Periodic Shock Waves Research Articles

MHD simulations of astrophysical and laboratory jets under different magnetic field configurations

This paper presents the results of MHD simulations of astrophysical and laboratory supersonic jets under a superposition of poloidal (Br, Bz) and toroidal (Bϕ) magnetic fields. It is shown that the escaping matter is quickly collimated by the magnetic field. A shock wave of an elongated shape is formed, which moves from the target to the boundary of the chamber, leaving behind a stable flow. A periodic shock wave structure is observed inside the main conical expanding shock wave. It is shown that the toroidal component of the magnetic field remains in the region throughout the entire calculation and plays a role in the collimation of the flow. The poloidal magnetic field decreases in the region of the jet cone, but remains in the simulation region throughout the calculation and also participates in flow collimation. Thus, both components Bz and Bϕ take part in the collimation of the flow by the magnetic field. The width of the jet and the opening angle of the cone Ɵ depend on the magnitude of the magnetic field induction. As the field increases, the jet becomes narrower and the cone angle decreases. Initially, we do not specify the rotation of the jet Ω. However, due to the presence of the Bϕ field, the substance acquires angular velocity and twists along the z axis. The simulation results are in agreement with laboratory jets arising in the experiment at the Neodymium laser installation, and with the previously obtained results of MHD modeling of jet formation separately, in poloidal or toroidal magnetic field.

Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation

Abstract In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Numerical study of roll wave development for non-uniform initial conditions using steep slope shallow water equations

Periodic shock waves of roll waves may cause the flow to jump over the canal bank, increase soil erosion, and enhance the destructive power of debris flow. This paper focuses on the spatiotemporal changes in waveform, velocity, discharge, shear stress, and Froude number during the development process of roll waves, taking the non-uniform flow as the initial condition. This work studied numerically the influence mechanism of different variables on the development process of roll waves based on steep slope shallow water equations. The roll wave is unstable when the disturbance amplitude is too large. By reducing the inlet velocity, increasing the disturbance wavelength, and reducing the roughness coefficient, it can make the roll wave stable. Reducing the inlet velocity not only reduces the frequency of the shock wave but also keeps the wave crest from rising. Under different boundary conditions of disturbance amplitude, the crest growth rate (crest water depth difference/crest distance difference) of the roll wave development segment at a certain time is almost consistent. For different initial water depths, there is overlap in the roll wave development segment at a certain time. Under the initial condition of non-uniform flow, the congestion at the inlet means that the initial water depth is less than the normal water depth, and the decline of the water depth at the inlet means that the initial water depth is greater than the uniform water depth.

Welding Residual Stress Elimination Technique in the Top Chord of Main Truss of Steel Truss Bridge

The large-amplitude fluctuations of ultrasound in high-energy ultrasonic stress relieving cause the crystal grains or lattices in the high residual stress zone to vibrate or creep alternately. This triggers secondary effects such as ultrasonic softening and dislocation movement. The sound field also produces periodic shock waves or intermittent shock waves, which form local pressure gradients at the wave front. These pressure gradients cause local heating of the grain boundary, accelerating material softening and promoting slip between grains, ultimately resulting in residual stress elimination. This technique was applied to detect the welding residual stress of the upper chord of the main truss of Sanguantang Bridge by using an ultrasonic stress meter. After the measurement, it was found that the welding residual stress in some areas was too large, and the welding residual stress needed to be eliminated. The welding seam was re-inspected after the residual stress relief operation was completed. The test results showed a maximum reduction rate of 63.91% and an average overall reduction rate ranging from 24.52% to 37.23%. The reduction effect is more significant in areas with higher welding residual stress.

Analytical analysis of the fourth-order Boussinesq equation by traveling wave solutions

To find periodic solutions of nonlinear equations, the Jacobi elliptic function expansion method is used. This strategy is more extensive than expanding the hyperbolic tangent series. This approach can produce periodic shock wave solutions such as solitary wave solutions.

Interaction of Shock Waves with Water Saturated by Nonreacting or Reacting Gas Bubbles.

A compressible medium represented by pure water saturated by small nonreactive or reactive gas bubbles can be used for generating a propulsive force in large-, medium-, and small-scale thrusters referred to as a pulsed detonation hydroramjet (PDH), which is a novel device for underwater propulsion. The PDH thrust is produced due to the acceleration of bubbly water (BW) in a water guide by periodic shock waves (SWs) and product gas jets generated by pulsed detonations of a fuel–oxidizer mixture. Theoretically, the PDH thrust is proportional to the operation frequency, which depends on both the SW velocity in BW and pulsed detonation frequency. The studies reported in this manuscript were aimed at exploring two possible directions of the improvement of thruster performances, namely, (1) the replacement of chemically nonreacting gas bubbles by chemically reactive ones, and (2) the increase in the pulsed detonation frequency from tens of hertz to some kilohertz. To better understand the SW-to-BW momentum transfer, the interaction of a single SW and a high-frequency (≈7 kHz) sequence of three SWs with chemically inert or active BW containing bubbles of air or stoichiometric acetylene–oxygen mixture was studied experimentally. Single SWs and SW packages were generated by burning or detonating a gaseous stoichiometric acetylene–oxygen or propane–oxygen mixture and transmitting the arising SWs to BW. The initial volume fraction of gas in BW was varied from 2% to 16% with gas bubbles 1.5–4 mm in diameter. The propagation velocity of SWs in BW ranged from 40 to 580 m/s. In experiments with single SWs in chemically active BW, a detonation-like mode of reaction front propagation (“bubbly quasidetonation”) was realized. This mode consisted of a SW followed by the front of bubble explosions and was characterized by a considerably higher propagation velocity as compared to the chemically inert BW. The latter could allow increasing the PDH operation frequency and thrust. Experiments with high-frequency SW packages showed that on the one hand, the individual SWs quickly merged, feeding each other and increasing the BW velocity, but on the other hand, the initial gas content for each successive SW decreased and, accordingly, the SW-to-BW momentum transfer worsened. Estimates showed that for a small-scale water guide 0.5 m long, the optimal pulsed detonation frequency was about 50–60 Hz.

Similarity reductions and wave solutions for the 3D-Kudryashov–Sinelshchikov equation with variable-coefficients in gas bubbles for a liquid

The variable-coefficients 3D-Kudryashov–Sinelshchikov (vcKS) equation is an important mathematical model that describes the pressure of gas bubbles in the liquid, and is considered as an impressive phenomenon with many applications in medicine, ocean dynamics and fluid mechanics. In this paper, the vcKS is investigated via the enlarged direct similarity technique and reduced to a sixth-order nonlinear differential equation. The Jacobi elliptic wave function method is used to construct many novel periodic and solitary wave solutions for the vcKS equation. Additionally, a physical application of bubble-liquid in medicine was given and some new graphical representation for the dynamic behavior of the obtained periodic shock waves and solitonic waves were constructed to show how the different variable functions choices affect on the wave propagation. Moreover, this effect appears like a higher shock wave by increasing independent variables, which is very important in the drug delivery process.

Mathematical tricks for pseudopotentials in the theories of nonlinear waves in plasmas

In the analysis of nonlinear waves in plasma, especially for the search for periodic waves, shock waves, and solitons, mechanical analogy methods are widely applicable. The most famous of them is the Sagdeev pseudopotential method. However, sometimes mathematical difficulties arise when deriving formulas for pseudopotentials. The author proposes three mathematical tricks to get around these difficulties and obtain exact formulas for pseudopotentials in cases where the direct, Sagdeev method is considered inapplicable: a trick based on the Lambert W-function, a trick based on the inverse function integration, and a trick based on reducing the theory equations to the Bernoulli differential equation (the Bernoulli pseudopotential method). This article, which is methodological by nature, provides detailed examples of the application of each of these tricks when deriving formulas for pseudopotentials.

Exact solutions of the combined Hirota-LPD equation with variable coefficients

In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.

Non-linear Acoustic Emissions from Therapeutically Driven Contrast Agent Microbubbles

Non-linear emissions from microbubbles introduced to the vasculature for exposure to focused ultrasound are routinely monitored for assessment of therapy and avoidance of irreversible tissue damage. Yet the bubble-based mechanistic source for these emissions, under subresonant driving at typical therapeutic pressure amplitudes, may not be well understood. In the study described here, dual-perspective high-speed imaging at 210,000 frames per second (fps), and shadowgraphically at 10 Mfps, was used to observe cavitation from microbubbles flowing through a 500-µm polycarbonate capillary exposed to focused ultrasound of 692 kHz at therapeutically relevant pressure amplitudes. The acoustic emissions were simultaneously collected via a broadband calibrated needle hydrophone system. The observations indicate that periodic bubble-collapse shock waves can dominate the non-linear acoustic emissions, including subharmonics at higher driving amplitudes. Contributions to broadband emissions through variance in shock wave amplitude and emission timings are also identified. Possible implications for in vivo microbubble cavitation detection, mechanisms of therapy and the conventional classification of cavitation activity as stable or inertial are discussed.

On traveling wave solutions: The Wu–Zhang system describing dispersive long waves

In this paper, we construct exact families of traveling wave (periodic wave, solitary wave, shock wave, singular-wave, singular-periodic wave, and singular-solitary wave) solutions of a well-known system of nonlinear PDEs, the Wu–Zhang system, which describes (1[Formula: see text]+[Formula: see text]1)-dimensional dispersive long waves. This system is solved by using the generalized [Formula: see text] expansion method, where G satisfies the Jacobi elliptic equation of fourth order. Meanwhile, the mechanical features of some families are explained through three-dimensional figures.

Non-linear acoustic emissions from therapeutically driven microbubbles

Acoustic detection of contrast agent microbubbles, infused to the vasculature for exposure to focused ultrasound, is now routinely undertaken to evaluate therapy and avoid irreversible tissue damage. Harmonic, subharmonic (to the frequency of the focused ultrasound, f0) and broadband emissions are often used to distinguish between stable and inertial cavitation activity, and associated bioeffects. The driven microbubble dynamic responsible for the generation of non-linear emissions, however, may not be well understood. Results from an investigation of single SonoVue microbubbles flowing through a capillary, for exposure to focused ultrasound at f0 = 692 kHz, will be presented. Dual high-speed imaging from orthogonal perspectives at circa 2 × 105, and shadowgraphically at 10 million frames per second, capture microbubble activity and shock wave generation. Acoustic emissions are simultaneously collected with a calibrated broadband needle hydrophone, and high frequency imaging array on passive receive, for cavitation mapping. The results indicate that non-linear emissions are mediated by periodic bubble collapse shock waves, with subharmonic emission occurring above a threshold driving pressure amplitude. Implications for detection and quantification of driven microbubble cavitation activity, as well as conventional classification as stable or inertial, will be discussed. [This work was supported by ERC Starting Grant TheraCav, 336189.]

Performance characterisation of a passive cavitation detector optimised for subharmonic periodic shock waves from acoustic cavitation in MHz and sub-MHz ultrasound

We describe the design, construction and characterisation of a broadband passive cavitation detector, with the specific aim of detecting low frequency components of periodic shock waves, with high sensitivity. A finite element model is used to guide selection of matching and backing layers for the shock wave passive cavitation detector (swPCD), and the performance is evaluated against a commercially available device. Validation of the model, and characterisation of the swPCD is achieved through experimental detection of laser-plasma bubble collapse shock waves. The final swPCD design is 20 dB more sensitive to the subharmonic component, from acoustic cavitation driven at 220 kHz, than the comparable commercial device. This work may be significant for monitoring cavitation in medical applications, where sensitive detection is critical, and higher frequencies are more readily absorbed by tissue.

Simulating energy cascade of shock wave formation process in a resonator by gas kinetic scheme

The temporal-spatial evolution of gas oscillation was simulated by gas kinetic scheme (GKS) in a cylindrical resonator, driven by a piston at one end and rigidly closed at the other end. Periodic shock waves propagating back and forth were observed in the resonator under finite amplitude of gas oscillation. The studied results demonstrated that the acoustic pressure is a saw-tooth waveform and the oscillatory velocity is a square waveform at the central position of the resonant tube. Moreover, it was found by harmonic analysis that there was no presence of obvious feature for pressure node in such a typical standing wave resonator, and the distribution of acoustic fields displayed a one-dimensional feature for the acoustic pressure while a quasi-one-dimensional form for oscillatory velocity, which demonstrated the nonlinear effects. The simulation results for axial distribution of acoustic intensity showed a good consistency with the published experimental data in the open literature domain, which provides a verification for the effectiveness of the GKS model proposed. The influence of displacement amplitude of the driving piston on the formation of shock wave was numerically investigated, and the simulated results revealed the cascade process of harmonic wave energy from the fundamental wave to higher harmonics. In addition, this study found that the acoustic intensity at the driving end of the resonant tube would increase linearly with the displacement amplitude of the piston due to nonlinear effects, rather than the exponential variation by linear theory. This research demonstrates that the GKS model is strongly capable of simulating nonlinear acoustic problems.

Simulating the energy cascade of shock wave formation process in a resonator by gas-kinetic scheme

The temporal-spatial evolution of the sound field in a cylindrical resonator driven by a piston was simulated by gas-kinetic scheme (GKS), and periodic shock waves propagating back and forth were observed in the resonator with finite displacement amplitude of the driving piston. It can been seen from the instantaneous spatial distribution of the oscillatory pressure and velocity in the resonator that the oscillatory pressure is a one-dimensional,but the oscillatory velocity exhibits a quasi-one-dimensional distribution due to the effect of the boundary layer.The evolution process of shock wave from the sinusoidal wave to the zigzag wave was demonstrated with the increasing of the driving displacement amplitude of the piston. Especially, simulation results revealed that the acoustic intensity was linearly proportional to the driving displacement amplitude of the piston, rather than the square type. The studied results demonstrated well the process of harmonic energy cascade by the FFT analysis, at the same time, the physical explanations has been also given. The simulation results of acoustic intensity were well consistent with the existing experimental results, which verifies the validity of GKS modeling method.

Covert cavitation: Spectral peak suppression in the acoustic emissions from spatially configured nucleations.

Dual laser-nucleation is used to precisely configure two cavitation bubbles within a focused ultrasound field of f0 = 692 kHz, in proximity to the tip of a needle hydrophone. With both bubbles responding in the f0/2 sub-harmonic regime, confirmed via ultra-high speed shadowgraphic imaging, an emission spectrum with no sub-harmonic content is demonstrated, for an inter-bubble spacing ≈λ0. A spectral model for periodic shock waves from multiple nucleations demonstrates peak suppressions at nf0/2 when applied to the experiment, via a windowing effect in the frequency domain. Implications for single-element passive detection of cavitation are discussed.

An analysis of the acoustic cavitation noise spectrum: The role of periodic shock waves.

Research on applications of acoustic cavitation is often reported in terms of the features within the spectrum of the emissions gathered during cavitation occurrence. There is, however, limited understanding as to the contribution of specific bubble activity to spectral features, beyond a binary interpretation of stable versus inertial cavitation. In this work, laser-nucleation is used to initiate cavitation within a few millimeters of the tip of a needle hydrophone, calibrated for magnitude and phase from 125 kHz to 20 MHz. The bubble activity, acoustically driven at f0 = 692 kHz, is resolved with high-speed shadowgraphic imaging at 5 × 106 frames per second. A synthetic spectrum is constructed from component signals based on the hydrophone data, deconvolved within the calibration bandwidth, in the time domain. Cross correlation coefficients between the experimental and synthetic spectra of 0.97 for the f0/2 and f0/3 regimes indicate that periodic shock waves and scattered driving field predominantly account for all spectral features, including the sub-harmonics and their over-harmonics, and harmonics of f0.

Deconvolution of acoustically detected bubble-collapse shock waves

The shock wave emitted by the collapse of a laser-induced bubble is detected at propagation distances of 30, 40and50mm, using a PVdF needle hydrophone, with a non-flat end-of-cable frequency response, calibrated for magnitude and phase, from 125kHz to 20MHz. High-speed shadowgraphic imaging at 5×106 frames per second, 10nstemporal resolution and 256 frames per sequence, records the bubble deflation from maximum to minimum radius, the collapse and shock wave generation, and the subsequent rebound in unprecedented detail, for a single sequence of an individual bubble. The Gilmore equation for bubble oscillation is solved according to the resolved bubble collapse, and simulated shock wave profiles deduced from the acoustic emissions, for comparison to the hydrophone recordings. The effects of single-frequency calibration, magnitude-only and full waveform deconvolution of the experimental data are presented, in both time and frequency domains. Magnitude-only deconvolution increases the peak pressure amplitude of the measured shock wave by approximately 9%, from single-frequency calibration, with full waveform deconvolution increasing it by a further 3%. Full waveform deconvolution generates a shock wave profile that is in agreement with the simulated profile, filtered according to the calibration bandwidth. Implications for the detection and monitoring of acoustic cavitation, where the role of periodic bubble collapse shock waves has recently been realised, are discussed.

Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi’s elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.