Edge Wave Modes Research Articles

Second-order topological corner states in two-dimensional elastic wave metamaterials with nonsymmorphic symmetries

This paper investigates the higher-order topological behavior of elastic wave metamaterials composed of L-shaped pillars attached to a plate. In this system, the multi-dimensional topological transitions can be developed for physical transfer among the corner, edge, and bulk elastic wave modes by tuning the rotation angle of the L-shaped pillars. Different from the four degenerate corner states in the symmorphic system, these corner states in this metamaterial evolve into two doublets because of the lack of mirror symmetries. With the energy highly trapped at the corners, it thus can immediately realize elastic resonators/sensors with high-quality factors. Besides, the coupling effect between corner and edge states is analyzed in the multi-dimensional topological system. The edge–corner coupling effect could be applied to relevant fields, such as high-sensitive detection, elastic energy harvesting, and optomechanical sensing.

Complex dispersion relations and evanescent waves in periodic magneto-electro curved phononic crystal plates

This paper presents a novel semi-analytical method to consider complex dispersion relations and evanescent wave characteristics in periodic magneto-electro-electric curved phononic crystal plates, based on the plate theory and Bloch-Floquet theorem. Generalized elastodynamic wave equation considering the magneto-electric effects in cylindrical coordinate system is established and discretized with the curved higher-order spectral elements. Then, the proposed approach is validated by the analysis of accuracy and convergence. The influence of magneto-electro coupling effect on dispersion relations are discussed. Significant parameters of the periodic curved phononic crystal are studied systematically, which fundamentally determines the tunability of complex dispersion relations of various guided wave modes including propagative wave modes, purely evanescent wave modes, evanescent edge wave modes, and complex wave modes. It is witnessed that propagative wave modes lead the bandgap to the emergence, variation, and closure with the change of significant parameters. And evanescent wave modes dominate two individual patterns of bandgaps. When the radius angle or curvature changes, the polarization of bandgaps appears. Furthermore, complex wave modes are generated with the increase of curvature resulting in strengthening the attenuation of propagative wave modes. The variation of lattice ratio can separate the intersection point of two propagative wave modes, which induces the opening of two bandgaps and the attenuation of evanescent wave modes. The results of this work provides may benefit the multi-functional development and application of smart materials and structures.

How sea ice can affect coastal swells, infragravity waves and leaky wave modes: Spectral adaptation from modulation

Several months of seafloor-mounted, pressure-transducer data collected in Mordvinova Bay off the south-east coast of Sakhalin Island in the Sea of Okhotsk are used to interpret surface-gravity waves across a wide range of periods, during intervals when the sea surface was free of sea ice, covered by drift ice of 0.5–1 m thickness, and when both drift ice and consolidated shore fast sea ice was present. A substantial fraction of the data set is shown to be frequency modulated by the diurnal tide and, to a lesser extent, by the semidiurnal tide. Whereas amplitude modulation by the tide has been found in the past in the open ocean, frequency modulation is less common and neither tidal amplitude modulation nor angle modulation have been observed and recorded under sea ice to the authors’ knowledge. Three theoretical oceanographic modalities that could excite the modulation are discussed. Although the sea ice affected the wind-generated waves and swells in the measured energy spectra as predicted, namely by preferential attenuation that causes spectra to become narrower by progressively reducing shorter periods with distance travelled, its effect on waves of longer periods, i.e. 1–10 min, was unanticipated. Oscillations in this band, present as infragravity (IG) waves, leaky waves and an edge wave mode are shown to be affected as well; especially spectral peaks corresponding to IG/leaky waves, which respond to the sea ice by shifting to a slightly longer mean period and becoming broader. The period shift is explained by the spectral sharpening previously mentioned, whilst the broadening is conjectured to be associated with changes to the frequency modulation.

The fundamental ultrasonic edge wave mode: Propagation characteristics and potential for distant damage detection

Engineering structures are often composed of thin elements containing features such as free edges, welds, ribs, and holes, which makes distant safety inspections based on guided waves difficult due to wave scattering. However, these features can themselves generate so-called ‘feature-guided’ waves, which can potentially be utilised for damage detection. One such example are flexural wedge waves, which have been investigated extensively both theoretically and experimentally in the past. Another example is edge waves. These waves, which are a natural analogue of Rayleigh waves propagating in a finite thickness plate, have received relatively little attention, specifically with respect to their possible use in distant damage inspections and Structural Health Monitoring systems. The current paper is aimed to address this gap, and it is focused on the investigation of the fundamental mode of edge waves (ES0), which is the most promising for practical applications. The features of the transient ES0 mode are investigated experimentally and numerically, and compared with previous theoretical studies. It was demonstrated that the ES0 mode can be effectively excited with the wedge excitation method, and distant damage detection with this wave mode at low frequency-thickness values (FTV < 5) is readily achievable. In particular, in a laboratory environment the ES0 mode propagated several meters with almost no decay. However, at higher frequency-thickness values, a wave amplitude modulation, significant energy decay and strong coupling between the ES0 and S0 wave modes were observed. These phenomena may restrict the defect resolution as well as the range of damage inspections based on the fundamental edge wave mode.

Energetic Rupture and Tsunamigenesis during the 2020 Mw 7.4 La Crucecita, Mexico Earthquake

AbstractThe La Crucecita earthquake ruptured on the megathrust, generating strong shaking and a modest but long-lived tsunami. This is a significant earthquake that illuminates important aspects of the behavior of the megathrust as well as the potential related hazards. The rupture is contained within 15–30 km depth, ground motions are elevated, and the energy to moment ratio is high. We argue that it represents a deep megathrust earthquake, the 30 km depth is the down-dip edge of slip. The inversion is well constrained, ruling out any shallow slip. It is the narrow seismogenic width and the configuration of the coastline that allow for deformation to occur offshore. The minor tsunamigenesis can be accounted for by the deep slip patch. There is a significant uplift at the coast above it, which leads to negative maximum tsunami amplitudes. Finally, tide-gauge recordings show that edge-wave modes were excited and produce larger amplitudes and durations in the Gulf of Tehuantepec.

Effect of Coriolis force on edge waves (I) Investigation of the normal modes

Essentially two classes of free edge waves can exist on a sloping continental shelf in the presence of Coriolis force. For small longshore wave length, fundamental waves of the first class behave like Stokes edge waves. However, for great wave lengths (of several hundred kilometers or more) the characteristics of the first class are significantly altered. In the northern hemisphere the phase speed for waves moving to the right (facing shore from the sea) exceeds the speed for waves which move to the left. Also, the group velocity for a given edge wave mode has a finite upper limit. Waves of the second class are essentially quasigeostrophic boundary waves with very low frequency and, like Kelvin waves, move only to the left (again facing shore from the sea). Unlike Stokes edge waves, those of the quasigeostrophic class are associated with large vorticity. Examination of the formal solution for forced edge waves indicates that those of the second class may be excited significantly by a wind stress vortex. Also, in contrast to the conclusion of Greenspan (1956), it is proposed that a hurricane can effectively excite the higher order edge wave modes in addition to the fundamental if wind stress is considered.

A Numerical Approach for Calculation of Characteristics of Edge Waves in Three-Dimensional Plates

Surface waves have been extensively studied in earthquake seismology. Surface waves are trapped near an infinitely large surface. The displacements decay exponentially with depth. These waves are also named Rayleigh and Love waves. Surface waves are also used for nondestructive testing of surface defects. Similar waves exist in finite width three-dimensional plates. In this case, displacements are no longer constant in the direction perpendicular to the wave propagation plane. Wave energy could still be trapped near the edge of the three-dimensional plate, and hence the term edge waves. These waves are thus different to the two-dimensional Rayleigh and Love waves. This paper presents a numerical model to study dispersion properties of edge waves in plates. A two-dimensional semi-analytical finite element method is developed, and the problem is closed by a perfectly matched layer adjacent to the edge. The numerical model is validated by comparing with available analytical and numerical solutions in the literature. On this basis, higher order edge waves and mode shapes are presented for a three-dimensional plate. The characteristics of the presented edge wave modes could be used in nondestructive testing applications.

Damage detection with the fundamental mode of edge waves

Detection of mechanical damage using Lamb or Rayleigh waves is limited to relatively simple geometries, yet real structures often incorporate features such as free or clamped edges, welds, rivets, ribs and holes. All these features are potential sources of wave reflections and scattering, which make the application of these types of guided waves for damage detection difficult. However, these features can themselves generate so-called ‘feature-guided’ waves. This article details the first application of the fundamental mode of transient edge waves for detection of mechanical damage. The fundamental edge wave mode (ES0) – a natural analogue to Rayleigh waves – is weakly dispersive and may decay with propagation distance. The phase and group velocities of the ES0 wave mode are close to the fundamental shear horizontal (SH0) and symmetric Lamb (S0) wave modes, at low and high frequencies, respectively. It is therefore quite challenging to excite a single ES0 mode and avoid wave coupling. However, it was found experimentally that at medium range frequencies the ES0 mode can be decoupled from SH0 and S0 modes, and its decay is small, allowing for distant detection of defects and damage along free edges of slender structural components. This article provides a brief theory of edge waves, excitation methodology and successful examples of distant detection of crack-like and corrosion damage in I-beam sections, which are widely applied in engineering and construction.

Generation of higher harmonics with the fundamental edge wave mode

The measurement and evaluation of nonlinear elastic properties represents a great interest in materials science and engineering. These measurements can be conducted using the internal resonance of higher harmonics generated with ultrasonic bulk, Rayleigh, and Lamb waves. However, these waves are spatially dispersive, making the evaluation very difficult. In this paper, we present the outcomes of an experimental study on the generation of higher harmonics with the fundamental mode of edge waves. This wave mode—an analog of the Rayleigh wave—propagates along the apex of an elastic plate, where the lateral plate surfaces naturally avoid spatial dispersion. In addition, this mode is weakly dispersive in the frequency domain enabling approximate internal resonance, which is required for the measurement of material nonlinearities. The current experimental results indicate that the second harmonic of the fundamental edge wave mode grows linearly with the propagation distance. The linear rate of accumulation of this harmonic can then be related to the nonlinear elastic properties as procured by previous theoretical studies.

Infragravity Wave Energy Partitioning in the Surf Zone in Response to Wind-Sea and Swell Forcing

An alongshore array of pressure sensors and a cross-shore array of current velocity and pressure sensors were deployed on a barred beach in southwestern Australia to estimate the relative response of edge waves and leaky waves to variable incident wind wave conditions. The strong sea breeze cycle at the study site (wind speeds frequently &gt; 10 m s−1) produced diurnal variations in the peak frequency of the incident waves, with wind sea conditions (periods 2 to 8 s) dominating during the peak of the sea breeze and swell (periods 8 to 20 s) dominating during times of low wind. We observed that edge wave modes and their frequency distribution varied with the frequency of the short-wave forcing (swell or wind-sea) and edge waves were more energetic than leaky waves for the duration of the 10-day experiment. While the total infragravity energy in the surf zone was higher during swell forcing, edge waves were more energetic during wind-sea periods. However, low-frequency (0.005–0.023 Hz) edge waves were found to be dominant in absence of wind-sea conditions, while higher-frequency (0.023–0.050 Hz) edge waves dominated when wind-sea conditions were present.

Effect of Dynamical Phase on the Resonant Interaction Among Tsunami Edge Wave Modes

Different modes of tsunami edge waves can interact through nonlinear resonance. During this process, edge waves that have very small initial amplitude can grow to be as large or larger than the initially dominant edge wave modes. In this study, the effects of dynamical phase are established for a single triad of edge waves that participate in resonant interactions. In previous studies, Jacobi elliptic functions were used to describe the slow variation in amplitude associated with the interaction. This analytical approach assumes that one of the edge waves in the triad has zero initial amplitude and that the combined phase of the three waves φ = θ1 + θ2 − θ3 is constant at the value for maximum energy exchange (φ = 0). To obtain a more general solution, dynamical phase effects and non-zero initial amplitudes for all three waves are incorporated using numerical methods for the governing differential equations. Results were obtained using initial conditions calculated from a subduction zone, inter-plate thrust fault geometry and a stochastic earthquake slip model. The effect of dynamical phase is most apparent when the initial amplitudes and frequencies of the three waves are within an order of magnitude. In this case, non-zero initial phase results in a marked decrease in energy exchange and a slight decrease in the period of the interaction. When there are large differences in frequency and/or initial amplitude, dynamical phase has less of an effect and typically one wave of the triad has very little energy exchange with the other two waves. Results from this study help elucidate under what conditions edge waves might be implicated in late, large-amplitude arrivals.

Numerical study of edge waves using extended Boussinesq equations

An edge wave numerical model was developed based on extended Boussinesq equations with the internal wave-generation method. The form of edge waves near a seawall was chosen as the input signal in order to avoid treatment of the moving shoreline on a sloping beach. As there was an energy transfer between different edge wave modes, not only the target mode but also other modes appeared in the simulations. Due to the nonlinear effect, the simulation results for mode-0 edge waves were slightly modulated by mode-1 and mode-2 waves. As the magnitudes of these higher-mode waves are not significantly related to those of the target mode, the internal wave-generation method in Boussinesq equations can produce high-quality edge waves. The numerical model was used to investigate the nonlinear properties of standing edge waves, and the numerical results were in strong agreement with theory.

Wavenumber–frequency analysis of the landslide-generated tsunamis at a conical island

In this paper we investigate the propagation of tsunamis generated by landslides around the coast of an island to understand the propagation and trapping mechanisms of the waves. Records of shoreline displacement have been processed using the wavenumber–frequency analysis (k–f). We identify the dispersion relation followed by the waves that propagate alongshore. It appears that the 0th-order edge wave mode is the only one relevant for shoreline run-up. Furthermore, it is shown that the edge wave dispersion relation is a quantitative tool to estimate the phase and group wave celerities. A very good agreement is found when comparing the wave celerities as calculated from the experimental records, against those predicted by the edge wave theory. Furthermore the analysis of the tsunami around the island has been carried out by means of the two-dimensional k–f. Using as input data a high-space resolution dataset of free surface elevations it is found that other modes, both trapped (1st-order edge waves) and non-trapped (free radiating waves), occur during the propagation.

Measurement of Infragravity Edge Waves on the East Coast of Taiwan Using a Single Instrument

Abstract Traditionally, an edge wave is measured along a long and straight coast with pressure-gauge arrays to acquire the frequency-wavenumber for dispersion relationships. Ideal measurement locations are limited, and the cost of purchasing and operating some 20 instruments emerges as a threshold requirement. In the present study, one instrument measuring both the bottom pressure and the flow velocity was used to retrieve edge-wave information on the east coast of Taiwan outside the Hua-Lien Harbor. Using the progressive linear-wave assumption for irrotational and inviscid shallow water provided the pressure-flow relationships for both the infragravity edge and leaky waves. The relationship for edge waves was helpful in identifying the edge-wave mode, whereas the relationship for the leaky waves eliminated the possibility of incorrectly interpreting a leaky wave as an edge wave. Applying this approach to field measurements on the east coast of Taiwan shows that a Stokes edge wave is a significant compone...

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size (2a), storm speed (U), and beach slope (α). Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, Ucr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 ≤ y ≤ Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.

Relative role of subinertial and superinertial modes in the coastal long wave response forced by the landfall of a tropical cyclone

A set of numerical experiments has been performed in order to analyze the long-wave response of the coastal ocean to a translating mesoscale atmospheric cyclone approaching the coastline at a normal angle. An idealized two-slope shelf topography is chosen. The model is forced by a radially symmetric atmospheric pressure perturbation with a corresponding gradient wind field. The cyclone's translation speed, radius, and the continental shelf width are considered as parameters whose impact on the long wave period, modal structure, and amplitude is studied. Subinertial continental shelf waves (CSW) dominate the response under typical forcing conditions and on the narrower shelves. They propagate in the downstream (in the sense of Kelvin wave propagation) direction. Superinertial edge wave modes have higher free surface amplitudes and faster phase speeds than the CSW modes. While potentially more dangerous, edge waves are not as common as subinertial shelf waves because their generation requires a wide, gently sloping shelf and a storm system translating at a relatively high (∼10 m s −1 or faster) speed. A relatively smaller size of an atmospheric cyclone also favors edge wave generation. Edge waves with the highest amplitude (up to 60% of the forced storm surge) propagate upstream. They are produced by a storm system with an Eulerian time scale equal to the period of a zero-mode edge wave with the wavelength of the storm spatial scale. Large amplitude edge waves were generated during Hurricane Wilma's landfall (2005) on the West Florida shelf with particularly severe flooding occurring upstream of the landfall site.

On the representation of Rossby waves on the β-plane by a piecewise uniform potential vorticity distribution

To bridge quasi-geostrophic dynamics and its discrete representation by a series of piecewise constant potential vorticity (PV), the dispersion relation for the Rossby wave in the single-layer β-plane is compared with that for the normal mode of edge waves straddling an infinite series of PV discontinuities (‘PV staircase’). It is shown that the edge waves over evenly spaced, uniform-height PV steps converge to the Rossby wave on the β-plane as Δ → 0, L → 0, Δ/L = βeff (Δ, L and βeff are the step size, step separation and the effective β, respectively), whereas they reduce to the single-step edge wave in the short-wave limit. For sufficiently small step separations, the difference in the phase velocities of the edge wave and the Rossby wave scales as O(L2). Two effects of increasing L on the zonal propagation are identified: (i) increased phase and group velocities in the short-wave limit due to an increased zonal wind at the PV steps and (ii) decreased phase and group velocities in the long-wave limit due to a decreased effective meridional tilt of the mode. The reduced tilt also severely limits the meridional group propagation. The relationship between the edge wave mode and the finite-difference approximation to the Rossby wave is also discussed.

Large‐scale edge waves generated by hurricane landfall

Direct observations of the storm surge induced by Hurricane Wilma's landfall revealed a formation of a wave pulse propagating alongshore. The height of the wave pulse exceeded 1.5 m in the detided sea level. The duration of this wave pulse was ∼6 h and the propagation speed was of O(10) m s−1. This wave has been identified as an edge wave of large spatial and temporal scales. A set of numerical experiments has been conducted to delineate a generation of edge waves with large spatial and temporal scales by a fast‐moving storm system. The model of the coastal ocean was set in a two‐dimensional configuration with the continental shelf and slope topography reminiscent of the West Florida shelf. A moving cyclonic system in the gradient wind balance was prescribed analytically. In order to identify a long wave response in the model, a linear boundary problem was solved, yielding dispersion characteristics and a structure of the edge wave modes. A fast‐moving storm system crossing the shelf at a right angle produces a nearly symmetrical response of two edge wave trains propagating both downstream (in the direction of a Kelvin wave) and upstream. Typically, zero‐mode edge waves dominate the response. As the translation speed of the storm becomes lower, its Eulerian timescale becomes longer and the waves are more affected by the Earth's rotation. In that case, the wave energy propagates predominantly downstream. When the storm trajectory deviates from the normal approach, the edge wave response is not symmetric: most of the energy propagates in the direction of the alongshore component of the storm translation velocity.

Acoustic-Gravity Waves from Bolide Sources

We have developed a new approach to modeling the acoustic-gravity wave (AGW) radiation from bolide sources. This first effort involves entry modeling of bolide sources that have available satellite data through procedures developed in ReVelle (Earth Moon Planets 95, 441–476, 2004a; in: A. Milani, G. Valsecchi, D. Vokrouhlicky (eds) NEO Fireball Diversity: Energetics-based Entry Modeling and Analysis Techniques, Near-earth Objects: Our Celestial Neighbors (IAU S236), 2007b). Results from the entry modeling are directly coupled to AGW production through line source blast wave theory for the initial wave amplitude and period at \(x=10\) (at 10 blast wave radii and perpendicular to the trajectory). The second effort involves the prediction of the formation and or dominance of the propagation of the atmospheric Lamb, edge-wave composite mode in a viscous fluid (Pierce, J. Acoust. Soc. Amer. 35, 1798–1807, 1963) as a function of the source energy, horizontal range and source altitude using the Lamb wave frequency that was deduced directly during the entry modeling and that is used as a surrogate for the source energy. We have also determined that Lamb wave production by bolides at close range decreases dramatically as either the source energy decreases or the source altitude increases. Finally using procedures in Gill (Atmospheric-Ocean Dynamics, 1982) and in Tolstoy (Wave Propagation, 1973), we have analyzed two simple dispersion relationships and have calculated the expected dispersion for the Lamb edge-wave mode and for the excited, propagating internal acoustic waves. Finally, we have used the above formalism to fully evaluate these techniques for four large bolides, namely: the Tunguska bolide of June 30, 1908; the Revelstoke bolide of March 31, 1965; the Crete bolide of June 6, 2002 and the Antarctic bolide of September 3, 2004. Due to page limitations, we will only present results in detail for the Revelstoke bolide.

The California tsunami of 15 June 2005 along the coast of North America

An Mw = 7.2 earthquake occurred on 15 June 2005 (utc) seaward of northern California off the west coast of North America. Based on the earthquake location and source parameters, the West Coast and Alaska Tsunami Warning Center issued a tsunami warning for the region extending from the California‐Mexico border to northern Vancouver Island, British Columbia (the first tsunami warning for this region since the 1994 Mw = 8.2 Shikotan earthquake). Six tide gauges on the west coast recorded tsunami waves from this event, with a maximum trough‐to‐crest wave height of 27.7 cm observed at Crescent City, California. Waves of 2.5 to 6.5 cm were measured at the five other sites: Port Orford (Oregon), North Spit and Arena Cove (California), and Tofino and Bamfield (British Columbia). The open‐ocean Deep‐ocean Assessment and Reporting of Tsunami (DART) buoys, 46404 and 46405, recorded tsunami waves of 0.5 and 1.5 cm, respectively, closely matching wave heights derived from numerical models. Incoming tsunami wave energy was mainly at periods of 10 to 40 min. The observed tsunami wave field is interpreted in terms of edge (trapped) and leaky (non‐trapped) waves and a “trapping coefficient” is introduced to estimate the relative contribution of these two wave types. Due to the high (3000 m) water depth in the source area, approximately two‐thirds of the total tsunami energy went to leaky wave modes and only one‐third to edge wave modes. The improved response to and preparedness for the 2005 California tsunami compared to the 1994 Shikotan tsunami is attributable, in part, to the operational capability provided by the open‐ocean bottom‐pressure recorder (DART) system, higher quality coastal tide gauges, and the effective use of numerical models to simulate real‐time tsunamis.