Simple Waves characterizing wave propagation in a nonlinear elastic medium

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 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.

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.

On the KZK-type equation for modulated ultrasound fields

The generalized mixed nonlinear Schrödinger (GMNLS) equation is a completely integrable model under certain parameter conditions, which has important applications in nonlinear optics, weak nonlinear dispersive water wave and quantum field theory. In this work, we use the unified transformation method to construct a Riemann-Hilbert problem and discuss the initial-boundary value problems of the GMNLS equation on the half-line. Furthermore, we also obtain that some spectral functions are not independent of each other, but meet a paramount global relation.

Phase-resolved wave simulation and direct numerical simulation of turbulence are performed to investigate the surface wave effects on the energy transfer in overlying turbulent flow. The JONSWAP spectrum is used to initialize a broadband wave field. The nonlinear wave field is simulated using a high-order spectral method, and the resultant wave surface provides the bottom boundary conditions for direct numerical simulation of the overlying turbulent flow. Two wave ages of and 25 are considered, corresponding to slow and fast wave fields, respectively, where denotes the celerity of the peak wave and denotes the friction velocity. The energy transfer of turbulent motions in the presence of surface waves is investigated through the spectral analysis of the two-point correlation transport equation. It is found that the production term has an extra peak at the dominant wavelength scale in the vicinity of the surface, and the energy transported to the surface via viscous and spatial turbulent transport is enhanced in the region of . The presence of surface waves results in an inverse turbulent energy cascade in the near-surface region, where small-scale wave-related motions transfer energy back to the dominant wavelength scale. Pressure-related terms reflecting the spatial and inter-component energy transfer are strongly dependent on the wave age. Furthermore, triadic interaction analysis reveals that the energy influx at the dominant wavelength scale is due to the contribution of the neighbouring streamwise turbulent motions, and those at the harmonic wavelength scales contribute the most.

Time-reversal (TR) refocusing of waves is one of fundamental principles in wave physics. Using the TR approach, "Time-reversal mirrors" can physically create a time-reversed wave that exactly refocus back, in space and time, to its original source regardless of the complexity of the medium as if time were going backwards. Lately, laboratory experiments proved that this approach can be applied not only in acoustics and electromagnetism but also in the field of linear and nonlinear water waves. Studying the range of validity and limitations of the TR approach may determine and quantify its range of applicability in hydrodynamics. In this context, we report a numerical study of hydrodynamic TR using a uni-directional numerical wave tank, implemented by the nonlinear high-order spectral method, known to accurately model the physical processes at play, beyond physical laboratory restrictions. The applicability of the TR approach is assessed over a variety of hydrodynamic localized and pulsating structures' configurations, pointing out the importance of high-order dispersive and particularly nonlinear effects in the refocusing of hydrodynamic stationary envelope solitons and breathers. We expect that the results may motivate similar experiments in other nonlinear dispersive media and encourage several applications with particular emphasis on the field of ocean engineering.

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.

By means of direct numerical simulations (DNS) based on the integrodifferential Zakharov equation, we study the long-term evolution of nonlinear random water wave fields. For the first time, formation of powerlike Kolmogorov-type spectra corresponding to weak-turbulent inverse cascade is demonstrated by DNS, and the evolution in time of the resulting spectra is quantitatively investigated. The predictions of the statistical theory for water waves, both qualitative (formation of the direct and inverse cascades, self-similar behavior) and quantitative (the spectra exponents, specific shape of self-similar functions, the rate of time evolution) are found to be in good agreement with the DNS results, except for the initial part of the evolution, where the established statistical theory is not applicable yet and the evolution has a much faster time scale.

A comparative analysis of momentum equations for an ideal-liquid flow in rigid and elastic pipelines has been performed. A fundamental difference of these equations has been shown using individual examples. The influence of the character of the boundary has been analyzed. A general solution analogous to the Riemann solution for propagation of a simple nonlinear wave in an elastic pipeline has been obtained. It has been shown that the wave shape changes for any velocity of the liquid in the pipeline and a shock wave occurs. The formation length of the shock wave has been evaluated.

We examine the implementation of a wave-breaking mechanism into a nonlinear potential flow solver. The success of the mechanism will be studied by implementing it into the numerical model HOS-NWT, which is a computationally efficient, open source code that solves for the free surface in a numerical wave tank using the high-order spectral (HOS) method. Once the breaking mechanism is validated, it can be implemented into other nonlinear potential flow models. To solve for wave-breaking, first a wave-breaking onset parameter is identified, and then a method for computing wave-breaking associated energy loss is determined. Wave-breaking onset is calculated using a breaking criteria introduced by Barthelemy et al. (J Fluid Mech https://arxiv.org/pdf/1508.06002.pdf , submitted) and validated with the experiments of Saket et al. (J Fluid Mech 811:642–658, 2017). Wave-breaking energy dissipation is calculated by adding a viscous diffusion term computed using an eddy viscosity parameter introduced by Tian et al. (Phys Fluids 20(6): 066,604, 2008, Phys Fluids 24(3), 2012), which is estimated based on the pre-breaking wave geometry. A set of two-dimensional experiments is conducted to validate the implemented wave breaking mechanism at a large scale. Breaking waves are generated by using traditional methods of evolution of focused waves and modulational instability, as well as irregular breaking waves with a range of primary frequencies, providing a wide range of breaking conditions to validate the solver. Furthermore, adjustments are made to the method of application and coefficient of the viscous diffusion term with negligible difference, supporting the robustness of the eddy viscosity parameter. The model is able to accurately predict surface elevation and corresponding frequency/amplitude spectrum, as well as energy dissipation when compared with the experimental measurements. This suggests the model is capable of calculating wave-breaking onset and energy dissipation successfully for a wide range of breaking conditions. The model is also able to successfully calculate the transfer of energy between frequencies due to wave focusing and wave breaking. This study is limited to unidirectional waves but provides a valuable basis for future application of the wave-breaking model to a multidirectional wave field. By including parameters for removing energy due to wave-breaking into a nonlinear potential flow solver, the risk of developing numerical instabilities due to an overturning wave is decreased, thereby increasing the application range of the model, including calculating more extreme sea states. A computationally efficient and accurate model for the generation of a nonlinear random wave field is useful for predicting the dynamic response of offshore vessels and marine renewable energy devices, predicting loads on marine structures, and in the study of open ocean wave generation and propagation in a realistic environment.