Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide experimental evidence of nonlinear and distinct PB-type focusing in standing water waves describing the scenario of two counter-propagating wave trains. The collected collinear wave measurements are in excellent agreement with the hydrodynamic coupled NLSE (CNLSE) and suggest that MI can undisturbedly prevail during the interplay of several wave systems and emphasize the potential role of exact NLSE solutions in extreme wave formation beyond the formal narrowband and uni-directional limits. Our work may inspire further experimental investigations in various nonlinear wave guides governed by CNLSE frameworks as well as theoretical progress to predict strong wave coherence in directional fields.
A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipl…
In this paper, the closed-form analytic solutions of two new Faraday's standing solitary waves due to the parametric resonance of liquid in a vessel vibrating vertically with a constant frequency are given for the first time. Using a model based on the symmetry of wave elevation and the linearized Boussinesq equation, we gain the closed-form wave elevations of the two kinds of non-monotonically decaying standing solitary waves with smooth crest and the even or odd symmetry. All of them have never been reported, to the best of our knowledge. Besides, they can well explain some experimental phen…
We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation pat…
A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extr…
A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated.…
The separation of micron-sized bacterial spores (Bacillus cereus) from a steady flow of water through the use of ultrasonic standing waves is demonstrated. An ultrasonic resonator with cross-section of 0.0254 m x 0.0254 m has been designed with a flow inlet and outlet for a water stream that ensures laminar flow conditions into and out of the resonator section of the flow tube. A 0.01905-m diameter PZT-4, nominal 2-MHz transducer is used to generate ultrasonic standing waves in the resonator. The acoustic resonator is 0.0356 m from transducer face to the opposite reflector wall with the acoust…
Despite the sphericity of the collapsing stellar core, the birth conditions of neutron stars can be highly non spherical due to a hydrodynamical instability of the shocked accretion flow. Here we report the first laboratory experiment of a shallow water analogue, based on the physics of hydraulic jumps. Both the experiment and its shallow water modeling demonstrate a robust linear instability and nonlinear properties of symmetry breaking, in a system which is one million times smaller and about hundred times slower than its astrophysical analogue.
We consider the problem of short-term prediction of rare, extreme water waves in unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually-intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the dispersive mixing of different harmonics, can grow significantly due to localized nonlinear focusing. Here we show how the interplay between i) statistical properties captured through linear information such as the waves power spectrum and ii)…
Legionella are gram-negative, facultative intracellular, and pathogenic bacteria that pose a risk for human health and cause significant energy losses due to extensive preventive heating-up of water installations. We investigate acoustically-driven motion - acoustophoresis - of several Legionella species, Escherichia coli, Pseudomonas aeruginosa, and Acanthamoeba castellanii, a common Legionella host in water. All the investigated cells can be acoustically manipulated in an ultrasonic standing wave in water, as they possess a non-zero acoustic contrast that is positive for all of the cells, le…
It is demonstrated that a standard coupled-mode theory can successfully describe weakly-nonlinear gravity water waves in Bragg resonance with a periodic one-dimensional topography. Analytical solutions for gap solitons provided by this theory are in a reasonable agreement with accurate numerical simulations of exact equations of motion for ideal planar potential free-surface flows, even for strongly nonlinear waves. In numerical experiments, self-localized groups of nearly standing water waves can exist up to hundreds of wave periods. Generalizations of the model to the three-dimensional case …
We study the dynamics of surface waves on a semi-toroidal ring of water that is excited by vertical vibration. We create this specific fluid volume by patterning a glass plate with a hydrophobic coating, which confines the fluid to a precise geometric region. To excite the system, the supporting plate is vibrated up and down, thus accelerating and decelerating the fluid ring along its toroidal axis. When the driving acceleration is sufficiently high, the surface develops a standing wave, and at yet larger accelerations, a traveling wave emerges. We also explore frequency dependencies and other…
A hydrophilic floating sphere that is denser than water drifts to an amplitude maximum (antinode) of a surface standing wave. A few identical floaters therefore organize into antinode clusters. However, beyond a transitional value of the floater concentration $ϕ$, we observe that the same spheres spontaneously accumulate at the nodal lines, completely inverting the self-organized particle pattern on the wave. From a potential energy estimate we show (i) that at low $ϕ$ antinode clusters are energetically favorable over nodal ones and (ii) how this situation reverses at high $ϕ$, in agreement w…
Waves excited on the surface of deep water decay in time and/or space due to the fluid viscosity, and the momentum associated with the wave motion is transferred from the waves to Eulerian slow currents by the action of the virtual wave stress. Here, based on the conservation of the total momentum, we found the virtual wave stress produced by calm gravity waves under the assumption that the slow Eulerian currents are weak in the sense that the Froude number is small and the scattering of surface waves by slow flow inhomogeneities can be neglected. In particular, we calculated the virtual wave …
Inhibition of hydrogen explosion is crucial to realize its wide applications and fine water spray is an ideal mitigant due to numerous advantages. In this work, interactions between a propagating hydrogen/air detonation wave and circular water cloud are numerically studied. Eulerian-Lagrangian method involving two-way gas-droplets coupling is applied, with a two-dimensional configuration. Different droplet (diameter, concentration) and cloud (diameter) properties are considered. Our results show that droplet size, concentration and cloud radius have significant effects on peak pressure traject…
Interactions between a propagating hydrogen/air detonation wave and circular water cloud are studied. Eulerian Lagrangian method involving two-way gas-droplet coupling is applied. Different droplet (diameter, concentration) and cloud (diameter) properties are considered. Results show that droplet size, concentration and cloud radius have significant effects on peak pressure trajectory of the detonation wave. Three propagation modes are identified: perturbed propagation, leeward re-detonation, and detonation extinction. Leeward re-detonation is analyzed from unsteady evolutions of gas and liqui…
The aim of this work is to study solitary water wave interactions between two topographic obstacles for the forced Korteweg-de Vries equation (fKdV). Focusing on the details of the interactions, we identify regimes in which solitary wave interactions maintain two well separated crests and regimes where the number of local maxima varies according to the laws $2\rightarrow 1\rightarrow 2\rightarrow 1\rightarrow 2$ or $2\rightarrow 1\rightarrow 2$. It shows that the geometric Lax-categorization of Korteweg-de Vries (KdV) two-soliton interactions still holds for the fKdV equation.
We study water waves propagating over a smooth obstacle in a fluid of varying depth, motivated by the observation that submerged islands in the ocean act as effective lenses that increase the amplitude and destructive power of tsunami waves near focal points. We show that islands of substantial height (compared to the water depth) lead to strong focusing in their immediate vicinity, and generate caustics of either cusp or butterfly type. We highlight similarities and differences with focusing of (high-frequency) gravitational waves by a neutron star. In the linear regime, the comparison is mad…
Water entry of small hydrophobic spheres on an axisymmetric wavefield
Brown University（布朗大学）；Georgia Institute of Technology（佐治亚理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Naval Undersea Warfare Center（海军水下战中心）
A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener-Hopf technique. The derivation of the Wiener-Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. We illustrate the method with some numerical computation…
