Edge Wave Amplitude Research Articles

Subharmonic edge wave excitation by narrow-band, random incident waves

A monochromatic, small amplitude, normally incident standing wave on a sloping beach is unstable to perturbation by subharmonic (half the frequency) edge waves. At equilibrium, edge wave shoreline amplitudes can exceed incident wave amplitudes. Here, the effect of incident wave randomness on subharmonic edge wave excitation is explored following a weakly nonlinear stability analysis under the assumption of narrow-band incident random waves. Edge waves respond to variations in both incident wave phase and amplitude, and the edge wave amplitudes and incident wave groups vary on similar time scales. When bottom friction is included, intermittent subharmonic edge wave excitation is predicted due to the combination of bottom friction and wave phase. Edge wave amplitude can be near zero for long times, but for short periods reaches relatively large values, similar to amplitudes with monochromatic incident waves and no friction.

A numerical algorithm for computing tsunami wave amplitudes

A numerical multistep algorithm for computing tsunami wave front amplitudes is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved with Godunov’s approach and a bicharacteristic method. A qualitative comparison of the two methods is done. A change of variables is made with the eikonal equation solution at the second step. At the last step, using an expansion of the fundamental solution to the shallow water equations in the new variables, we obtain a Cauchy problem of lesser dimension for the leading edge wave amplitude. The results of numerical experiments are presented.

Physical modeling of resonance phenomena in the long wave dynamics

We have prepared two sets of experiments in a wave flume to model effects occurring in nature and to demonstrate resonance phenomena in laboratory conditions. The first set was performed to investigate non-linear wave run-up on the beach caused by harmonic wave maker located at some distance from the shore line. It is revealed that under certain wave excitation frequencies a significant increase in run-up amplification is observed [Ezersky et al. 2013]. It is found that this amplification is due to the excitation of resonant mode in the region between the shoreline and wave maker. Frequency and magnitude of the maximum amplification are in good correlation with the numerical calculation results represented in the recently published paper [Stefanakis et al. 2011]. The second set of experiments was performed to study resonance effects due to parametric excitation of edge waves. It is known that surface waves propagating toward the shore can excite edge waves propagating along the shore line. Although the edge wave amplitude decreases in an offshore direction they may contain enough energy to be responsible for erosion of the shore and generate so-called cusps [Buchan et al. 1995]. We investigate parametric mechanism of such generation when plane surface wave with frequency W excite edge wave with frequency W/2. It is show that parametric generation of edge waves can amplify run-up up to two times.

RING EDGE WAVES AND THE MASSES OF NEARBY SATELLITES

Moons embedded in gaps within Saturn’s main rings generate waves on the gap edges due to their gravitational disturbances. These edge waves can serve as diagnostics for the masses and, in some cases, orbital characteristics of the embedded moons. Although N-body simulations of the edges are far better in inferring masses from edge morphology, the long run-times of this technique often make it impractical. In this paper, we describe a faster approach to narrow the range of masses to explore with N-body simulations, to explore the multidimensional parameter space of edge/moon interactions, and to guide the planning of spacecraft observations. Using numerical, test-particle models and neglecting particle–particle interactions, we demonstrate that the simple analytic theory of the edge waves applies well to Pan in the Encke Gap but breaks down for smaller moons/ gaps like Daphnis in the Keeler Gap. Fitting an analytic model to our simulation results allows us to suggest an improved relationship between moon-mass and edge wave amplitude. Numerical methods also grant freedom to explore a wider range of moon and ring orbits than the circular, coplanar case considered by analytic theory. We examine how pre-encounter inclinations and eccentricities affect the properties of the edge waves. In the case where the moon or ring-edge particle orbits initially have eccentric radial variations that are large compared to the gap width, there is considerable variation in edge wave amplitude depending on the orbital phase of the encounter. Inclined moons also affect the edge wave amplitude, potentially significantly, as well as generate vertical waves on the gap-edges. Recent Cassini images acquired as Saturn approaches equinox and the Sun’s elevation on the ringplane is extremely low have revealed long shadows associated with the Keeler gap edge waves created by the embedded moon Daphnis. We interpret these as being cast by ∼1 km high vertical structure in the waves created by Daphnis’ out-of-plane perturbations on the ring particles.

Optimization of Hydrophone Centering in Circular Ultrasonic Transducer during Field Characterization Using Edge Waves: A Feasibility Study

In this article, we present a feasibility study using transducer edge waves to center a hydrophone during field characterization. A mathematical analysis is presented to study amplitude changes in edge waves; this analysis is based on the impulse response method and considers the transducer as a piston embedded in a rigid planar baffle. The experimental setup used to measure the edge waves is also presented. It has been observed, mathematically and experimentally, that edge wave amplitude reaches its maximum when the hydrophone is centered and that these amplitude changes are very sensitive to position, approximately 70% of its maximal amplitude; however, amplitude changes in front planar waves are less evident (10%). Agreement between experimental and calculated results demonstrates that changes in edge wave amplitude can be used to center the hydrophone in an objective way.

Optimal growth of Eady edge waves

The classical Eady model with interior perturbation potential vorticity q' = 0 and a modification with β ≠ 0 are treated in terms of edge waves, which are propagating on the oppositely directed pseudo-potential vorticity gradients induced by the rigid upper and lower lid. By this reformulation it is made possible to derive easily interpretable equations for optimal instantaneous growth. In the modified model the coupling between the amplitudes of the edge waves is not equal in contrast to the classical model. Nevertheless, it is proven that the optimally instantaneously growing disturbances (OIs) and their growth rates are identical in both models. The ability of the OIs to maintain their growth rates for longer time intervals is examined. This ability depends on the similarity between the normalmode (NM) structures and the OI structures and therefore differs between the two models. The singular vectors (SVs) of both models are computed, and it is shown that their growth rates and their evolution can be explained with the aid of the previous results, i.e. it is possible to deduce the SV growth rates qualitatively by modulating the OI growth rates with the ability of the SVs to maintain these optimal growth rates. Correspondingly, the SV evolution is determined by the OI and NM structures. The SVs evolve such that they transit exactly through the OI structures toward the growing NM structure or to its remnants near to the cutoffs, where a growing NM does not exist. The norm-dependency of the OIs and SVs is examined and on the basis of these results it is possible to classify all norms considered here into two groups with qualitatively different characteristics. The results suggest that the unequal coupling between the edge wave amplitudes, which is investigated here, might be relevant to rapid surface cyclone development and Petterssen type B development, in case the vertical windshear decreases with height.

Edge wave scattering by a coastal structure

The propagation of an incident edge wave on a straight beach with a coastal structure perpendicular to the coastline is analyzed. A simple idealized beach profile with a steep foreshore, a plane slope and a horizontal shelf on an infinitely long straight shoreline is considered. The edge wave is reflected and transmitted at the structure and, by assuming that its width is much smaller than the alongshore wave length, the solution to the problem is obtained by a mode matching method including the head loss at the structure. The existence of the steep foreshore and the shelf modifies the dispersion equation of the trapped modes obtained by Eckart (Wave Report 100, Ref. 51-12, Scripps Institute of Oceanography, University of California, La Jolla, 1951, 99pp). The reflection coefficient, defined by the ratio of the incident and reflected edge wave amplitudes, depends on the friction coefficient at the coastal structure.

Resonant scattering of edge waves by longshore periodic topography: finite beach slope

The resonant scattering of low-mode progressive edge waves by small-amplitude longshore periodic depth perturbations superposed on a plane beach has recently been investigated using the shallow water equations (Chen & Guza 1998). Coupled evolution equations describing the variations of edge wave amplitudes over a finite-size patch of undulating bathymetry were developed. Here similar evolution equations are derived using the full linear equations, removing the shallow water restriction of small (2N + 1)θ, where N is the maximum mode number considered and θ is the unperturbed planar beach slope angle. The present results confirm the shallow water solutions for vanishingly small (2N + 1)θ and allow simple corrections to the shallow water results for small but finite (2N + 1)θ. Additionally, multi-wave scattering cases occurring only when (2N + 1)θ = O(1) are identified, and detailed descriptions are given for the case involving modes 0, 1, and 2 that occurs only on a steep beach with θ = π/12.

Resonant scattering of edge waves by longshore periodic topography

The resonant scattering of topographically trapped, low-mode progressive edge waves by longshore periodic topography is investigated using a multiple-scale expansion of the linear shallow water equations. Coupled evolution equations for the slowly varying amplitudes of incident and scattered edge waves are derived for small-amplitude, periodic depth perturbations superposed on a plane beach. In ‘single-wave scattering’, an incident edge wave is resonantly scattered into a single additional progressive edge wave having the same or different mode number (i.e. longshore wavenumber), and propagating in the same or opposite direction (forward and backward scattering, respectively), as the incident edge wave. Backscattering into the same mode number as the incident edge wave, the analogue of Bragg scattering of surface waves, is a special case. In ‘multi-wave scattering’, simultaneous forward and backward resonant scattering results in several (rather than only one) new progressive edge waves. Analytic solutions are obtained for single-wave scattering and for a special case of multi-wave scattering involving mode-0 and mode-1 edge waves, over perturbed depth regions of both finite and semi-infinite longshore extent. In single-wave backscattering with small (subcritical) detuning (i.e. departure from exact resonance), the incident and backscattered wave amplitudes both decay exponentially with propagation distance over the periodic bathymetry, whereas with large (supercritical) detuning the amplitudes oscillate with distance. In single-wave forward scattering, the wave amplitudes are oscillatory regardless of the magnitude of the detuning. Multi-wave solutions combine aspects of single-wave backward and forward scattering. In both single- and multi-wave scattering, the exponential decay rates and oscillatory wavenumbers of the edge wave amplitudes depend on the detuning. The results suggest that naturally occurring rhythmic features such as beach cusps and crescentic bars are sometimes of large enough amplitude to scatter a significant amount of incident low-mode edge wave energy in a relatively short distance (O(10) topographic wavelengths).

Edge waves revisited

For edge waves on an infinite beach of constant slope comparison is made between the full linear solution and the shallow water approximation. Modifications to the shallow-water edge-wave dispersion relation accounting for an offshore shelf are obtained and compared with an earlier approximation. A physically comprehensible description of edge waves is given, using a geometrical optics approach, leading to approximate expressions for their dispersion relation. Both shallow water and general depth are considered. For the shallow water case comparison is made with the exact relation for an exponentially seaward decreasing slope. Finally the maximum possible edge wave amplitude is estimated by heuristic arguments.

Subharmonic generation of edge waves by nonuniform incident waves

Subharmonic resonance of edge waves by incident and reflected waves on a plane beach has been studied thus far either without any long-shore modulation or with long-shore modulation on the scale O( kϵ) −1 where k is the wave number of the edge wave and ϵ the characteristic wave slope. In the former case, the evolution equation of the edge wave amplitude is an ordinary differential equation involving a cubic term; in the latter it becomes coupled partial differential equations of the cubic Schrödinger type. We study here yet another kind of long-shore modulation as a result of nonuniformity in the envelope of the incident/reflected waves. In addition to the initial growth, nonlinear forced evolution is also examined. It is shown that incident reflected wave packet of long-shore width on the scale O( kϵ 3) −1 or longer will essentially leak O(1) subharmonic progressive edge waves from the sides.

Long-period waves on a natural beach

A field experiment is described in which three two-component electromagnetic flowmeters were used to measure simultaneously the longshore (υ) and onshore-offshore (u) velocity components along a line normal to the shoreline and up to 100 m offshore. Spectral analysis of the data from this experiment reveals the presence of a set of discrete spectral peaks of low frequency (0.014–0.05 Hz) which dominate over the wind wave peak close to the shoreline and which decay in amplitude with distance from the shore. The amplitudes of the velocity components for each of the four lowest-frequency (and clearest) peaks have been plotted against distance from the shoreline and are found to compare satisfactorily with the calculated edge wave amplitude variation for the beach. Phases between u, υ at one flowmeter and between u, u at different distances offshore have also been found from cross spectra and confirm that the peaks are due to edge waves. It is suggested that each of these low-frequency peaks corresponds to a progressive edge wave mode at the cutoff frequency for the beach. Ball (1967) and Guza and Inman (1975) calculate that the cutoff frequency υ for an edge wave of mode n should be proportional to [n(n + 1 )]1/2, where the constant of proportionality is dependent on the subaqueous beach profile. The frequencies of the four lowest-frequency peaks agree well with the predicted frequencies for the lowest modes of cutoff edge waves (n = 1, · · ·, 4), and the observed offshore decay of amplitude at each frequency indicates that assigning the peaks to these modes is correct. Energy exchange between these cutoff modes, through nonlinear interaction, is also suggested by the results. The significance of observing progressive edge waves on this beach is briefly discussed.

Edge waves and beach cusps

Genetically, beach cusps are of at least two types: those linked with incident waves which are surging and mostly reflected (reflective systems) and those generated on beaches where wave breaking and nearshore circulation cells are important (dissipative systems). The spacings of some cusps formed under reflective wave conditions both in the laboratory and in certain selected natural situations are shown to be consistent with models hypothesizing formation by either (1) subharmonic edge waves (period twice that of the incident waves) of zero mode number or (2) synchronous (period equal to that of incident waves) edge waves of low mode. Experiments show that visible subharmonic edge wave generation occurs on nonerodable plane laboratory beaches only when the incident waves are strongly reflected at the beach, and this observation is quantified. Edge wave resonance theory and experiments suggest that synchronous potential edge wave generation can also occur on reflective beaches and is a higher-order, weaker resonance than the subharmonic type. In dissipative systems, modes of longshore periodic motion other than potential edge waves may be important in controlling the longshore scale of circulation cells and beach morphologies. On reflective plane laboratory beaches, initially large subharmonic edge waves rear-rage sand tracers into shapes which resemble natural beach cusps, but the edge wave amplitudes decrease as the cusps grow. Cusp growth is thus limited by negative feedback from the cusps to the edge wave excitation process. Small edge waves can form longshore periodic morphologies by providing destabilizing perturbations on a berm properly located in the swash zone. In this case the retreating incident wave surge is channelized into breeches in the berm caused by the edge waves, and there is an initially positive feedback from the topography to longshore periodic perturbations.