the as-fabricated material properties, but very little published material property data exist for these materials. This is complicated by the fact that the properties vary as a function of build parameters (i.e., laser exposure time). Designers therefore have limited useful material property information for part design. Frequency-dependent material parameters are often determined by measuring the wave speed and attenuation of an ultrasonic pulse as it propagates through the material. This work employs laser Doppler vibrometry to detect extensional waves in a solid polymer rod of circular cross-section that is excited by a longitudinal ultrasonic contact transducer. Transverse motion associated with extensional waves propagating along the rod axis is measured at m...

Prediction of multivalued waveforms in media with power-law attenuation

The lossless Burgers equation predicts a multivalued waveform beyond a certain propagation distance. Inclusion of thermoviscous attenuation, which increases as frequency squared, prevents the occurrence of multivalued waveforms. The same is true for any attenuation law that is proportional to frequency raised to an exponent greater than unity. For exponents less than unity the situation is less clear. For example, when attenuation is constant with frequency (exponent equal zero) there is a critical value of the attenuation coefficient below which a multivalued waveform is predicted and above which it is not. To investigate the prediction of multivalued waveforms for power-law attenuation with exponents between zero and unity, a Burgers equation with the loss term expressed as a fractional derivative is used [Prieur and Holm, J. Acoust. Soc. Am. 130, 1125 (2011)]. Transformation of the equation into intrinsic coordinates following Hammerton and Crighton [J. Fluid Mech. 252, 585 (1993)] permits numerical so...