Introduction to Non-Linear Waves
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Type-I Hyperbolic Metasurfaces for Highly-Squeezed Designer
Type-I hyperbolic metasurfaces for highly-squeezed designer polaritons with negative group velocity Yihao Yang1,2,3,4, Pengfei Qin2, Xiao Lin3, Erping Li2, Zuojia Wang5,*, Baile Zhang3,4,*, and Hongsheng Chen1,2,* 1State Key Laboratory of Modern Optical Instrumentation, College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China. 2Key Lab. of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Hangzhou 310027, China. 3Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore. 4Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore. 5School of Information Science and Engineering, Shandong University, Qingdao 266237, China *[email protected] (Z.W.); [email protected] (B.Z.); [email protected] (H.C.) Abstract Hyperbolic polaritons in van der Waals materials and metamaterial heterostructures provide unprecedenced control over light-matter interaction at the extreme nanoscale. Here, we propose a concept of type-I hyperbolic metasurface supporting highly-squeezed magnetic designer polaritons, which act as magnetic analogues to hyperbolic polaritons in the hexagonal boron nitride (h-BN) in the first Reststrahlen band. Comparing with the natural h-BN, the size and spacing of the metasurface unit cell can be readily scaled up (or down), allowing for manipulating designer polaritons in frequency and in space at will. Experimental measurements display the cone-like hyperbolic dispersion in the momentum space, associating with an effective refractive index up to 60 and a group velocity down to 1/400 of the light speed in vacuum. -
Waves on Deep Water, I
Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the effect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (flat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ. -
Group Velocity
Particle Waves and Group Velocity Particles with known energy Consider a particle with mass m, traveling in the +x direction and known velocity 1 vo and energy Eo = mvo . The wavefunction that represents this particle is: 2 !(x,t) = Ce jkxe" j#t [1] where C is a constant and 2! 1 ko = = 2mEo [2] "o ! Eo !o = 2"#o = [3] ! 2 The envelope !(x,t) of this wavefunction is 2 2 !(x,t) = C , which is a constant. This means that when a particle’s energy is known exactly, it’s position is completely unknown. This is consistent with the Heisenberg Uncertainty principle. Even though the magnitude of this wave function is a constant with respect to both position and time, its phase is not. As with any type of wavefunction, the phase velocity vp of this wavefunction is: ! E / ! v v = = o = E / 2m = v2 / 4 = o [4] p k 1 o o 2 2mEo ! At first glance, this result seems wrong, since we started with the assumption that the particle is moving at velocity vo. However, only the magnitude of a wavefunction contains measurable information, so there is no reason to believe that its phase velocity is the same as the particle’s velocity. Particles with uncertain energy A more realistic situation is when there is at least some uncertainly about the particle’s energy and momentum. For real situations, a particle’s energy will be known to lie only within some band of uncertainly. This can be handled by assuming that the particle’s wavefunction is the superposition of a range of constant-energy wavefunctions: jkn x # j$nt !(x,t) = "Cn (e e ) [5] n Here, each value of kn and ωn correspond to energy En, and Cn is the probability that the particle has energy En. -
21 Non-Linear Wave Equations and Solitons
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Open Textbooks 8-2014 21 Non-Linear Wave Equations and Solitons Charles G. Torre Department of Physics, Utah State University, [email protected] Follow this and additional works at: https://digitalcommons.usu.edu/foundation_wave Part of the Physics Commons To read user comments about this document and to leave your own comment, go to https://digitalcommons.usu.edu/foundation_wave/2 Recommended Citation Torre, Charles G., "21 Non-Linear Wave Equations and Solitons" (2014). Foundations of Wave Phenomena. 2. https://digitalcommons.usu.edu/foundation_wave/2 This Book is brought to you for free and open access by the Open Textbooks at DigitalCommons@USU. It has been accepted for inclusion in Foundations of Wave Phenomena by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Foundations of Wave Phenomena, Version 8.2 21. Non-linear Wave Equations and Solitons. In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a well-localized* and unusually stable disturbance in the water that propagated for miles virtually unchanged. The disturbance was stimulated by the sudden stopping of a boat on the canal. He called it a “wave of translation”; we call it a solitary wave. As it happens, a number of relatively complicated – indeed, non-linear – wave equations can exhibit such a phenomenon. Moreover, these solitary wave disturbances will often be stable in the sense that if two or more solitary waves collide then after the collision they will separate and take their original shape. -
Double Negative Dispersion Relations from Coated Plasmonic Rods∗
MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation. -
Photons That Travel in Free Space Slower Than the Speed of Light Authors
Title: Photons that travel in free space slower than the speed of light Authors: Daniel Giovannini1†, Jacquiline Romero1†, Václav Potoček1, Gergely Ferenczi1, Fiona Speirits1, Stephen M. Barnett1, Daniele Faccio2, Miles J. Padgett1* Affiliations: 1 School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, UK 2 School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK † These authors contributed equally to this work. * Correspondence to: [email protected] Abstract: That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wavevectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam’s transverse spatial structure. Using time-correlated photon pairs we show a reduction of the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several microns over a propagation distance of the order of 1 m. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Introducing spatial structure to an optical beam, even for a single photon, reduces the group velocity of the light by a readily measurable amount. One sentence summary: The group velocity of light in free space is reduced by controlling the transverse spatial structure of the light beam. Main text The speed of light is trivially given as �/�, where � is the speed of light in free space and � is the refractive index of the medium. -
Shallow Water Waves and Solitary Waves Article Outline Glossary
Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other. -
Soliton and Some of Its Geophysical/Oceanographic Applications
Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Soliton and Some of its Geophysical/Oceanographic Applications M. Lakshmanan Centre for Nonlinear Dynamics Bharathidasan University Tiruchirappalli – 620024 India Discussion Meeting, ICTS-TIFR 21 July 2016 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Theme Soliton is a counter-intuitive wave entity arising because of a delicate balance between dispersion and nonlinearity. Its major attraction is its localized nature, finite energy and preservation of shape under collision = a remark- ably stable structure. Consequently, it has⇒ ramifications in as wider areas as hydrodynamics, tsunami dynamics, condensed matter, magnetism, particle physics, nonlin- ear optics, cosmology and so on. A brief overview will be presented with emphasis on oceanographic applications and potential problems will be indicated. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Plan 1 Introduction 2 Historical: (i) Scott-Russel (ii) Zabusky/Kruskal 3 Korteweg-de Vries equation and solitary wave/soliton 4 Other ubiquitous equations 5 Mathematical theory of solitons 6 Geophysical/Oceanographic applications 7 Solitons in higher dimensions 8 Problems and potentialities M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory -
Plasma Waves
Plasma Waves S.M.Lea January 2007 1 General considerations To consider the different possible normal modes of a plasma, we will usually begin by assuming that there is an equilibrium in which the plasma parameters such as density and magnetic field are uniform and constant in time. We will then look at small perturbations away from this equilibrium, and investigate the time and space dependence of those perturbations. The usual notation is to label the equilibrium quantities with a subscript 0, e.g. n0, and the pertrubed quantities with a subscript 1, eg n1. Then the assumption of small perturbations is n /n 1. When the perturbations are small, we can generally ignore j 1 0j ¿ squares and higher powers of these quantities, thus obtaining a set of linear equations for the unknowns. These linear equations may be Fourier transformed in both space and time, thus reducing the differential equations to a set of algebraic equations. Equivalently, we may assume that each perturbed quantity has the mathematical form n = n exp i~k ~x iωt (1) 1 ¢ ¡ where the real part is implicitly assumed. Th³is form descri´bes a wave. The amplitude n is in ~ general complex, allowing for a non•zero phase constant φ0. The vector k, called the wave vector, gives both the direction of propagation of the wave and the wavelength: k = 2π/λ; ω is the angular frequency. There is a relation between ω and ~k that is determined by the physical properties of the system. The function ω ~k is called the dispersion relation for the wave. -
Phase and Group Velocity
Assignment 4 PHSGCOR04T Sound Model Questions & Answers Phase and Group velocity Department of Physics, Hiralal Mazumdar Memorial College For Women, Dakshineswar Course Prepared by Parthapratim Pradhan 60 Lectures ≡ 4 Credits April 14, 2020 1. Define phase velocity and group velocity. We know that a traveling wave or a progressive wave propagated through a medium in a +ve x direction with a velocity v, amplitude a and wavelength λ is described by 2π y = a sin (vt − x) (1) λ If n be the frequency of the wave, then wave velocity v = nλ in this situation the traveling wave can be written as 2π x y = a sin (nλt − x)= a sin 2π(nt − ) (2) λ λ This can be rewritten as 2π y = a sin (2πnt − x) (3) λ = a sin (ωt − kx) (4) 2π 2π where ω = T = 2πn is called angular frequency of the wave and k = λ is called propagation constant or wave vector. This is simply a traveling wave equation. The Eq. (4) implies that the phase (ωt − kx) of the wave is constant. Thus ωt − kx = constant Differentiating with respect to t, we find dx dx ω ω − k = 0 ⇒ = dt dt k This velocity is called as phase velocity (vp). Thus it is defined as the ratio between the frequency and the wavelength of a monochromatic wave. dx ω 2πn λ vp = = = 2π = nλ = = v dt k λ T This means that for a monochromatic wave Phase velocity=Wave velocity 1 Where as the group velocity is defined as the ratio between the change in frequency and change in wavelength of a wavegroup or wave packet dω ∆ω ω1 − ω2 vg = = = dk ∆k k1 − k2 of two traveling wave described by the equations y1 = a sin (ω1 t − k1 x)and y2 = a sin (ω2 t − k2 x). -
Dispersion Relation & Index Ellipsoids
8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index –The magnitude of the wave vector conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 n k 0 free space wavelength 0 2. Direction –The direction of the wave is perpendicular to the wave fronts (more to be said later). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently.