On the KZK-type equation for modulated ultrasound fields

Abstract. A numerical model for long-term simulation of gravity surface waves is described. The model is designed as a component of a coupled Wave Boundary Layer/Sea Waves model, for investigation of small-scale dynamic and thermodynamic interactions between the ocean and atmosphere. Statistical properties of nonlinear wave fields are investigated on a basis of direct hydrodynamical modeling of 1-D potential periodic surface waves. The method is based on a nonstationary conformal surface-following coordinate transformation; this approach reduces the principal equations of potential waves to two simple evolutionary equations for the elevation and the velocity potential on the surface. The numerical scheme is based on a Fourier transform method. High accuracy was confirmed by validation of the nonstationary model against known solutions, and by comparison between the results obtained with different resolutions in the horizontal. The scheme allows reproduction of the propagation of steep Stokes waves for thousands of periods with very high accuracy. The method here developed is applied to simulation of the evolution of wave fields with large number of modes for many periods of dominant waves. The statistical characteristics of nonlinear wave fields for waves of different steepness were investigated: spectra, curtosis and skewness, dispersion relation, life time. The prime result is that wave field may be presented as a superposition of linear waves is valid only for small amplitudes. It is shown as well, that nonlinear wave fields are rather a superposition of Stokes waves not linear waves. Potential flow, free surface, conformal mapping, numerical modeling of waves, gravity waves, Stokes waves, breaking waves, freak waves, wind-wave interaction.

The modulation instability (MI) is a universal mechanism that is responsible for the disintegration of weakly nonlinear narrow-banded wave fields and the emergence of localized extreme events in dispersive media. The instability dynamics is naturally triggered, when unstable energy sidebands located around the main energy peak are excited and then follow an exponential growth law. As a consequence of four wave mixing effect, these primary sidebands generate an infinite number of additional sidebands, forming a triangular sideband cascade. After saturation, it is expected that the system experiences a return to initial conditions followed by a spectral recurrence dynamics. Much complex nonlinear wave field motion is expected, when the secondary or successive sideband pair that is created is also located in the finite instability gain range around the main carrier frequency peak. This latter process is referred to as higher-order MI. We report a numerical and experimental study that confirms observation of higher-order MI dynamics in water waves. Furthermore, we show that the presence of weak dissipation may counterintuitively enhance wave focusing in the second recurrent cycle of wave amplification. The interdisciplinary weakly nonlinear approach in addressing the evolution of unstable nonlinear waves dynamics may find significant resonance in other nonlinear dispersive media in physics, such as optics, solids, superfluids, and plasma.

The popular KZK equation has been widely used to model the nonlinear acoustic wave field. However, since it is based on a parabolic approximation, it has various limitations, both at close distances from the transducer and at large radial extents from the beam axis or, in case of phased arrays, at steering angles larger than, say, 20/spl deg/. The purpose of this work is to model the nonlinear acoustic wave field without using the parabolic approximation. The model is based on the full nonlinear wave equation known as the Westervelt equation. This equation is solved numerically using an explicit finite difference scheme with fourth order centred space differences and second order centered or second order backward time differences. To check the validity of our model, numerical results for the lossless linear case were compared to analytical results. Moreover, the nonlinear beam profiles were compared to the results of a numerical solution of the KZK equation and to hydrophone measurements performed with a single element, unfocused transducer with a radius of 6.35 mm and a centre frequency of 1 MHz. We used a pulsed excitation of 6 cycles with pressure levels of 40, 220 and 500 kPa. The axial beam profiles and the lateral beam profiles at several distances were measured. For a low drive level, our numerical predictions show a good agreement with the analytic solution as well as with the measured data for all fundamental frequency beam profiles. At higher excitations, our simulations give a significant second harmonic component at 15 db below the fundamental level, which is in agreement with the measurements. The numerical data reproduces all the features at the fundamental and harmonic frequency beam profiles. For regions where the KZK equation is known to be less accurate, our model performs better. The sidelobes created at larger observation angles (above 20/spl deg/) and the nearfield variations are accurately reproduced. The performance of our model is good since it predicts accurately the harmonic field at any position with respect to the transducer. It can be used for the whole scanning sector that is required for harmonic beam profile simulations for phased arrays.

The advantages of nonlinear ultrasound propagation are exploited in an ever increasing number of applications. As a consequence, much research is devoted to the development of numerical methods capable of modeling nonlinear pressure wave fields propagating through (inhomogeneous) biomedical tissue. These tools are essential to design and optimize ultrasound transducers and to investigate novel ultrasound modalities or devices. The iterative nonlinear contrast source (INCS) method is an accurate full-wave method for modeling three-dimensional nonlinear acoustic wave fields propagating through soft biomedical tissue containing arbitrary inhomogeneities in the coefficient of nonlinearity, attenuation and speed of sound. This method is directionally independent and can deal with large three-dimensional domains, measuring hundreds of wavelengths and periods, and scattering. Here, we present the first synthetic harmonic ultrasound images generated with INCS. The images are created using a delay-and-sum aperture focusing scheme. Results show the applicability of the INCS method as a simulator for ultrasound harmonic imaging systems.

Biomedical tissues usually show inhomogeneity in their acoustic medium parameters. These inhomogeneities cause refraction and scattering of diagnostic and therapeutic ultrasound waves. A method that is able to model the effects of inhomogeneity in the attenuation and in the nonlinearity is essential for the design of transducers for new ultrasound modalities and the development of novel ultrasound applications. The Iterative Nonlinear Contrast Source (INCS) method has originally been designed for the accurate modeling of nonlinear acoustic wave fields in homogeneous media. It considers the nonlinear term from the Westervelt equation as a distributed contrast source, and the corresponding integral equation is solved using an iterative Neumann scheme. This paper presents an extension of the INCS method that can handle inhomogeneity in the attenuation and in the coefficient of nonlinearity. Results are presented for the one-dimensional case. These show that in this case the presented method correctly predicts the effects related to nonlinear propagation and scattering by inhomogeneities in the attenuation and the coefficient of nonlinearity.

In the field of nonlinear waves, there exist two kinds of waves, i.e., one is Non-Coupled Simple Wave (nonlinear P wave) and Coupled Simple Wave (nonlinear S wave). In this paper, the problem is limitted to the two-dimensional case. After numerical computations, we have then found that the behavior of waves in the nonlinear wave field is governed by the theory of Simple Waves. Even complex wave form can be analyzed as composite Simple Waves.

Acoustic radiation force (ARF) creep imaging applies step ARF excitation to induce creep displacement of soft tissue, and the corresponding time-dependent responses are used to estimate soft tissue viscoelasticity or its contrast. Single degree of freedom (SDF) and homogeneous analytical models have been used to characterize soft tissue viscoelasticity in ARF creep imaging. The purpose of this study is to investigate the fundamental limitations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. In this paper, finite element (FE) models are developed to simulate the dynamic behavior of viscoelastic soft tissue subjected to step ARF. Both homogeneous and heterogeneous models are studied with different soft tissue viscoelasticity and ARF configurations. The results indicate that the SDF model can provide good estimations for homogeneous soft tissue with high viscosity, but exhibits poor performance for low viscosity soft tissue. In addition, a smaller focal region of the ARF is desirable to reduce the estimation error with the SDF models. For heterogeneous media, the responses of the focal region are highly affected by the local heterogeneity, which results in deterioration of the effectiveness of the SDF and homogeneous simplifications.

The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of transducers for novel ultrasound modalities. Starting from the Westervelt equation, nonlinear pressure wave fields can be modeled via a contrast source formulation, as has been done with the INCS method. An extension of this method will be presented that can handle inhomogeneities in the coefficient of nonlinearity. The contrast source formulation results in an integral equation, which is solved iteratively using a Neumann scheme. The convergence of this scheme has been investigatedfor relevant media (e.g., blood, brain, and liver). Further, as an example, the method has been applied to compute the 1D nonlinear acoustic wave field in an inhomogeneous medium insonified by a 1 MHz Gaussian pulse propagatingup to 100 mm. The results show that the method is able to predict the propagation and the scattering effects of nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity. This motivates a similar extension ofthe 3D INCS method.

The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of transducers for novel ultrasound modalities. Starting from the Westervelt equation, nonlinear pressure wave fields can be modeled via a contrast source formulation, as has been done with the INCS method. An extension of this method will be presented that can handle inhomogeneities in the coefficient of nonlinearity. The contrast source formulation results in an integral equation, which is solved iteratively using a Neumann scheme. The convergence of this scheme has been investigated for relevant media (e.g., blood, brain, and liver). Further, as an example, the method has been applied to compute the 1-D nonlinear acoustic wave field in an inhomogeneous medium insonified by a 1 MHz Gaussian pulse propagating up to 100 mm. The results show that the method is able to predict the propagation and the scattering effects of nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity. This motivates a similar extension of the 3-D INCS method.

Humans rely on the temporal processing ability of the auditory system to perceive speech during everyday communication. The temporal envelope of speech is essential for speech perception, particularly envelope modulations below 20 Hz. In the literature, the neural representation of this speech envelope is usually investigated by recording neural phase-locked responses to speech stimuli. However, these phase-locked responses are not only associated with envelope modulation processing, but also with processing of linguistic information at a higher-order level when speech is comprehended. It is thus difficult to disentangle the responses into components from the acoustic envelope itself and the linguistic structures in speech (such as words, phrases and sentences). Another way to investigate neural modulation processing is to use sinusoidal amplitude-modulated stimuli at different modulation frequencies to obtain the temporal modulation transfer function. However, these transfer functions are considerably variable across modulation frequencies and individual listeners. To tackle the issues of both speech and sinusoidal amplitude-modulated stimuli, the recently introduced Temporal Speech Envelope Tracking (TEMPEST) framework proposed the use of stimuli with a distribution of envelope modulations. The framework aims to assess the brain's capability to process temporal envelopes in different frequency bands using stimuli with speech-like envelope modulations. In this study, we provide a proof-of-concept of the framework using stimuli with modulation frequency bands around the syllable and phoneme rate in natural speech. We evaluated whether the evoked phase-locked neural activity correlates with the speech-weighted modulation transfer function measured using sinusoidal amplitude-modulated stimuli in normal-hearing listeners. Since many studies on modulation processing employ different metrics and comparing their results is difficult, we included different power- and phase-based metrics and investigate how these metrics relate to each other. Results reveal a strong correspondence across listeners between the neural activity evoked by the speech-like stimuli and the activity evoked by the sinusoidal amplitude-modulated stimuli. Furthermore, strong correspondence was also apparent between each metric, facilitating comparisons between studies using different metrics. These findings indicate the potential of the TEMPEST framework to efficiently assess the neural capability to process temporal envelope modulations within a frequency band that is important for speech perception.

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

A method of studying nonlinear random fields of surface gravity waves by direct numerical simulation is developed by combining the “high-order spectral method” of West et al. (J. Geophys. Res. 92 (1987) 11,803–11,824) and the complex amplitude function b(k) introduced by Zakharov (J. Appl. Mech. Tech. Phys. (Engl. Transl.) 2 (1968) 190–194) in his Hamiltonian formalism of the water wave problem. By computing the rate of change of b(k) for a given wave field by both the high-order spectral method and Zakharov's equation, it is confirmed that the high-order spectral method, when the order of nonlinearity is appropriately chosen, gives exactly the same result as Zakharov's equation within a much shorter computation time. Methods of evaluating the frequency spectrum, directional spectrum and the component phase speed are given. Results of some preliminary computations are also presented to prove the great potentialities of the present approach as a tool for the examination of various characteristics of nonlinear random wave fields with continuous spectra, such as ocean waves.

Purpose: To present a multimodal imaging system that combines three modalities, optical spectroscopy, ultrasonography and acoustic radiation force (ARF) for improved diagnosis of breast cancer based on noninvasive interrogation of vasculature. Method and Materials: The scanhead incorporates two 1MHz focused transducers aligned orthogonally to deliver ARF in a 2.4cm region. A commercial ultrasound probe is positioned to image the focal plane. The ultrasound probe and focused transducers have two degrees‐of‐freedom, so that the focal zone can be scanned in tissue depth‐wise and laterally. Three force sensors are mounted on the breast compression plate to ensure consistent, slight compression. A fiber array (6 sources and 12 detectors) connects to the concave bottom plate of the scanhead. Sources are illuminated by two laser diodes (680 and 830nm) through a multi‐channel optical switch. Diffuse reflectance signals are collected by 12 detectors, amplified by avalanche photodiodes, and transmitted through a multi‐channel data acquisition card for analysis. The scanhead is mounted on a 360° ring gantry, which can rotate around the breast and move towards or away from the patient. A software system was implemented in Visual C++ to perform real‐time tasks, including acquisition of optical and force signals, control of optical switch and function generator, and data display. Results: The maximum exposure to laser was controlled within 0.2W⋅cm−2 (ANSI Z136.1). The ARF fields were maintained below FDA diagnostic limits (0.72W⋅cm−2) for manipulating blood flow and oxyhemoglobin concentrations at the time and location of detection, causing measurable differences in the dynamic behavior of tissue blood supply as reported by optical spectroscopy. This technique was demonstrated to be highly diagnostic in a murine tumor model. Conclusions: A multimodal system incorporating ARF, optical spectroscopy and ultrasonography has been developed to characterize breast cancers. Pilot clinical study is being carried out.Acknowledgement: Work supported by NCI Grant CA107860.

Previous investigations of the relativistic Weibel instability provide motivation to consider the nonlinear domain because, for asymmetric particle distributions, there is only an isolated unstable Weibel mode—reminiscent of nonlinear wave-types of behavior. From the collisionless Boltzmann equation together with Maxwell’s equations, a nonlinear, self-consistent wave equation is derived that is solvable for a broad range of distribution functions. For monochromatic electrons the nonlinear equation can be solved exactly, but results in an unphysical behavior of the magnetic field due to the compact support required of the distribution function. The general equation can be solved by asymptotic representation producing physically correct nonlinear wave solutions over bounded domains with varying internal structure of the electric and magnetic fields that range from nearly Gaussian to “sawtooth” in shape. A lower limit on the nonlinear wave amplitude is required in order that the nonlinear wave be of limited extent and so not represent a sinusoidal disturbance with no bounding domain. Limits for the nonlinear wave maximum magnetic field, and particle number density within the nonlinear wave, are given by considering the constraints on the nonlinear wave due to radiation processes, electron collision effects, and electron degeneracy pressure. The basic physical scale results are depicted mostly conducive for astrophysical applications involving relativistic flows and γ-ray emission, for which detailed investigations will be given elsewhere.

1. Acoustic communication requires not only detection of the signal but also discrimination of differences among signals by the receiver. Attenuation and degradation of acoustic signals during transmission through the atmosphere will impose limits on acoustic communication. Attenuation of sound during atmospheric transmission results primarily from atmospheric absorption, ground attenuation, scattering of a sound beam, and deflection of sound by stratified media. For maximum range of detection, therefore, animals should favor optimal positions in their habitat and optimal weather conditions. Frequency-dependent attenuation seems not to differ consistently among major classes of terrestrial habitats, such as forests and fields. Increased scattering of higher frequencies from vegetation in forests is in part matched by scattering from micrometerological heterogeneities in the open. 2. In addition to frequency-dependent attenuation, two kinds of degradation during atmospheric transmission will limit a receiver's ability to resolve differences among acoustic signals: the accumulation of irregular amplitude fluctuations from nonstationary heterogeneities, often atmospheric turbulence, and reverberation. Both types of degradation affect temporal patterns of amplitude or intensity modulation more than patterns of frequency modulation. Both effects should increase with carrier frequency, as they depend on the relationship between wavelength and the dimensions of scattering heterogeneities. Irregular amplitude fluctuations are more severe in open habitats and primarily mask low frequencies of amplitude modulation; reverberations are more severe in forested habitats and primarily mask high frequencies of amplitude modulation and rapid, repetitive frequency modulation. This difference between forested and open habitats could explain previous reports that birds in the undergrowth of tropical forests avoid rapid frequency modulation in their long-range vocalizations. 3. Maximum range of detection is probably not the primary selection pressure on many animal vocalizations, even for territorial advertisement, except perhaps in tropical forests. Instead, acoustic signals might incorporate features that degrade predictably with range to permit a receiver to estimate the signaler's distance. Future investigations might explore the propagation of animal vocalizations in relation to the usual spacing of animals in their habitat. Features that encode different kinds of information, such as individual and species identity, might propagate to different distances. 4. Measurements of the transmission of sound in natural environments have often not controlled several important parameters. First, the effects of gound attenuation and scattering are not linear with range; consequently measurements of excess attenuation over different ranges in the same environment might differ. Second, the directionality of speakers and microphones will affect measurements of attenuation and reverberations in scattering environments. Third, as stationary waves shift with frequency, any single microphone placement will lie in a null for some frequencies and in a maximum for others.