The existence of Rayleigh–Bloch surface waves (original) (raw)

Quarterly of Applied Mathematics, 2003

Existence of surface waves of non-Rayleigh type propagating on some anisotropic elastic half-spaces is proved. Conditions for originating the non-Rayleigh type waves are analyzed. An example of a transversely isotropic material admitting a surface wave of the non-Rayleigh type is constructed.

Rayleigh–Bloch waves trapped by a periodic perturbation: exact solutions

Zeitschrift für angewandte Mathematik und Physik

Exact solutions describing the Rayleigh-Bloch waves for the two-dimensional Helmholtz equation are constructed in the case when the refractive index is a sum of a constant and a small amplitude function which is periodic in one direction and of finite support in the other. These solutions are quasiperiodic along the structure and exponentially decay in the orthogonal direction. A simple formula for the dispersion relation of these waves is obtained.

On Bloch waves for the Stokes equations

2016

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in R d. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

Two-dimensional steady edge waves. Part I: Periodic waves

Wave Motion, 2009

We prove existence and uniqueness for two-dimensional steady water waves propagating along the beach. For small periodic shoreline data, global solutions vanishing in the seaward direction are found. In addition, we prove a priori properties of solutions, well-adapted to the physical background.

Wave Propagation in Infinite Periodic Structures

2016

This paper explores the possibility of generalised periodic structure waves (PSW) that include the well-known Bloch-Floquet (BF) waves as a special case. We consider two types of structure waves (SW) in an infinite, uniform, one dimensional structure of equally spaced scatterers that also absorb energy. For the first structure wave type (SW1), forward transmission and backward reflection phase shifts are independent of wave propagation direction. For a second structure wave type (SW2), the phase shifts have opposite signs for opposite directions of propagation. Examples of SW1 are bending waves, such as flexural waves of a plate, and for SW2 longitudinal waves, such as acoustic waves in a fluid. The differences in amplitudes and phases of the forward and backward SW within any "cell" between adjacent scatterers are found to be equivalent to continuous PSW convolved with a periodic structure function. Finding the PSW dispersion relations requires a function that is the solution of a quadratic equation derived from imposing the same relative SW amplitudes and phases in all cells. Conservation of energy identifies physically acceptable PSW. For no energy absorption and backward and forward scatter phase shifts differing by 2 /   , PSW of the first type (PSW1) are BF waves that propagate unattenuated in passing bands and are evanescent in stopping bands. Including energy absorption for the same phase shifts, PSW1 propagation occurs at all wavenumbers but is attenuated. This extends the BF dispersion relations to include energy absorption which blurs the distinction between passing and stopping bands. For other scatterer phase shifts, PSW1 may still be possible but only at discrete wavenumbers. In contrast PSW of the second type (PSW2) are only consistent with conservation of energy at discrete stopping wavenumbers that are the Bragg reflection condition. PSW1 also exhibit Bragg reflection, but as a narrow stopping band for small scatterer reflectivity and energy absorption. A theory for incoherent wave energy scattering in an infinite periodic structure is also developed, and its results for energy reflection, transmission and absorption are similar to those of PSW1 except for coherence effects.

The zero surface tension limit two-dimensional water waves

Communications on Pure and Applied Mathematics, 2005

We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.

On a new type of solitary surface waves in finite water depth

Many models of shallow water waves, such as the famous Camassa-Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e. the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.

Wave propagation in infinite periodic structures taking into account energy absorption

2015

Interfacial periodic waves of permanent form with free-surface boundary conditions

preprint, 2000

In a two-fluid system where the upper surface of the upper fluid is free, there are two independent modes of oscillation about the state of equilibrium, an ‘internal’ mode and an ‘external’ mode, which are described by two distinct dispersion curves. An efficient numerical scheme based on Fourier series expansions is used to calculate periodic waves of permanent form and of finite amplitude. Three kinds of waves are calculated: combination waves resulting from the interaction between an ‘internal’ mode and an ‘external’ mode with the same phase speed but wavelengths in a ratio of 2 (1:2 resonance), combination waves resulting from the interaction between a long ‘internal’ mode and a short ‘external’ mode with the same phase speed, and pure ‘external’ waves. It is shown that the 1:2 resonance, which is well-known for capillary – gravity surface waves and can profoundly affect wave field evolution, can affect pure gravity waves in a two-fluid system, but not in oceanic conditions. On the other hand, it is shown that the long/short wave resonance can occur in ocean-type conditions. Finally it is confirmed that pure external waves of finite amplitude behave like surface waves.

The existence of Rayleigh–Bloch surface waves (original) (raw)

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