Nonlinear Acoustic Pulses Research Articles

Nonlinear pulses at the interface and its relation to state and temperature

.Environmental temperature has a well-conserved effect on the pulse velocity and excitability of excitable biological systems. The consistency suggests that the cause originates from a fundamental principle. A physical (hydrodynamic) approach has proposed that the thermodynamic state of the hydrated interface (e.g., plasma membrane) determines the pulse behavior. This implies that the temperature effect happens because the environmental temperature affects the state of the interface in any given system. To test the hypothesis, we measured temperature-dependent phase diagrams of a lipid monolayer and studied the properties of nonlinear acoustic pulses excited along the membrane. We observed that the membrane in the fluid-gel transition regime exhibited lower compressibility (i.e., stiffer) overall with increasing temperature. Nonlinear pulses excited near the transition state propagated with greater velocity with increasing temperature, and these observations were consistent with the compressibility profiles. Excitability was suppressed significantly or ceased completely when the state departed too far from the transition regime either by cooling or by heating. The overall correlation between the pulses in the membrane and in living systems as a function of temperature supports the view that the thermodynamic state of the interface and phase transition are the key to understanding pulse propagation in excitable systems.Graphical abstract

Similarities between action potentials and acoustic pulses in a van der Waals fluid

An action potential is typically described as a purely electrical change that propagates along the membrane of excitable cells. However, recent experiments have demonstrated that non-linear acoustic pulses that propagate along lipid interfaces and traverse the melting transition, share many similar properties with action potentials. Despite the striking experimental similarities, a comprehensive theoretical study of acoustic pulses in lipid systems is still lacking. Here we demonstrate that an idealized description of an interface near phase transition captures many properties of acoustic pulses in lipid monolayers, as well as action potentials in living cells. The possibility that action potentials may better be described as acoustic pulses in soft interfaces near phase transition is illustrated by the following similar properties: correspondence of time and velocity scales, qualitative pulse shape, sigmoidal response to stimulation amplitude (an ‘all-or-none’ behavior), appearance in multiple observables (particularly, an adiabatic change of temperature), excitation by many types of stimulations, as well as annihilation upon collision. An implication of this work is that crucial functional information of the cell may be overlooked by focusing only on electrical measurements.

It sounds like an action potential: unification of electrical, chemical and mechanical aspects of acoustic pulses in lipids.

In an ongoing debate on the physical nature of the action potential (AP), one group adheres to the electrical model of Hodgkin and Huxley, while the other describes the AP as a nonlinear acoustic pulse propagating within an interface near a transition. However, despite remarkable similarities, acoustics remains a non-intuitive mechanism for APs for the following reason. While acoustic pulses are typically associated with the propagation of density, pressure and temperature variation, APs are most commonly measured electrically. Here, we show that this discrepancy is lifted upon considering the electrical and chemical aspects of the interface, in addition to its mechanical properties. Specifically, we demonstrate how electrical and pH aspects of acoustic pulses emerge from an idealized description of the lipid interface, which is based on classical physical principles and contains no fit parameters. The pulses that emerge from the model show similarities to APs not only in qualitative shape and scales (time, velocity and voltage), but also demonstrate saturation of amplitude and annihilation upon collision.

Collision and annihilation of nonlinear sound waves and action potentials in interfaces.

Nerve impulses, previously proposed as manifestations of nonlinear acoustic pulses localized at the plasma membrane, can annihilate upon collision. However, whether annihilation of acoustic waves at interfaces takes place is unclear. We previously showed the propagation of nonlinear sound waves that propagate as solitary waves above a threshold (super-threshold) excitation in a lipid monolayer near a phase transition. Here we investigate the interaction of these waves. Sound waves were excited mechanically via a piezo cantilever in a lipid monolayer at the air-water interface and their amplitude is reported before and after a collision. The compression amplitude was observed via Förster resonance energy transfer between donor and acceptor dyes, measured at fixed points along the propagation path in the lipid monolayer. We provide direct experimental evidence for the annihilation of two super-threshold interfacial pulses upon head-on collision in a lipid monolayer and conclude that sound waves propagating in a lipid interface can interact linearly, nonlinearly, or annihilate upon collision depending on the state of the system. Thus we show that the main characteristics of nerve impulses, i.e. solitary character, velocity, couplings, all-or-none behaviour, threshold and even annihilation are also demonstrated by nonlinear sound waves in a lipid monolayer, where they follow directly from the thermodynamic principles applied to an interface. As these principles are equally unavoidable in a nerve membrane, our observations strongly suggest that the underlying physical basis of action potentials and the observed nonlinear-pules is identical.

Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators.

The nonlinear progressive wave equation (NPE) is a time-domain formulation of the Euler fluid equations designed to model low-angle wave propagation using a wave-following computational domain. The wave-following frame of reference permits the simulation of long-range propagation and is useful in modeling blast wave effects in the ocean waveguide. Existing models do not take into account frequency-dependent sediment attenuation, a feature necessary for accurately describing sound propagation over, into, and out of the ocean sediment. Sediment attenuation is addressed in this work by applying lossy operators to the governing equation that are based on a fractional Laplacian. These operators accurately describe frequency-dependent attenuation and dispersion in typical ocean sediments. However, dispersion within the sediment is found to be a secondary process to absorption and effectively negligible for ranges of interest. The resulting fractional NPE is benchmarked against a Fourier-transformed parabolic equation solution for a linear case, and against the analytical Mendousse solution to Burgers' equation for the nonlinear case. The fractional NPE is then used to investigate the effects of attenuation on shock wave propagation.

Nonlinear acoustic pulse propagation in ocean waveguides containing an elastic seabed

High intensity underwater sources, such as explosions, generate nonlinear finite-amplitude pulses that behave differently than linear acoustic pulses within shallow water waveguides. The nonlinearity is known to decrease the critical angle for total internal reflection from that of the linear case when the seafloor is approximated as a fluid. However, this result has not been extensively studied for elastic seafloors where shear waves are present. In this work, a time-domain model is developed assuming an isotropic linear elastic bottom, inviscid water column, and allowing for nonlinear advective acceleration and a nonlinear equation of state. The model is numerically implemented using a high-order Godunov scheme and then benchmarked against tank experiment data for the linear case. The nonlinear model is used to study the critical grazing angle for ocean bottoms of varying shear speeds to determine the combined effect of nonlinearity and elasticity on bottom penetration.

Detection of nonlinear picosecond acoustic pulses by time-resolved Brillouin scattering

In time-resolved Brillouin scattering (also called picosecond ultrasonic interferometry), the time evolution of the spatial Fourier component of an optically excited acoustic strain distribution is monitored. The wave number is determined by the momentum conservation in photon-phonon interaction. For linear acoustic waves propagating in a homogeneous medium, the detected time-domain signal of the optical probe transient reflectivity shows a sinusoidal oscillation at a constant frequency known as the Brillouin frequency. This oscillation is a result of heterodyning the constant reflection from the sample surface with the Brillouin-scattered field. Here, we present an analytical theory for the nonlinear reshaping of a propagating, finite amplitude picosecond acoustic pulse, which results in a time-dependence of the observed frequency. In particular, we examine the conditions under which this information can be used to study the time-evolution of the weak-shock front speed. Depending on the initial strain pulse parameters and the time interval of its nonlinear transformation, our theory predicts the detected frequency to either be monotonically decreasing or oscillating in time. We support these theoretical predictions by comparison with available experimental data. In general, we find that picosecond ultrasonic interferometry of nonlinear acoustic pulses provides access to the nonlinear acoustic properties of a medium spanning most of the GHz frequency range.

Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low-angle wave propagation using a wave-following computational domain [ McDonald et al., J. Acoust. Soc. Am. 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. However, the current model does not take into account sediment attenuation, a feature necessary for accurately describing sound propagation into and out of the ocean sediment. These attenuating, dispersive sediments are naturally captured with linear, frequency-domain solutions through use of complex wavespeeds, but a comparable treatment is nontrivial in the time-domain. Recent developments in fractional loss operator methods allow for frequency-dependent loss mechanisms to be applied in the time-domain providing physically realistic results [Prieur et al., J. Acoust. Soc. Am. 130]. Using these approaches, the governing equations used to describe the NPE are modified to use fractional derivatives in order to develop a fractional NPE. The updated model is then benchmarked against a Fourier-transformed parabolic equation solution for the linear case using various sediment attenuation factors.

Nonlinear acoustic pulse propagation in range-dependent underwater environments

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low angle wave propagation using a wave-following computational domain [B. E. McDonald et al., JASA 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. The standard formulation consists of four separate mathematical quantities that physically represent refraction, nonlinear steepening, radial spreading, and diffraction. The latter two of these effects are linear whereas the steepening and refraction are nonlinear. This formulation recasts pressure, density, and velocity into a single variable, a dimensionless pressure perturbation, which allows for greater efficiency in calculations. Nonlinear effects such as weak shock formation are accurately captured with the NPE. The numerical implementation is a combination of two numerical schemes: a finite-difference Crank-Nicholson algorithm for the linear terms of the NPE and a flux-corrected transport algorithm for the nonlinear terms. While robust, solutions are not available for sloping seafloors. In this work, range-dependent environments, characterized by sloping bathymetry, are investigated and benchmarked using a rotated coordinate system approach.

Solitary acoustic pulses in quantum semiconductor plasmas

We report properties of solitary acoustic pulses that propagate in electron-hole quantum semiconductor plasmas. We show that the dynamics of nonlinear acoustic pulses is governed by the Korteweg–de Vries equation, which includes contributions of the electron and hole quantum recoil effects, quantum statistical pressures of the plasma species, as well as exchange and correlation effects. Typical values for GaAs, GaSb, GaN, and InP semiconductors are used to estimate the speed and profiles of solitary acoustic pulses. The nonlinear solitary pulses depict intrinsic localization of electrostatic wave energies in semiconductor plasmas.

Finite-amplitude acoustic pulses in a waveguide layer with longitudinal inhomogeneity

This paper considers the propagation of a weakly nonlinear acoustic pulse in a slightly curved waveguide layer which is strongly inhomogeneous in the transverse direction and weakly inhomogeneous in the longitudinal direction. The basic system of hydrodynamic equations reduces to a nonlinear wave equation, whose coefficients are determined using the equations of state of the medium. It is established that as the adiabatic exponent passes through the value γ = 3/2, the nature of the pulse propagation changes: for large values of γ, the medium is focusing, and for smaller values, it is defocusing. It is shown that the pulse propagation process is characterized by three scales: the high-frequency filling is modulated by the envelope, whose evolution, in turn, is determined by the moderate-rate evolution of the envelope phase and slow amplitude variation. A generalized nonlinear Schrodinger equation with the coefficients dependent on the longitudinal coordinate is derived for the pulse envelope. An explicit soliton solution of this equation is constructed for some types of longitudinal inhomogeneity.

Finite Volume Formulation of Compact Upwind and Central Schemes with Artificial Selective Damping

The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes. The different schemes are analyzed and tested on several numerical experiments. A new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes is presented. Results are shown for a 1D advection equation, a 2D rotating Gaussian pulse and a subsonic inviscid vortical flow on uniform and non-uniform meshes and for a non-linear acoustic pulse.

Increase in the efficiency of the shear wave generation in gelatin due to the nonlinear absorption of a focused ultrasonic beam

Experimental results and theoretical estimates are presented to demonstrate the prospects of using the acoustic nonlinearity of a gel-like medium for increasing the efficiency of the shear wave generation in it by a pulsed ultrasonic beam. The experiment is based on the propagation of a focused beam of longitudinal acoustic waves at a frequency of 1.1 MHz in a gelatin sample and on the detection of shear waves by the optical method [1]. It is demonstrated that the amplitude of the shear wave excited by a nonlinear acoustic pulse can be increased by an order of magnitude owing to the formation of shock fronts in the profile of this pulse.

Possibility of controlled ejection of ferrofluid grains from a magnetically ordered ferrofluid using high frequency non-linear acoustic pulses – a particle dynamical study

We consider a model dilute ferrofluid that is subjected to a strong, homogeneous magnetic field directed perpendicular to the surface of the ferrofluid, such that there is a chain formation in the direction perpendicular to the surface of the liquid. We study the propagation of impulses generated at high-frequency across finite times through the ferrofluid chains. Our numerical analysis shows that a very high-frequency sequence of non-linear acoustic pulses of appropriate magnitudes, initiated at the base of the container, can lead to the ejection of desired number of ferrofluid grains through the liquid–air interface. The proposed mechanism, if successfully realized in the laboratory, could help design a nozzle-free, ultrafast, ink-jet printer of unparalleled resolution.

Influence of relaxation on the profile of a nonlinear acoustic pulse

The influence of transverse relaxation on a nonlinear acoustic wave generated in the presence of the acoustic self-induced transparency (SIT) phenomenon is investigated. An explicit analytical equation for the profile of the nonlinear acoustic wave is derived by a perturbation theory developed on the basis of the inverse scattering problem. This equation provides a possible means for investigating experimentally the profile of an acoustic SIT pulse with allowance for relaxation effects, so that these results can be used to construct new types of acoustoelectronic devices.

Axi-symmetric nonlinear waves in two-dimensional anharmonic lattices

We consider two-dimensional elastic nonlinear waves with cylindrical symmetry. We show that the propagation of axisymmetric nonlinear acoustic pulses is governed by a radial Boussinesq equation which is a cylindrical analogue of the one-dimensional Boussinesq equation. We investigate soliton-like and self-similar solutions and we show that there is a connection between the radial Boussinesq equation and the cylindrical KdV equation.

Time-domain solution of the parabolic equation including nonlinearity

This paper develops a two-dimensional (range and depth) formulation for small-angle propagation of nonlinear acoustic pulses and weak shocks in a refracting medium which can be range dependent. The formulation readily extends to three dimensions. The nonlinearity is most efficiently treated in the time domain, so the signal is not Fourier decomposed into component frequencies. Beginning with a second-order wave equation which includes the lowest-order nonlinearity, an approximate first-order nonlinear progressive wave equation (NPE) is derived. The NPE is the nonlinear time-domain counterpart of the linear frequency-domain parabolic equation (PE). The derivation is accomplished by transforming the wave equation to a pulse-following frame and perturbing about a unidirectional plane-wave signal. The NPE manifests a natural separation of terms governing refraction, diffraction, spreading and nonlinear steepening. Numerical methods are outlined for the solution of the time-domain problem. Calculations using the formulation developed here successfully follow the development of initially smooth pulses into N waves and the reflection of weak shocks from a caustic.

Time domain approach to the propagation of nonlinear acoustic pulses near caustics

Nonlinear acoustic ray tracing in the ocean fails whenever a wave front converges to a caustic of focus, and neighboring rays intersect. It is the purpose of this paper to construct a numerical algorithm for dealing with propagation of large amplitude pulses in the neighborhood of a caustic. A time‐straining transformation of the nonlinear acoustic equation is introduced that enables a perturbation expansion of the pressure in powers of signal strength to be attempted. An essential ingredient of the procedure is the knowledge of the solution of the corresponding linearized problem. An efficient procedure for calculating this for incident pulses of N waveform of arbitrary two‐dimensional wave front geometry is described. The corresponding calculation of the nonlinear time straining is also presented. The procedure could readily be adapted to pulses of more general shape. [Work done in collaboration with Applied Research Laboratories, University of Texas at Austin, and supported by ONR.]

Self Refraction of Nonlinear Ion Acoustic Waves

Self refraction and focussing of nonlinear ion acoustic pulses with transverse amplitude inhomogeneity are studied experimentally. The nonlinearity causes self-refraction, however it does not last further since the amplitude inhomogeneity is weakened by diffraction. The nonlinear rarefactive pulse tends to focus under a certain condition but it is also interrupted to last by both dispersion and diffraction.

Integrated grating acoustic scanner

This letter describes a new type of electronically scanned and focused acoustic imaging device called the integrated grating acoustic scanner. In this device, a nonlinear chirped surface acoustic wave pulse expansion/compression filter and an array of grooves were integrated on the same piece of piezoelectric single-crystal plate. The periodical grooves scattered a chirped surface-wave pulse coherently into a focused bulk-wave spot, which scans at the surface-wave velocity. A 150-MHz integrated LiNbO3 grating scanner with 1-cm field of view and 40-μm resolution has been constructed and studied.