Nonlinear Surface Waves Research Articles

Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids

The evolution of the free surface of a three-dimensional conducting fluid in the presence of gravity, surface tension and vertical electric field due to parallel electrodes, is considered. Based on the analysis of the Dirichlet–Neumann operators, a series of fully nonlinear models is derived systematically from the Euler equations in the Hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength. For special cases, well-known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms. It is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface: (i) when the separation distance between two electrodes is small compared with typical wavelength, the Boussinesq, Benney–Luke (BL) and Kadomtsev–Petviashvili (KP) equations with modified coefficients are obtained, and electric forces can turn KP-I to KP-II and vice versa; (ii) as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength, we generalize the BL and KP equations by adding pseudo-differential operators resulting from the electric field; (iii) for a quasi-monochromatic plane wave in deep fluid, we derive the cubic nonlinear Schrödinger (NLS) equation, but its type (focusing or defocusing) is strongly influenced by the value of the electric parameter. For sufficient surface tension, numerical studies reveal that lump-type solutions exist in the aforementioned three regimes. Particularly, even when the associated NLS equation is defocusing for a wave train, lumps can exist in fully nonlinear models.

Measuring Alkali-Silica Reaction (ASR) Microscale Damage in Large-Scale Concrete Slabs Using Nonlinear Rayleigh Surface Waves

The second harmonic generation (SHG) of nonlinear Rayleigh waves is applied to characterize the damage state induced by alkali-silica reaction (ASR) in large-scale plain concrete slabs. A measurement setup that uses a wedge transmitter and a non-contact, air-coupled receiver is implemented to generate and detect nonlinear Rayleigh surface waves and to obtain both the linear and nonlinear acoustic parameters, while complementary measurements track expansion of the concrete slabs. These results demonstrate the potential of SHG using nonlinear Rayleigh waves to assess the evolution of microscale damage induced by ASR in large-scale, in-service concrete components.

Possible management of near shore nonlinear surging waves through bottom boundary conditions

We propose an alternative way for managing near shore surging waves, including extreme waves like tsunamis, going beyond the conventional passive measures like the warning system. We study theoretically the possibility of influencing the nonlinear surface waves through a leakage boundary effect at the bottom. It has been found through analytic result, that the controlled leakage at the bottom might regulate the amplitude of the surface solitary waves. This could lead to a possible decay of the surging waves to reduce its hazardous effects near the shore. Our theoretical results are estimated by applying it to a real coastal bathymetry of the Bay of Bengal in India.

Investigation of a New Waveguide Structure Based on Negative Index Material for Optoelectronic Applications

In this work, a waveguide structure consisting of a new artificial negative index material (NIM) surrounded by a nonlinear cover and a ferrite (YIG) substrate has been designed and investigated. We apply the boundary conditions and impose the condition of negative effective permeability of the ferrite slab to derive the dispersion relation related to the proposed structure. The NIM permittivity and permeability are not constant and depend on the operating frequency. The dispersion properties of the nonlinear electromagnetic surface waves (NEM) are analyzed and the associated propagation index is calculated. Results show that the dispersion could be tuned and controlled by selecting the NIM film thickness and the film-cover interface nonlinearity. The proposed structure is supporting unusual types of NEM surface waves having a non-reciprocal behavior widely used in designing optoelectronic devices.

English

The detailed mathematical study of the recent paper by Sajjadi, Hunt and Drullion (2014) is pre- sented. The mathematical developement considered by them, for unsteady growing monochro- matic waves is also extended to Stokes waves. The present contribution also demonstrates agree- ment with the pioneering work of Belcher and Hunt (1993) which is valid in the limit of the complex part of the wave phase speed \c_i \downarrow 0. It is further shown that the energy-transfer parameter and the surface shear stress for a Stokes wave reverts to a monochromatic wave when the second harmonic is excluded. Furthermore, the present theory can be used to estimate the amount of energy transferred to each component of nonlinear surface waves on deep water from a turbulent shear flow blowing over it. Finally, it is demonstrated that in the presence of turbulent eddy viscosity the Miles (1957) critical layer does not play an important role. Thus, it is concluded that in the limit of zero growth rate the effect of the wave growth arises from the elevated critical layer by finite turbulent diffusivity, so that the perturbed flow and the drag force is determined by the asymmetric and sheltering flow in the surface shear layer and its matched interaction with the upper region.

Drying shrinkage in concrete assessed by nonlinear ultrasound

This research develops a nonlinear ultrasonic (NLU) technique, second harmonic generation (SHG), to monitor the time-dependent microstructural evolution and shrinkage in concrete over a period from 28 to 55days of age. Drying shrinkage by moisture migration from concrete to its environment causes stress and microcracking and can lead to larger crack formation, which compromises performance. Here, the process of drying is monitored by the SHG method using nonlinear Rayleigh surface waves to obtain the acoustic nonlinearity parameter. The results show large changes in the measured acoustic nonlinearity parameter which is attributed to damage generated during drying shrinkage. Finally, the measured acoustic nonlinearity parameter is used to compare the microstructural condition in hardened concrete as affected by shrinkage mitigation (through the use of shrinkage-reducing admixture) and crack filling (or self-healing) by carbonation.

Phased Array Beam Fields of Nonlinear Rayleigh Surface Waves**Supported by the National Natural Science Foundation of China under Grant Nos 61271356 and 51575541, the National Research Foundation of Korea under Grant Nos 2013-M2A2A9043241 and 2013-R1A2A2A01016042, and the Hunan Provincial Innovation Foundation For Postgraduate under Grant No CX2016B046.

This study concerns calculation of phased array beam fields of the nonlinear Rayleigh surface waves based on the integral solutions for a nonparaxial wave equation. Since the parabolic approximation model for describing the nonlinear Rayleigh waves has certain limitations in modeling the sound beam fields of phased arrays, a more general model equation and integral forms of quasilinear solutions are introduced. Some features of steered and focused beam fields radiated from a linear phased array of the second harmonic Rayleigh wave are presented.

Rogue events in spatiotemporal numerical simulations of unidirectional waves in basins of different depth

The evolution of unidirectional nonlinear sea surface waves is calculated numerically by means of solutions of the Euler equations. The wave dynamics corresponds to quasi-equilibrium states characterized by JONSWAP spectra. The spatio-temporal data are collected and processed providing information about the wave height probability and typical appearance of abnormally high waves (rogue waves). The waves are considered at different water depths ranging from deep to relatively shallow cases ($k_p h > 0.8$, where $k_p$ is the peak wavenumber, and $h$ is the local depth). The asymmetry between front and rear rogue wave slopes is identified; it becomes apparent for sufficiently high waves in rough sea states at all considered depths. The lifetimes of rogue events may reach up to 30-60 wave periods depending on the water depth. The maximum observed wave has height of about 3 significant wave heights. A few randomly chosen in-situ time series from the Baltic Sea are in agreement with the general picture of the numerical simulations.

An operator expansion method for computing nonlinear surface waves on a ferrofluid jet

We present a new numerical method to simulate the time evolution of axisymmetric nonlinear waves on the surface of a ferrofluid jet. It is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To do so, we describe the associated Dirichlet–Neumann operator in terms of a Taylor series expansion where each term can be efficiently computed by a pseudo-spectral scheme using the fast Fourier transform. We show detailed numerical tests on the convergence of this operator and, to illustrate the performance of our method, we simulate the long-time propagation and pairwise collisions of axisymmetric solitary waves. Both depression and elevation waves are examined by varying the magnetic field. Comparisons with weakly nonlinear predictions are also provided.

Method and analysis for determining yielding of titanium alloy with nonlinear Rayleigh surface waves

Methods for determining yielding of titanium (Ti) alloy material with second harmonic Rayleigh ultrasonic wave are investigated. Both piezoelectric angle beam transducers and high frequency laser scanning vibrometer (LSV) are used to detect ultrasonic signals in the Ti alloy specimens with different plastic strain levels. Technical features and outcomes with use of piezoelectric transducers and LSV are compared. The method using piezoelectric transducers, with much higher signal-to-noise ratio than LSV, has been further improved by deploying two transducers with central frequencies corresponding to the fundamental and second order harmonic signals respectively to improve the testing reliability and accuracy. Both the techniques using piezoelectric transducer and LSV demonstrate consistently that the acoustic nonlinearity increases with plastic strain, and the second harmonic Rayleigh ultrasonic wave can be utilized for effective determination of yielding in Ti alloy. Our experiments further show that the acoustic nonlinearity increases gradually with plastic strain at small plastic strain level, and there is a more significant increase of acoustic nonlinearity when the plastic strain reaches a higher level. Microscopic investigations using scanning electron microscopy (SEM) and high resolution transmission electron microscopy (HRTEM) are conducted for clarifying the relationship between the observed acoustic nonlinearity and micro-structural changes.

English

The original investigation of Lamb (1932, §349) for the effect of viscosity on monochromatic surface waves is extended to account for second-order Stokes surface waves on deep water in the presence of surface tension. This extension is used to evaluate interfacial impedance for Stokes waves under the assumption that the waves are growing and hence the surface waves are unsteady. Thus, the previous investigation of Sajjadi et al. (2014) is further explored in that (i) the surface wave is unsteady and nonlinear, and (ii) the effect of the water viscosity, which affects surface stresses, is taken into account. The determination of energy-transfer parameter, from wind to waves, are calculated through a turbulence closure model but it is shown the contribution due to turbulent shear flow is some 20% lower than that obtained previously. A derivation leading to an expression for the closed streamlines (Kelvin cat-eyes), which arise in the vicinity of the critical height, is found for unsteady surface waves. From this expression it is deduced that as waves grow or decay, the cats-eye are no longer symmetrical. Also investigated is the energy transfer from wind to short Stokes waves through the viscous Reynolds stresses in the immediate neighborhood of the water surface. It is shown that the resonance between the Tollmien-Schlichting waves for a given turbulent wind velocity profile and the free-surface Stokes waves give rise to an additional contribution to the growth of nonlinear surface waves.

Development of attenuation and diffraction corrections for linear and nonlinear Rayleigh surface waves radiating from a uniform line source

In recent studies with nonlinear Rayleigh surface waves, harmonic generation measurements have been successfully employed to characterize material damage and microstructural changes, and found to be sensitive to early stages of damage process. A nonlinearity parameter of Rayleigh surface waves was derived and frequently measured to quantify the level of damage. The accurate measurement of the nonlinearity parameter generally requires making corrections for beam diffraction and medium attenuation. These effects are not generally known for nonlinear Rayleigh waves, and therefore not properly considered in most of previous studies. In this paper, the nonlinearity parameter for a Rayleigh surface wave is defined from the plane wave displacement solutions. We explicitly define the attenuation and diffraction corrections for fundamental and second harmonic Rayleigh wave beams radiated from a uniform line source. Attenuation corrections are obtained from the quasilinear theory of plane Rayleigh wave equations. To obtain closed-form expressions for diffraction corrections, multi-Gaussian beam (MGB) models are employed to represent the integral solutions derived from the quasilinear theory of the full two-dimensional wave equation without parabolic approximation. Diffraction corrections are presented for a couple of transmitter-receiver geometries, and the effects of making attenuation and diffraction corrections are examined through the simulation of nonlinearity parameter determination in a solid sample.

Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves (Hamilton et al. in J Acoust Soc Am 97:891–897, 1995), surface waves on current-vortex sheets in incompressible MHD (Ali and Hunter in Q Appl Math 61(3):451–474, 2003; Ali et al. in Stud Appl Math 108(3):305–321, 2002) and on the incompressible plasma–vacuum interface (Secchi in Q Appl Math 73(4):711–737, 2015). The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in Hunter (J Hyperbolic Differ Equ 3(2):247–267, 2006), Secchi (Q Appl Math 73(4):711–737, 2015). In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

Nonlinear surface waves at ferrite-metamaterial waveguide structure

A new ferrite slab made of a metamaterial (MTM), surrounded by a nonlinear cover cladding and a ferrite substrate, was shown to support unusual types of electromagnetic surface waves. We impose the boundary conditions to derive the dispersion relation and others necessary to formulate the proposed structure. We analyse the dispersion properties of the nonlinear surface waves and we calculate the associated propagation index and the film–cover interface nonlinearity. In the calculation, several sets of the permeability of the MTM are considered. Results show that the waves behaviour depends on the values of the permeability of the MTM, the thickness of the waveguide and the film–cover interface nonlinearity. It is also shown that the use of the singular solutions to the electric field equation allows to identify several new properties of surface waves which do not exist in conventional waveguide.

Detuning and wave breaking during nonlinear surface wave focusing

Using an energy focusing technique, several transient wave trains were generated in a two-dimensional wave flume to investigate their nonlinear evolution. By increasing the initial wave steepness (S0), while all other parameters remained the same, non-breaking through breaking conditions were achieved. The experimental results demonstrated that the waves’ initial steepness affected the focusing process significantly. As expected, the nonlinear effects on wave packets with a low initial steepness were very small. However, for cases with large initial steepness, nonlinearity increased causing the wave packet to focus and break prematurely, a “detuning” process. As a result of the detuning process more than one breaker may occur, and more importantly when this ensues, energy loss was altered. It is found that for S0<0.274, only one breaker occurred, and the energy loss increased with increased initial steepness. For larger steepness, two breakers occurred and the total energy loss remained nearly constant (~17%) as the initial steepness was increased. As the initial steepness was increased still further, a large plunging breaking was obtained with additional energy loss (4%). These results suggest that the “detuning” has a pronounced influence not only on the resulting wave field, but on the resulting energy loss.

On the multi-symplectic structure of the Serre–Green–Naghdi equations

In this short note, we present a multi-symplectic structure of the Serre–Green–Naghdi (SGN) equations modelling nonlinear long surface waves in shallow water. This multi-symplectic structure allow the use of efficient finite difference or pseudo-spectral numerical schemes preserving exactly the multi-symplectic form at the discrete level.

Apodized waveguide arrays induced by photorefractive nonlinear surface waves.

A new type of nonlinear waveguides, photorefractive surface optical waveguides is suggested, which can be induced by photorefractive surface waves on the boundary of photorefractive crystal. The disturbed refractive index distribution of such waveguides behaves as a periodic lattice with apodized envelope, thus we call them photorefractive surface apodized waveguide arrays. Moreover, the dispersion relation and corresponding modes are analyzed. It is very interesting that the dispersion curves of index-guided modes and Bragg-guided modes couple and intertwine with each other, and anti-crossings instead of crossings between them hence generate some mini-gaps. Moreover there exists a type of extraordinary modes constituted by the splice of index-guided modes and Bragg-guided modes.

Combination of nonlinear ultrasonics and guided wave tomography for imaging the micro-defects

The use of guided wave tomography has become an attractive alternative to convert ultrasonic wave raw data to visualized results for quantitative signal interpretation. For more accurate life prediction and efficient management strategies for critical structural components, there is a demand of imaging micro-damages in early stage. However, there is rarely investigation on guided wave tomographic imaging of micro-defects. One of the reasons for this might be that it becomes challenging to monitor tiny signal difference coefficient in a reliable manner for wave propagation in the specimens with micro-damages. Nonlinear acoustic signal whose frequency differs from that of the input signal can be found in the specimens with micro-damages. Therefore, the combination of guided wave tomography and nonlinear acoustic response induced by micro-damages could be a feasibility study for imaging micro-damages. In this paper, the nonlinear Rayleigh surface wave tomographic method is investigated to locate and size micro-corrosive defect region in an isotropic solid media. The variations of acoustic nonlinear responses of ultrasonic waves in the specimens with and without defects are used in guided wave tomographic algorithm to construct the images. The comparisons between images obtained by experimental signals and real defect region induced by hydrogen corrosion are presented in this paper. Results show that the images of defect regions with different shape, size and location are successfully obtained by this novel technique, while there is no visualized result constructed by conventional linear ultrasonic tomographic one. The present approach shows a potential for inspecting, locating and imaging micro-defects by nonlinear Rayleigh surface wave tomography.

Nonlinear surface waves on the plasma-vacuum interface

In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagates along the plasma-vacuum interface when it is linearly weakly stable. Following the approach of Ali and Hunter (2003), we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.

Nonlinear s-polarized quasi-surface waves in the symmetric structure with a metamaterial core

The theory of nonlinear s-polarized surface waves, which propagate in a symmetric planar threelayered structure with a metamaterial core and nonlinear coverings, has been constructed. It has been shown that the occurrence of only nonlinear quasi-surface waves is possible in such a structure. Dispersion laws and energy fluxes of symmetric, even, and odd modes at different parameters have been found and investigated. The qualitative behavior of dispersion laws substantially depends on the core thickness.