Visualization and interpretation for the electromagnetic derivation of Snell's laws

Acoustic energy propagates effectively to long ranges in the ocean interior because of the physical properties of the marine environment. Sound propagation in the ocean is relevant to a variety of studies in communication, climatology and marine biology. Examples drawn from ocean acoustics, therefore, are compelling to students with a variety of interests. The dependence of sound speed on depth results in a waveguide that permits the detection of acoustic energy at ranges, in some experiments, of thousands of kilometres. This effect serves as an illustration of Snell's law with a continuously variable index of refraction. Acoustic tomography also offers a means for imaging the ocean's thermal structure, because of the dependence of sound speed on temperature. The ability to perform acoustic thermometry for large transects of the ocean provides an effective means of studying climate change. This application in an area of substantial popular attention allows for an effective introduction to concepts in ray propagation. Aspects of computational ocean acoustics can be productive classroom examples in courses ranging from introductory physics to upper-division mathematical methods courses.

The problem of reflection and refraction of elastic waves for an incident transverse wave at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids has been investigated. Using the theory developed by Iesan (Acta Mech 60:67–89, 1986), the formulae corresponding to the amplitude and energy ratios of reflected and refracted elastic waves have been obtained. The results similar to Singh and Tomar (Mech Materials 39:932–940, 2007) are recovered from the present analysis. The amplitude and energy ratios are computed numerically for a particular model.

Preface. Intent of the Book. 1. Plane Electromagnetic Waves. Introduction. 1.1 Propagating Plane Waves. 1.2 Polarized Plane Waves. 1.3 Doppler Shift. 1.4 Plane Waves in a Lossy Medium. 1.5 Dispersion and Group Velocity. 1.6 Power and Energy Propagation. 1.7 Momentum Propagation. Endnotes. 2. Plane Waves in Compound Media. Introduction. 2.1 Plane Wave Propagating in a Material as It Orthogonally Interacts with a Second Material. 2.2 Electromagnetic Boundary Conditions. 2.3 Plane Wave Propagating in a Material as It Orthogonally Interacts with Two Boundaries. 2.4 Plane Wave Propagating in a Material as It Orthogonally Interacts with Multiple Boundaries. 2.5 Polarized Plane Waves Propagating in a Material as They Interact Obliquely with a Boundary. 2.6 Brewster's Law. 2.7 Applications of Snell's Law and Brewster's Law. Endnote. 3. Transmission Lines and Waveguides. 3.1 Infi nitely Long Transmission Lines. 3.2 Governing Equations. 3.3 Special Cases. 3.4 Power Transmission. 3.5 Finite Transmission Lines. 3.6 Harmonic Waves in Finite Transmission Lines. 3.7 Using AC Spice Models. 3.8 Transient Waves in Finite Transmission Lines. 4. Ideal Models vs Real-World Systems. Introduction. 4.1 Ideal Transmission Lines. 4.2 Ideal Model Transmission Line Input and Output. 4.3 Real-World Transmission Lines. 4.4 Effects of Surface Roughness. 4.5 Effects of the Propagating Material. 4.6 Effects of Grain Boundaries. 4.7 Effects of Permeability. 4.8 Effects of Board Complexity. 4.9 Final Conclusions for an Ideal versus a Real-World Transmission Line. Endnotes. 5. Complex Permittivity of Propagating Media. Introduction. 5.1 Basic Mechanisms of the Propagating Material. 5.2 Permittivity of Permanent Polar Molecules. 5.3 Induced Dipole Moments. 5.4 Induced Dipole Response Function, G(tau ) . 5.5 Frequency Character of the Permittivity. 5.6 Kramers-Kronig Relations for Induced Moments. 5.7 Arbitrary Time Stimulus. 5.8 Conduction Electron Permittivity. 5.9 Conductivity Response Function. 5.10 Permittivity of Plasma Oscillations. 5.11 Permittivity Summary. 5.12 Empirical Permittivity. 5.13 Theory Applied to Empirical Permittivity. 5.14 Dispersion of a Signal Propagating through a Medium with Complex Permittivity. Endnotes. 6. Surface Roughness. Introduction. 6.1 Snowball Model for Surface Roughness. 6.2 Perfect Electric Conductors in Static Fields. 6.3 Spherical Conductors in Time-Varying Fields. 6.4 The Far-Field Region. 6.5 Electrodynamics in Good Conducting Spheres. 6.6 Spherical Coordinate Analysis. 6.7 Vector Helmholtz Equation Solutions. 6.8 Multipole Moment Analysis. 6.9 Scattering of Electromagnetic Waves. 6.10 Power Scattered and Absorbed by Good Conducting Spheres. 6.11 Applications of Fundamental Scattering. Endnotes. 7. Advanced Signal Integrity. Introduction. 7.1 Induced Surface Charges and Currents. 7.2 Reduced Magnetic Dipole Moment Due to Field Penetration. 7.3 Infl uence of a Surface Alloy Distribution. 7.4 Screening of Neighboring Snowballs and Form Factors. 7.5 Pulse Phase Delay and Signal Dispersion. Chapter Conclusions. Endnotes. 8. Signal Integrity Simulations. Introduction. 8.1 Defi nition of Terms and Techniques. 8.2 Circuit Simulation. 8.3 Transient SPICE Simulation. 8.4 Emerging SPICE Simulation Methods. 8.5 Fast Convolution Analysis. 8.6 Quasi-Static Field Solvers. 8.7 Full-Wave 3-D FEM Field Solvers. 8.8 Conclusions. Endnotes. Bibliography. Index.

Applied Electromagnetics and Electromagnetic Compatibility

Transmission of longitudinal waves through a plane interface between two dissimilar porous elastic solid half-spaces

Transmission of longitudinal wave at a plane interface between micropolar elastic and chiral solid half-spaces: Incidence from micropolar half-space

Based on Qian's theory, we investigated the reflection and refraction of finite amplitute sound wave on a plane boundary. Under the continuity conditions of the pressure and the normal velocity, some interesting results are obtained. There exists two kinds of refraction waves. One obeys Snell's law; the other quantitatively agrees to the steeply dipping wave observed by Muir et al. A further analysis shows that a double refraction may occur in an extent of the incident angle when ρ1/ρ2 > √C022/C012 - 1.

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A single-layer reflecting element is proposed based on the principle of linear-to-circular polarization conversion focusing metasurface, which can independently control the phases of x-polarized and y-polarized reflecting waves and operate in a broadband of 10-14 GHz. Following the generalized Snell's laws of reflection, a super cell is designed with a phase-gradient of -60 for x-polarized waves and 60 for y-polarized waves, and the simulation results show the well wideband anomalous reflection as expected. In the design of the multifunctional metasurface, the 1313 unit cells are used to satisfy the parabolic profile and the focal-distance-to-diameter ratio is set to be 0.5. The phase compensation for forming a constant aperture phase is provided by the individual reflected elements with different structure parameters and x-y=90 is used to realize polarization conversion. The designed sample is simulated in CST Microwave Studio and the results show that both of the x-polarized and y-polarized plane waves are well focused through the reflection of the focusing metasurface in a broadband of 10-14 GHz. Traditionally, multi-layer element is used to broaden phase coverage and bandwidth, the single-layer design in this paper greatly reduces the cost, processing difficulty and thickness of the lens. For further application, a linearly polarized Vivaldi antenna with a highest gain of 10 dB is located at the focal point of metasurface and the angle included between its polarization direction and x-axis is 45 in order to acquire right-handed circularly polarized reflecting wave. According to the reversibility principle of electromagnetic wave propagation, the spherical wave radiated by the feed antenna is converted into plane wave by the reflection of the focusing metasurface so that the antenna gain is remarkably enhanced. Simultaneously, the linearly polarized wave can be transformed into circularly polarized wave. Finally, the feed antenna and the metasurface are fabricated, assembled and measured. Numerical and experimental results are in good agreement with each other, which shows that the -1 dB gain bandwidth of the high-gain antenna is 24% (11-14 GHz) and the 3 dB axial ratio bandwidth is 29.8% (10-13.5 GHz). In addition, the gain at 12 GHz reaches a highest value of 19.6 dBic, and the aperture efficiency is more than 54%. The good performances indicate that the proposed broadband high-gain circularly polarized antenna has a well promising application in various communication systems. It is worth noting that the horizontally polarized, vertically polarized, right-handed circularly polarized and left-handed circularly polarized high-gain antenna can be realized with the rotation of feed antenna. In this case the idea is more versatile and valuable for designing the polarization reconfigurable antenna systems.

We have calculated, within the framework of the Boltzmann equation, a dc electric current and emf in a two-dimensional system induced by the high-frequency field of an electromagnetic wave or the electric field of a plasmon wave. It is established that the generated current consists of two contributions, one of which is proportional to the real part of the wave vector projection of the exciting wave onto the interface plane and represents the electron drag effect, and the other contribution is proportional to the extinction coefficient of the wave in the interface plane. It is shown that the main cause of the second contribution is a nonuniform electron heating created by the wave and controlled by the energy relaxation time of the electron gas. In FET the heating mechanism of the electric-current formation can remarkably exceed the current calculated neglecting the heating.

This article describes and proves the basic phenomena which take place when thermal waves approach an interface between two media: the reflection and the refraction. In synthesis the Snell law for plane thermal waves is proved, both theoretically and experimentally by means of the mirage technique.

In 1968, Veselago posed that media with simultaneous negative values of the permittivity and permeability would exhibit a negative index of refraction (Soviet Physics Uspekhi, vol. 10, pp. 509-514, 1968.). Due to the left-handed relationship between the electric field, magnetic field, and wave vector in these materials he termed them left-handed media (LHM). These LHM are expected to have many unusual properties such as inverted Snell's law, Doppler shift and Cherenkov radiation. However, until recently, such materials were unavailable and these predictions remained uncorroborated for nearly thirty years. As with any new field, the topic of the LHM suffers from some misconceptions and misunderstandings. One such misconception has to do with the notion of negative group velocity which has been closely linked with these media (R.A. Shelby et al, Appl. Phys. Lett., vol. 78, pp. 489-91, 2001) and in fact has been used as equivalent to the term LHM. In this paper, the existence and meaning of negative group velocity in LHM are discussed. It is shown that similar to normal right-handed media (RHM), LHM can also exhibit a negative group velocity and group delay within the region of anomalous dispersion (/spl delta/n//spl delta//spl omega/<0). On the other hand, in the experiments presented so far, the researchers have only measured positive group velocity and negative phase velocity corresponding to so called backward waves (BWs).

When an incident bulk wave interacts with an interface, such as the boundary between a fluid and a solid, as found in immersion testing, both reflected and transmitted bulk waves are generated. Chapter 6 describes general properties of reflected and refracted waves for the case when a planar wave is incident on a plane interface. In the process a variety of concepts are introduced, such as acoustic impedance, Snell’s law, critical angles, pulse distortion, and inhomogeneous waves.

Absorbing boundary conditions for acoustic and elastic waves in stratified media

This paper presents a numerical scheme to nest a nearshore wave model (SWAN) in an ocean wave model (WAM). It allows users to run the WAM model on a coarse grid (deep or intermediate water depth) and to run the SWAN model on a nested fine grid (shallow water) to achieve computational efficiency. The system was used to hindcast waves in February 1993 along the Belgian and Dutch coasts. Model results from SWAN and WAM are intercompared and are also compared with buoy measurements. Good agreement was found between WAM, SWAN and observations. In addition, a numerical investigation of depth refraction of waves was made for a uniform 1:1100 sloping coast. A wave directional resolution of the order of 15° was found to be sufficient for SWAN to reproduce the wave refraction given by Snell's law while for WAM a directional resolution of 5 was required to achieve the same accuracy.