Elementary Plane Waves Research Articles

Simple Wave Transformations in the Channel

The Riemann wave describes the plane flow behind an infinite wavefront, though it is used to describe its propagation in channels. The influence of the walls on the process of wave propagation is not taken into account. However, friction against the walls slows the flow, and the limited diameter of the elementary plane waves, which compensate for the friction, undergo diffraction divergence. As a result, the entropy constancy and laminar flow structure are violated. This means that a simple wave as such may not exist in the channel. In this case, analysis of the friction and diffraction divergence make it possible to explain the turbulence occurrence in the flow and to find a way to form a simple wave of finite aperture with a large Rayleigh length.

Fecal Associated Microbiome (FAM) in Children: the differet distribution distinguised by genders

We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near certain algebraic plane curves. Our solutions are complex contour integral superpositions of elementary plane wave solutions with polynomial phase, the desired localisation being associated with the coalescence of saddle points. Our solutions provide a unified framework in which to describe some classical phenomena in two-dimensional high frequency wave propagation, including smooth and cusped caustics, whispering gallery and creeping waves, and tangent ray diffraction by a smooth boundary. We also study a subclass of solutions exhibiting localisation near a cubic parabola, and discuss their possible relevance to the study of the canonical inflection point problem governing the transition from whispering gallery waves to creeping waves.

Contour integral solutions of the parabolic wave equation

We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near certain algebraic plane curves. Our solutions are complex contour integral superpositions of elementary plane wave solutions with polynomial phase, the desired localisation being associated with the coalescence of saddle points. Our solutions provide a unified framework in which to describe some classical phenomena in two-dimensional high frequency wave propagation, including smooth and cusped caustics, whispering gallery and creeping waves, and tangent ray diffraction by a smooth boundary. We also study a subclass of solutions exhibiting localisation near a cubic parabola, and discuss their possible relevance to the study of the canonical inflection point problem governing the transition from whispering gallery waves to creeping waves.

Finite-Aperture Riemann’s Wave

The Riemann compression wave describes a 2D flow behind an infinite wave front, but is used for describing its propagation in channels. The effect of walls on the process of its propagation is disregarded in this case. However, friction against the walls and diffraction divergence of elementary plane waves compensating the friction change the conditions of wave propagation. These effects, which are inevitable in a channel, violate the constancy of entropy and the jet nature of the flow, which contradicts the definition of a “simple wave.” Analysis of diffraction divergence makes it possible to solve the problem of formation of a finite-aperture simple wave (a wave beam with a large Rayleigh length). The evolution of friction processes explains the emergence of a turbulent flow and makes it possible to describe the mechanism of a spontaneous transition of deflagration into detonation more accurately.

Analytical modelling for predicting the sound field of planar acoustic metasurface

An analytical model is built to predict the acoustic fields of acoustic metasurfaces. The acoustic fields are investigated for a Gaussian sound beam incident on the acoustic metasurfaces. The Gaussian sound beam is decomposed into a set of discrete elementary plane waves. The diffraction caused by the acoustic metasurfaces can be obtained using this analytical model, which is validated with the numerical simulations for the different incident angles of the Gaussian sound beam. This model overcomes the limitation of the method based on the generalised Snell's law which can only predict the direction of a specific diffracted order. Actually, this analytical model can be also used to predict the sound fields of acoustic metasurfaces under any incident sound if its Fourier transforms exist. This conclusion is demonstrated by studying the sound field for a point sound source incident on the acoustic metasurface. The acoustic admittances of acoustic metasurfaces are required in the calculation of the analytical model. Therefore, a numerical method for obtaining the effective acoustic admittances is proposed for the structurally complex metasurfaces without the analytical expressions of material properties, such as equivalent density and sound speed.

Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method.

A rigorous theoretical treatment to analyze the electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a sphere buried in a lossy half-space is presented. To consider the losses in the media an inhomogeneous plane wave is considered. The incident and the scattered electric field components are expanded in series of vectorial spherical harmonics using the Legendre functions generalized via hypergeometrical and gamma functions, with unknown expansion coefficients. The spectral-domain method to represent the scattered electric field is used in order to compute the scattered-reflected and scattered-transmitted fields, considering the reflection and transmission of each elementary plane wave by the interface. Finally, the unknown coefficients of the scattered field are computed by imposing the boundary condition on the spherical surface. In order to validate the model, a homemade code has been implemented. Comparisons with the simulations performed with a commercial software and the results in the literature are presented.

A new method for direction finding based on Markov random field model

AbstractInvestigating the characteristics of plasma waves observed by scientific satellites in the Earth's plasmasphere/magnetosphere is effective for understanding the mechanisms for generating waves and the plasma environment that influences wave generation and propagation. In particular, finding the propagation directions of waves is important for understanding mechanisms of VLF/ELF waves. To find these directions, the wave distribution function (WDF) method has been proposed. This method is based on the idea that observed signals consist of a number of elementary plane waves that define wave energy density distribution. However, the resulting equations constitute an ill‐posed problem in which a solution is not determined uniquely; hence, an adequate model must be assumed for a solution. Although many models have been proposed, we have to select the most optimum model for the given situation because each model has its own advantages and disadvantages. In the present study, we propose a new method for direction finding of the plasma waves measured by plasma wave receivers. Our method is based on the assumption that the WDF can be represented by a Markov random field model with inference of model parameters performed using a variational Bayesian learning algorithm. Using computer‐generated spectral matrices, we evaluated the performance of the model and compared the results with those obtained from two conventional methods.

Electromagnetic scattering by two concentric spheres buried in a stratified material

In this paper, a rigorous method to analyze the electromagnetic scattering of an elliptically polarized plane wave by two concentric spheres buried in a dielectric stratified medium is presented. The interaction of the electromagnetic radiation with the stratified material is taken into account by means of the transfer matrix approach, in this way we can consider the stratified medium as an effective single interface. All the electromagnetic fields are expanded in series of spherical vector harmonics. The transmitted field through the stratified medium is obtained by means of the effective transmission coefficient. This field is scattered by the two concentric spheres, and the scattered field interacts again with the stratified material. The scattered-reflected and scattered-transmitted fields by the layered medium are computed by exploiting the plane-wave spectrum of the scattered field, considering the reflection and transmission of each elementary plane wave by the effective interface. The boundary conditions imposition on the spheres' surfaces leads to a linear system that returns the unknown coefficients of the problem. A numerical code has been implemented to compute the field over all the space. In order to compute the scattered fields, a truncation criterion has been proposed for the numerical evaluation of the series. Finally, to validate the presented method, comparisons between the results of the proposed code and the results of simulations with a software based on the finite element method have been implemented, showing very good agreement.

Spectral domain method for the electromagnetic scattering by a buried sphere

A rigorous method to analyze the electromagnetic scattering of an elliptically polarized plane wave by a sphere buried in a dielectric half-space, is presented. The electric field components of the incident and the scattered monochromatic plane waves are expanded in series of vectorial spherical harmonics, with unknown expansion coefficients. The scattered-reflected and scattered-transmitted fields are computed by exploiting the plane-wave spectrum of the scattered field, considering the reflection and transmission of each elementary plane wave by the interface. The boundary-condition imposition leads to a linear system that returns the unknown coefficients of the scattered field. To achieve a numerical solution, a code has been implemented, and a truncation criterion for the involved series has been proposed. Comparisons with the literature and simulations performed with a commercial software are presented. A generalization of the method to the case of a short pulse scattered by a buried sphere is presented, taking into account the dispersive properties of the involved media.

Elementary water waves

New elementary farfield and nearfield time-harmonic water waves, observed from a Galilean reference frame, are given. These two related classes of waves are modifications of classical elementary plane waves that are obtained in a simple way, using only elementary fundamental considerations, by analyzing waves that slowly grow from rest at time T = −∞ and are bounded in the farfield. This approach circumvents the need for a radiation condition, which may indeed be regarded as (indirectly) satisfied ab initio by the elementary farfield and nearfield waves given here. The farfield waves are the product of classical elementary waves by a function, called radiation function, that is defined explicitly in terms of the dispersion function. Thus, this radiation function is valid not only for water waves, but more generally for a broad class of linear dispersive waves. For illustration and verification purposes, elementary stationary-phase considerations are used to determine the main properties (phase and group velocities, wave patterns, asymptote and cusp lines) of farfield waves, and two particular classes of water waves—steady ship waves and time-harmonic waves generated by an offshore structure—in uniform finite water depth are considered. The elementary nearfield waves can readily be used to construct free-surface Green functions or in a spectral representation of nearfield flows.

Diffuse reflection by rough surfaces: an introduction

In this introductory paper, we present with some details the (mathematically) simplest methods proposed to compute the electromagnetic field scattered by a rough surface separating two homogeneous media. These methods remain largely used both in propagation and remote sensing problems. The methods described in the paper are: – the geometrical optics approximation, in which the wave is considered as a set of rays obeying the laws of reflection and refraction; – the small perturbation method, due to Rayleigh and Rice, in which the field is given as an expansion on a set of elementary harmonic plane waves, the coefficients of which are determined so as to satisfy the boundary conditions; – the Kirchhoff approximation, in which the field is given as an integral on the rough surface; in this method one needs to know some components of the field on the surface, and an approximation is substituted to the unknown true value. We end with a short discussion of some problems not adequately solved by these methods, namely self-shadowing, multiple scattering and some inadequacies of the Gaussian model for random surfaces. To cite this article: M. Sylvain, C. R. Physique 6 (2005).

Measurable parameters of electromagnetic waves in a hot plasma: The extension of the WDF direct problem

A method to study waves in a hot plasma is presented. It is based on the concept of the Wave Distribution Function (WDF). In this approach, the wave field is considered as a continuum of elementary plane waves of different frequencies propagated in different directions. The complex dispersion relation of each elementary wave is solved for a fixed frequency taking into account the Doppler effect. Experimental data of given devices are estimated. As a test of the method, whistler waves in a plasma near the geostationary orbit are studied. The results well reproduce the recent calculations made by Sazin & Horne. Some new details about the wave dispersion have been found.

Diffraction of a Gaussian beam from a periodic planar screen

The diffraction of electromagnetic radiation from one-dimensional planar transmittive screens illuminated by Gaussian-profile beams is examined. The incident Gaussian beam is expressed as a superposition of elementary plane waves that propagate along different directions, with the aid of the plane-wave spectrum technique based on Fourier optics. For each elementary plane wave, the diffracted field is obtained by applying the spectral domain method combined with the method of moments. Results for the cases of normal and oblique incidence are in agreement with theoretically expected properties of such planar screens and are found to be very sensitive to the aspect ratio γ (strip width to period). The diffracted beam is composed of distinct subbeams that propagate along specific directions. These directions coincide with the Floquet harmonics generated by an incident plane wave propagating along the same direction as the initial beam.

Coupling efficiency of two anisotropic single‐mode slab waveguides butt‐joined via a gap

AbstractThe fundamental characteristics of the coupling efficiency between two anisotropic slab waveguides with different configurations butt‐joined via a gap are analyzed in the case of a TM mode incidence by means of a single‐mode slab waveguide model made of equatorial dielectric tensor. the electromagnetic field in the gap is represented by a combination of the forward elementary plane wave propagating from the incident side to the output waveguide and the backward elementary plane wave propagating in the opposite direction.By applying the boundary conditions, the power transmission and reflection coefficients in the guided‐wave system are computed. Further, the effects of the lateral misalignment and inclination between the wave‐guides and the index of refraction and spacing of the gap on the coupling efficiency are studied based on the numerical results. an optimum condition to maximize the coupling efficiency is obtained.

Full wave calculation of ELF/VLF propagation from a dipole source located in the lower ionosphere

A full wave technique has been developed to calculate the field intensities both in the upper ionosphere and on the ground when a dipole source immersed in the lower ionosphere radiates ELF/VLF waves. The radiated wave is divided into a large number of elementary plane waves, for each of which the propagation in the horizontally stratified model consisting of the ionosphere, the free space and the ground is calculated by the full wave technique. Then the plane waves are summed up to give a horizontal distribution of the radiated wave intensities at any altitudes. This technique is applied to investigate the propagation characteristics of radiation from a polar electrojet (PEJ) antenna modulated at an ELF/VLF frequency, which was created by the Tromsø heating facility. Comparison between the results of experiments and the calculation is given in the companion paper [Kimura et al., this issue]. We discuss several propagation characteristics, such as the frequency dependence of the radiated wave propagation up to the upper ionosphere and down to the ground. Under the conditions that the frequency is 2.5 kHz and the ionospheric dc electric field is 5 mV/m, the calculated results are roughly consistent with the experimental results and the radiation efficiency of the PEJ antenna is found to be extremely small, 2.5×10−6 upward and 3×10−8 downward.

On the Estimation of Wave Energy Distribution of Magnetospheric VLF Waves at the Ionospheric Base with Ground-Based Multiple Electromagnetic Field Components.

An inversion method is proposed to estimate the wave energy distribution of magnetospheric VLF radio waves (VLF/ELF emissions and whistlers) at the ionospheric base by using the simultaneously observed two horizontal magnetic and a vertical electric field components. The simulation experiments are extensively carried out to examine the effectiveness of the method. Two models for the wave source (a single source and two simultaneous sources) have been used, in which the observed signal is simulated to be composed of a number of elementary plane waves whose arriving directions are distributed around a specific direction in the case of a single source (two specific directions for the case of two simultaneous sources) and whose polarizations are distributed around right-handed circular. The distributions how the wave energy density is distributed at the ionospheric base have been obtained as an inversion problem in which the entropy of the wave distribution function is made maximum for the spectral matrix composed of the auto-and cross-power spectra of the three field components. It is then found that the present new method is very promising especially for the situation when a few wave sources (ducts) are present or when the ionospheric exit region is widely spread for which the previous direction finding systems are ineffective. Finally, the present method is applied to the actual data of whistlers observed at a low latitude, and it is found that the method is very effective in locating the wave source distribution, being in good agreement with the results by the previous direction finding methods. Further useful information on the mechanism of magnetospheric duct propagation and the ionospheric transmission is obtainable from the present method.

Transmission of High-Frequency Sound Waves Through a Slug Flow Jet

An analysis has been performed of sound waves which propagate in a pipe with gas flow. At the pipe exit these waves are partially reflected and the remainder are diffracted. The analysis is carried out by resolving the sound at the exit into its Fourier components and then continuing the solution, which is a combination of elementary plane waves, beyond the exit. These waves are of two types: homogeneous waves which propagate to infinity, and inhomogeneous waves with complex wave numbers which decay. The reflected waves are evaluated from the inhomogeneous waves. At the boundary of the jet, refraction of the elementary plane waves is accounted for and the far field sound is evaluated by the method of stationary phase. Comparisons of the theoretical calculations are made with experimental results and with calculations of other theories.

Spectral expansion method in problems of laser-beam propagation in the turbulent atmosphere

A mixed spectral expansion over elementary spherical and plane waves is suggested for use as an approximate solution of the stochastic wave equation describing propagation of optical waves in a turbulent medium. In this case, the complex amplitude of an elementary plane wave is calculated as a solution of a shortened equation, considering only the phase fluctuations of this wave. It is shown that such an approximate solution uniformly approximates statistical moments of the field (up to the fourth order, inclusive) under arbitrary conditions of wave propagationin a turbulent medium and conditions of wave diffraction on the transmitting aperture.

Shapes of Growing Domains in Ferroelectric Triglycine Sulfate and Triglycine Selenate

In TGS and TGSe, shapes of the domains growing under electric field have been investigated on the basis of Nakamura's theory which offers the envelope constructed from elementary plane waves of growth. In the formula v = v ∞ exp (-δ/ E ) for the normal velocity v of sidewise wall motion under electric field E , the activation field δ has been assumed to be proportional to 3/2 powers of wall-energy density of σ'. The values of v in the various directions have been calculated with σ' obtained from the Zhirnov-type continuum theory. The domain shapes and their dependence on the applied field have been determined with the calculated values of v . The results satisfactorily explain the experimentally observed lenticular and elliptical shapes in TGS and the restricted lenticular, lenticular and elliptical shapes in TGSe. It has been shown that there exist critical fields for the classification of these kinds of shape.

Reflection of pulse by multiple-dielectric layers

The reflection of a pulse by two dielectric slabs is treated both analytically and numerically. The reflected wave is obtained by expanding the reflection coefficient of an elementary plane wave in a series, including the special case for which total reflection occurs. The effect of the carrier frequency of a pulse-modulated carrier wave on the reflected wave form is also discussed.