Finite Depth Research Articles

The Phase Structure of Wave Disturbances Excited by a Pulsating Source at the Interface of a Liquid Flow of Finite Depth and an Ice Sheet

The floating ice sheet determines the dynamic interaction between the ocean and the atmosphere, affects the dynamics of the sea surface and subsurface waters, since the ice sheet and the entire mass of liquid under it participate in the general vertical movement. The paper investigates the phase structure of wave fields arising at the interface between ice and a flow of homogeneous liquid of finite thickness when flowing around a localized pulsating source of disturbances. The ice sheet is modeled by a thin elastic plate, the deformations of which are small, and the plate is physically linear. An integral representation of the solution is obtained, and the results of calculations of dispersion dependencies and phase patterns for various parameters of wave generation are presented. It is shown that the main parameters determining the characteristics of the amplitude-phase structure of wave disturbances of the ice sheet surface are ice thickness, flow velocity, and pulsation frequency. Numerical calculations demonstrate that when the flow velocities, ice thickness, and frequency change, there is a noticeable qualitative restructuring of the phase patterns of the excited long-range wave fields at the ice-liquid interface.

Numerical PMM test in finite depth

This study conducted a numerical planar motion mechanism (PMM) test for a containership, considering the effects of finite depth. An OpenFOAM-based computational fluid dynamics (CFD) approach was employed to conduct the simulations. Before conducting a series of numerical computations, uncertainty analysis for the grid size and time step was performed to ensure the reliability of the computational results. Two PMM tests, static drift and pure yaw, were conducted, and the results were validated against experimental data. The comparison demonstrated good agreement between forces, moments, and hydrodynamic coefficients when compared with both experimental and other computational results. Furthermore, a comparison between a first- and second-order combination model and a first- and third-order combination model revealed that the latter showed better alignment with experimental data in deep water, while the former performed better in shallow water, emphasizing the role of crossflow. This study contributes to understanding the differences in maneuvering performance between deep and shallow water conditions.

Remote sensing of wave-orbital velocities in the surfzone

Wave-orbital velocities are estimated with particle image velocimetry (PIV) applied to rapid sequences of images of the surfzone surface obtained with a low-cost camera mounted on an amphibious tripod. Time series and spectra of the remotely sensed cross-shore wave-orbital velocities are converted to the depth of colocated acoustic Doppler velocimeters (ADVs), using linear finite depth theory. These converted velocities are similar to the velocities measured in situ (mean nRMSE for time series=16% and for spectra=10%). Small discrepancies between depth-attenuated surface and in situ currents may be owing to errors in the surface velocity measurements, uncertainties in the water depth, the vertical elevation of the ADVs, and the neglect of nonlinear effects when using linear finite depth theory. These results show the potential to obtain spatially dense estimates of wave velocities using optical near-field remote methods during field campaigns and continuous monitoring operations.

Preliminary results of the non-invasive cardiac radioablation for ventricular tachycardia study; an in silico dosimetric comparison of photon vs proton radiotherapy for ventricular arrhythmias

Abstract Background Stereotactic arrhythmia radio-ablation (STAR) has been proposed as an alternative treatment for refractory ventricular arrhythmias (VA) in patients who are unsuitable or refractory to standard catheter ablation (CA). It consists in the application of external beam radiotherapy in a single dose of 25 Gy to the target areas. Most of the STAR treatments performed so far used photon radiotherapy. As compared to photons, particle therapy with protons, based on its finite penetration depth in tissue, might have the potential to reduce the dose to surrounding healthy tissues thus lowering the risk of side effects. Purpose The purpose of this in silico simulation study was to evaluate the feasibility of STAR with protons in patients with VA and to compare the doses of the organs at risk (OARs) in photon and proton treatment plans. Methods 14 patients (11 males) candidates to standard CA for VA were enrolled. ECG gated CT scans in expiratory breath-hold, and standard electroanatomical mapping were performed in each patients to reconstruct a 3D bipolar voltage map and identify the ablation target. Based on the information obtained by the contouring of the ablation target three treatment plans were simulated: one with photons, and two with protons - with and without cardiac gating- to assess the doses at the OARs. Results The median age was 68.5 (63.25-71.5), all patients had a history of recurrent VA and the indication for treatment with CA. 7 patients had an ischemic cardiomyopathy, 4 an idiopathic cardiomyopathy and 3 ventricular ectopic beats without structural heart disease. The mean left ventricular ejection fraction was 40±11% and 8 out of 14 patients were ICD carriers. Both photons and protons treatment plans reached the goals for target coverage (25 Gy on 98% of the target volume). A statistically significant reduction in the mean maximum dose (Dmax) to some OARs was observed in proton compared to photon plans (see table 1). Specifically, the Dmax delivered to the mitral valve, vena cava, descending aorta, esophagus and to the lungs have been largely reduced with the proton plans. However, in some specific cases, a higher maximum dose was observed for protons in structures close or overlapping with the target, mainly due to the need of an additional margin in the treatment plan to account for proton range uncertainty. The use of cardiac gating did not show a significant dose reduction to OARs compared to the non-gated proton plan. Conclusion The preliminary results of our in silico simulation study suggest that STAR therapy with the use of protons might be associated with a significant dose reduction to the surrounding organs. Nevertheless, the choice of the best treatment should be evaluated on a case by-case strategy since some specific cases may be better suitable for photon treatment.Table 1

Rapid fabrication of gold microsphere arrays with stable deep-pressing anisotropic conductivity for advanced packaging

Smooth metal microspheres with uniform sizes are ideal for constructing particle-arrayed anisotropic conductive films (ACF), but synthesis is hindered by challenges in controlling anisotropic metal growth. Here, we present a positioned transient-emulsion self-assembly and laser-irradiation strategy to fabricate pure gold microsphere arrays with smooth surfaces and uniform sizes. The fabrication involves assembling gold nanoparticles into uniform colloidosomes within a pre-designed microhole array, followed by rapid transformation into well-defined microspheres through laser heating. The gold nanoparticles melt and merge in a layer-by-layer manner due to the finite skin depth of the laser, leading to a localized photothermal effect. This strategy circumvents anisotropic growth, enables tunable control of microsphere size and positioning, and is compatible with conventional lithography. Importantly, these pure gold microspheres exhibit stable conductivity under deep compression, offering promising applications in soldering micro-sized chips onto integrated circuits.

EFFECT OF NONUNIFORM POROSITY IN CURVED BREAKWATER ON WATER WAVES

Abstract Assuming linear theory, the phenomenon of scattering of waves by a circular arc shaped barrier with nonuniform porosity is studied. The water region is considered to be of infinite or finite depth. Based on a judicious application of Green’s integral theorem, the corresponding boundary value problem is reduced to a hypersingular integral equation of second kind. The boundary element method and the collocation method are adopted to solve the hypersingular integral equation, and we ensure a good matching of the solutions obtained by the two methods. The reflection coefficient and energy dissipation are evaluated by using the solution of the integral equation which is then studied graphically. Different choices of distributions of pores on the barrier are considered, and we observe that the nonuniform porosity of the barrier has significant effect on the reflected wave and the energy dissipation.

Modeling and Hybrid Inversion of Mineral Deposits Using the Dipping Dike Model with Finite Depth Extent

The dipping dike model has shown to be a useful approximation for mineral deposits. To make this model more realistic, we include the thickness, which yields the depth to the bottom, as an additional parameter. The magnetic anomaly is obtained by combining the anomalies of two infinite dikes, so that the resulting expression is simpler than the classical prismatic models with polygonal cross section. We employ a Metropolis-Hasting (MH) algorithm coupled with the Levenberg-Marquardt (LM) method to invert magnetic profiles assuming a model of multiple dike-like sources. We use a few iterations of the LM method to improve the candidate solutions at the end of each random walk generated by MH. The following parameters are obtained: depth to the top, thickness, half-width, horizontal location of the top center, geological dip, in addition to two effective parameters that depend on the intensity of magnetization and the directions of the induced and remanent fields. For synthetic anomalies, both noise-free and noisy magnetic data are considered, with examples presented for each scenario. These examples highlight the discrepancy between models with finite and infinite sources. They also illustrate the higher accuracy of the hybrid MH-LM method over the pure MH approach. Moreover, two field examples related to mineral exploration have been considered: the Pima copper mine, United States, where the relative differences between the parameters obtained by our algorithm and those known from drilling are not higher than 10%, and a magnetic profile over iron ore deposits located in Laje, northeast Brazil, where the inverted parameters were useful for detailing previous studies.

On Small-Amplitude Asymmetric Water Waves

AbstractWe generalize the method used by Mæhlen and Seth (Asymmetric traveling wave solutions of the capillary–gravity Whitham equation, 2023, arxiv:2312.15343) used to prove the existence of small-amplitude asymmetric solutions to the capillary–gravity Whitham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetric solutions for this class of equations. The result is then applied to an infinite depth capillary–gravity Whitham equation and an infinite depth capillary–gravity Babenko equation to show a nonexistence result of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary–gravity Babenko equation.

Effects of nonlinearity on the crest shape of extreme irregular sea waves: Nonlinear harmonic separation and analysis

This work highlights the effects of nonlinearity on the crest shape of extreme directional water wave events in infinite and finite water depths. HOS-ocean, a High-Order Spectral code developed for solving the deterministic propagation of nonlinear wavefields in open ocean [Ducrozet et al., “HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method,” Comput. Phys. Commun. 203, 245–254 (2016)], is used for comparisons with linear simulations. Nonlinear wave harmonics were extracted combining and extending previous methods [Fitzgerald et al., “Phase manipulation and the harmonic components of ringing forces on a surface-piercing column,” Proc. R. Soc. A 470, 20130847 (2014) and Hann et al., “A new set of focused wave linear combinations to extract non-linear wave harmonics,” in Proceedings of 29th International Workshop on Water Waves and Floating Bodies, 2014], showing the contribution of various orders of nonlinearity. The work shows that “walls of water,” [Adcock et al., “Nonlinear dynamics of wave-groups in random seas: Unexpected walls of water in the open ocean,” Proc. R. Soc. A 471, 20150660 (2015) and Adcock et al., “On the shape of large wave-groups on deep water—The influence of bandwidth and spreading,” Phys. Fluids 28, 106601 (2016)], a significant increase in the crest width, are also formulated in finite water depths. Mainly responsible for this is the first order harmonic, with significant energy transfers away from components propagating at an angle, which are more pronounced in finite water depths. As nonlinearity increases, higher order harmonics (>second) become, as expected, more significant. In deep water, rapid energy transfers lead to a broadening of the spectrum which is reflected in the increase in the first order amplitude sum and hence increased crest elevations at the point of the large event. However, in finite water depths, energy transfers are gradually leading to a narrowing of the spectrum around the spectral peak and to a general broadening of the spectrum at the higher harmonics; reflected in the significant decrease in the first order amplitude sum and hence a decrease in crest elevations at the point of the large event, also aided by the larger increase in the out-of-phase difference terms (zeroth harmonic).

Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents

A new modified three-dimensional (3D) nonlinear Schrödinger equation (3DMNLSE) is derived for gravity waves in the presence of wind, dissipation, and two-dimensional slowly varying currents, which include transverse and longitudinal currents in finite water depth. The effect of currents on modulational instability (MI) is investigated. The divergence (convergence) effect of longitudinal favorable (adverse) currents decreases (increases) the MI growth rate and region of instability while suppressing (enhancing) the formation of freak waves. Meanwhile, the transverse currents hardly affect the occurrence of MI. Furthermore, some results from the simulations with the space evolution of 3DMNLSE are presented. The results show the ubiquitous occurrence of freak waves in 3D wave fields under certain sets of initial conditions. We demonstrate that larger waves can be triggered when a weakly modulated wave train enters a region of adverse currents. The maximum amplitude of a freak wave depends on the ratio of the current velocity to the wave phase velocity.

A theoretical exploration of the electronic structure and single photoionization of the many-electron system confined in Gaussian potential

This manuscript investigates the electronic structures, spectral properties, and photoionization processes of the confined atomic system. For this purpose, a relativistic methodology employing the Dirac–Coulomb Hamiltonian within the context of relativistic configuration interaction is suggested, utilizing independent particle basis wavefunctions. The key idea of this approach is to place the atom inside a Gaussian potential, which gives a realistic description of the spatial confinement in quantum dots due to a smooth change at the quantum dot boundaries and has a finite range and depth for the spatial confinement. As a result, the local central potential is modified, which is determined by a self-consistent process. The solutions to the Dirac equation, incorporating the aforementioned central potential, yield both the continuous and bound state wave functions. The photoionization process is determined through the application of the distorted wave approach within the context of relativistic Dirac theory. As an application, the electronic structures of the confined Li atom, including energies, ionization potentials, transition rates, and photoionization dynamical properties such as wave functions, cross sections, and photoelectron angular distributions, are systematically investigated within the dipole approximation for a wide range of potential depths and confining radii. A systematic comparison of the present outcomes is made with other available results. The present study is not only meaningful for fundamental research in atomic and molecular physics, but also has implications for a range of disciplines, such as nanochemistry, materials science, and other related fields.

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves

We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or 1:-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1:-1$$\\end{document} semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or 1:-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1:-1$$\\end{document} semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.

Hamiltonian dynamics on digital quantum computers without discretization error

We introduce an algorithm to compute expectation values of time-evolved observables on digital quantum computers that requires only bounded average circuit depth to reach arbitrary precision, i.e. produces an unbiased estimator with finite average depth. This finite depth comes with an attenuation of the measured expectation value by a known amplitude, requiring more shots per circuit. The average gate count per circuit for simulation time t is O(t2μ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{O}}({t}^{2}{\\mu }^{2})$$\\end{document} with μ the sum of the Hamiltonian coefficients, without dependence on precision, providing a significant improvement over previous algorithms. With shot noise, the average runtime is O(t2μ2ϵ−2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{O}}({t}^{2}{\\mu }^{2}{\\epsilon }^{-2})$$\\end{document} to reach precision ϵ. The only dependence in the sum of the coefficients makes it particularly adapted to non-sparse Hamiltonians. The algorithm generalizes to time-dependent Hamiltonians, appearing for example in adiabatic state preparation. These properties make it particularly suitable for present-day relatively noisy hardware that supports only circuits with moderate depth.

A modified experimental approach for generating internal solitary waves using the dual paddle technique

A modification to the dual paddle wave generation method for internal solitary waves (ISWs) has been achieved through the implementation of the adjusted high-order unidirectional (aHOU) model and the Miyata–Choi–Camassa (MCC) model. This modified method utilizes layer-averaged horizontal velocities from ISWs in both the upper and lower fluid layers to control the operational velocities of the corresponding paddles. Experimental investigations conducted in a two-layer fluid with varied stratification conditions were employed to evaluate the applicability of the aHOU and MCC models. The validation of this approach involved examining the stability of generated ISWs by comparing experimental waveforms with theoretical predictions, demonstrating good agreement with prior research. Furthermore, evaluation of the modified wave generation method included an analysis of dimensionless phase velocity and characteristic frequency. The study also explored the quasi-linear relationship between the designed wave amplitude and the actual wave amplitude, establishing a polynomial fit between the actual wave amplitude and the layer thickness ratio. This approach offers a new method for stable and controllable laboratory generation of ISWs. Under finite water depth conditions, the method can produce ISWs that propagate stably over long distances and effectively control parameters such as wave amplitude and wave profile across a broad range of wave amplitudes and stratifications. These results confirm the effectiveness of the modified method and underscore its practical applicability in ISWs generation.

Modulational instability of a pair of collinear wave trains

A coupled system of three nonlinear evolution equations is derived for a pair of collinear wave packets over finite depth fluid. The wave packets are narrow banded having different carrier wave frequencies. The evolution equations are employed to perform stability analysis of a pair of collinear wave trains. It is observed that the region of instability as well as the growth rate of instability for counter-propagating waves is greater than that for co-propagating waves when everything else remains same. The region of instability increases with the increase in the depth of the medium. It is also found that the growth rate of instability of a wave train increases with the increase in wave-steepness of the second wave train.

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Load motion on an ice cover in the presence of a liquid layer with velocity shear

The behavior of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift in the upper layer is studied. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The critical velocities, the deflection of ice cover, and the wave forces are studied depending on the current velocity gradient, the shear layer thickness, the direction of motion, and the compression ratio.

Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a general phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form.In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

The study of water wave scattering by series of submerged elastic scatterers

A submerged flexible breakwater can be used to control waves in shallow water as an advanced alternative to traditional rigid submerged structures. Submerged flexible breakwaters not only cost less to build than conventional submerged breakwaters but also allow ships and marine life to bypass them if they are deep enough. These marine structures decrease the intensity of the shock wave and prevent standing waves from forming. We present a solution for transmission/reflection using a series of submerged identical elastic scatterers. Scattering and transfer matrices will be used to compute the transmission and reflection coefficients for N-submerged elastic scatterers. The behavior of the transmission for different values of physical parameters such as submergence depth, length of the scatterer, the gap between two successive scatterers, stiffness of the scatterer, and the number of submerged elastic scatterers is given. In doing so, we can achieve the frequencies and the incident amplitudes at which these submerged elastic scatterers can be effectively used as breakwaters in water of finite depth. We have looked at the transmission and reflection behaviors for a range of physical parameter values, such as the submergence depth, the length of the scatterer, the spacing between successive scatterers, the stiffness of the elastic scatterer, and the number of scatterers over a wide frequency range. We have given a spectrum of frequencies for which, given certain values of physical parameters, we can achieve zero transmission. Furthermore, even at high frequencies, we have demonstrated an incidence amplitude range for which zero propagation of transmission is feasible.