# Fourier Series Analysis

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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.
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• Test of Self-Organization in Beach Cusp Formation Giovanni Coco,1 T
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C3, 3101, doi:10.1029/2002JC001496, 2003 Test of self-organization in beach cusp formation Giovanni Coco,1 T. K. Burnet, and B. T. Werner Complex Systems Laboratory, Cecil and Ida Green Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California, USA Steve Elgar Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA Received 4 June 2002; revised 6 November 2002; accepted 13 November 2002; published 29 March 2003. [1] Field observations of swash flow patterns and morphology change are consistent with the hypothesis that beach cusps form by self-organization, wherein positive feedback between swash flow and developing morphology causes initial development of the pattern and negative feedback owing to circulation of flow within beach cusp bays causes pattern stabilization. The self-organization hypothesis is tested using measurements from three experiments on a barrier island beach in North Carolina. Beach cusps developed after the beach was smoothed by a storm and after existing beach cusps were smoothed by a bulldozer. Swash front motions were recorded on video during daylight hours, and morphology was measured by surveying at 3–4 hour intervals. Three signatures of self- organization were observed in all experiments. First, time lags between swash front motions in beach cusp bays and horns increase with increasing relief, representing the effect of morphology on flow. Second, differential erosion between bays and horns initially increases with increasing time lag, representing the effect of flow on morphology change because positive feedback causes growth of beach cusps. Third, after initial growth, differential erosion decreases with increasing time lag, representing the onset of negative feedback that stabilizes beach cusps.
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• Chapter 16: Fourier Series
Chapter 16: Fourier Series 16.1 Fourier Series Analysis: An Overview A periodic function can be represented by an infinite sum of sine and cosine functions that are harmonically related: Ϧ ͚ʚͨʛ Ɣ ͕1 ƍ ȕ ͕) cos ͢!*ͨ ƍ ͖) sin ͢!*ͨ Fourier Coefficients: are)Ͱͥ calculated from ͕1; ͕); ͖) ͚ʚͨʛ Fundamental Frequency: ͦ_ where multiples of this frequency are called !* Ɣ ; ͢! * harmonic frequencies Conditions that ensure that f(t) can be expressed as a convergent Fourier series: (Dirichlet’s conditions) 1. f(t) be single-values 2. f(t) have a finite number of discontinuities in the periodic interval 3. f(t) have a finite number of maxima and minima in the periodic interval 4. the integral /tͮ exists Ȅ/ |͚ʚͨʛ|ͨ͘; t These are sufficient conditions not necessary conditions; therefore if these conditions exist the functions can be expressed as a Fourier series. However if the conditions are not met the function may still be expressible as a Fourier series. 16.2 The Fourier Coefficients Defining the Fourier coefficients: /tͮ 1 ͕1 Ɣ ǹ ͚ʚͨʛͨ͘ ͎ /t /tͮ 2 ͕& Ɣ ǹ ͚ʚͨʛ cos ͟!*ͨ ͨ͘ ͎ /t /tͮ 2 ͖& Ɣ ǹ ͚ʚͨʛ sin ͟!*ͨ ͨ͘ ͎ / t Example 16.1 Find the Fourier series for the periodic waveform shown. Assessment problems 16.1 & 16.2 ECEN 2633 Spring 2011 Page 1 of 5 16.3 The Effects of Symmetry on the Fourier Coefficients Four types of symmetry used to simplify Fourier analysis 1. Even-function symmetry 2. Odd-function symmetry 3.
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• Coating Provides Infrared Camouflage
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• Sound and Vision: Visualization of Music with a Soap Film, and the Physics Behind It C Gaulon, C Derec, T Combriat, P Marmottant, F Elias
Sound and Vision: Visualization of music with a soap film, and the physics behind it C Gaulon, C Derec, T Combriat, P Marmottant, F Elias To cite this version: C Gaulon, C Derec, T Combriat, P Marmottant, F Elias. Sound and Vision: Visualization of music with a soap film, and the physics behind it. 2017. hal-01487166 HAL Id: hal-01487166 https://hal.archives-ouvertes.fr/hal-01487166 Preprint submitted on 11 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sound and Vision: Visualization of music with a soap ﬁlm, and the physics behind it C. Gaulon1, C. Derec1, T. Combriat2, P. Marmottant2 and F. Elias1;3 1 Laboratoire Mati`ere et Syst`emesComplexes UMR 7057, Universit´eParis Diderot, Sorbonne Paris Cit´e,CNRS, F-75205 Paris, France 2 Laboratoire Interdisciplinaire de Physique LIPhy, Universit´eGrenoble Alpes and CNRS UMR 5588 3 Sorbonne Universit´es,UPMC Univ Paris 06, 4 place Jussieu, 75252 PARIS cedex 05, France ﬂ[email protected] Abstract A vertical soap ﬁlm, freely suspended at the end of a tube, is vi- brated by a sound wave that propagates in the tube.
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• Guidance for Flood Risk Analysis and Mapping
Guidance for Flood Risk Analysis and Mapping Coastal Notations, Acronyms, and Glossary of Terms May 2016 Requirements for the Federal Emergency Management Agency (FEMA) Risk Mapping, Assessment, and Planning (Risk MAP) Program are specified separately by statute, regulation, or FEMA policy (primarily the Standards for Flood Risk Analysis and Mapping). This document provides guidance to support the requirements and recommends approaches for effective and efficient implementation. Alternate approaches that comply with all requirements are acceptable. For more information, please visit the FEMA Guidelines and Standards for Flood Risk Analysis and Mapping webpage (www.fema.gov/guidelines-and-standards-flood-risk-analysis-and- mapping). Copies of the Standards for Flood Risk Analysis and Mapping policy, related guidance, technical references, and other information about the guidelines and standards development process are all available here. You can also search directly by document title at www.fema.gov/library. Coastal Notation, Acronyms, and Glossary of Terms May 2016 Guidance Document 66 Page i Document History Affected Section or Date Description Subsection Initial version of new transformed guidance. The content was derived from the Guidelines and Specifications for Flood Hazard Mapping Partners, Procedure Memoranda, First Publication May 2016 and/or Operating Guidance documents. It has been reorganized and is being published separately from the standards. Coastal Notation, Acronyms, and Glossary of Terms May 2016 Guidance Document 66
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• Modes of Wave-Chaotic Dielectric Resonators
Modes of wave-chaotic dielectric resonators H. E. Tureci, H. G. L. Schwefel, and A. Douglas Stone Department of Applied Physics, Yale University, New Haven, CT 06520, USA Ph. Jacquod D¶epartement de Physique Th¶eorique, Universit¶e de Geneve, CH-1211 Geneve 4, Switzerland (Dated: October 29, 2003) Dielectric optical micro-resonators and micro-lasers represent a realization of a wave-chaotic sys- tem, where the lack of symmetry in the resonator shape leads to non-integrable ray dynamics. Modes of such resonators display a rich spatial structure, and cannot be classi¯ed through mode indices which would require additional constants of motion in the ray dynamics. Understanding and controlling the emission properties of such resonators requires the investigation of the correspon- dence between classical phase space structures of the ray motion inside the resonator and resonant solutions of the wave equations. We ¯rst discuss the breakdown of the conventional eikonal approx- imation in the short wavelength limit, and motivate the use of phase-space ray tracing and phase space distributions. Next, we introduce an e±cient numerical method to calculate the quasi-bound modes of dielectric resonators, which requires only two diagonalizations per N states, where N is approximately equal to the number of half-wavelengths along the perimeter. The relationship be- tween classical phase space structures and modes is displayed via the Husimi projection technique. Observables related to the emission pattern of the resonator are calculated with high e±ciency. I. INTRODUCTION A promising approach to making compact and high-Q optical resonators is to base them on \totally internally reected” modes of dielectric micro-structures.
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• Pattern Formation, Wave Propagation and Stability in Conservation Laws with Slow Diﬀusion and Fast Reaction
Pattern Formation, Wave Propagation and Stability in Conservation Laws with Slow Diﬀusion and Fast Reaction Haitao Fan ∗ and Hailiang Liu † Contents 1 Introduction 2 2 Convergence and pattern formation 4 3 Propagation speed of fronts 7 3.1Lawofmotionoffronts........................................................ 7 3.2LaxShockcondition.......................................................... 8 3.3Variationalformula.......................................................... 10 4 Existence and stability of layer solutions 11 4.1Connectingorbit............................................................ 11 4.2 Dynamic Stability and Approaching to a Wave . 15 5 Existence and Structure of the limit lim→0+ u (x, t) 19 Abstract The limiting behavior of the solution of scalar conservation laws with slow diﬀu- sion and fast bistable reaction is considered. In a short time the solution develops transition patterns connected by shock layers and rarefaction layers, when the initial data has ﬁnitely many monotone pieces. The existence and uniqueness of the front proﬁles for both shock layers and rarefaction layers are established. A variational characterization of the wave speed of these proﬁles is derived. These proﬁles are shown to be stable. Furthermore, it is proved that solutions with monotone intimal data approach to shock layer or rarefaction layer waves as time goes to inﬁnity. Key Words: Reaction-diﬀusion-convection equations, Pattern formation, transition fronts, asymptotic stability. AMS subject classiﬁcation: 35K65; 35K57; 41A60. Date: August 14, 2001 ∗Department of Mathematics, Georgetown University Washington, DC 20057. Email: [email protected]. Web: http://www.georgetown.edu/faculty/fanh †Department of Mathematics, University of California, Los Angeles, CA 90095-1555. Email: [email protected]. Web: http://www.math.ucla.edu/∼hliu 1 1 Introduction Front propagation and interface motion occur in many scientiﬁc areas such as chemical kinetics, combustion, biology, transport in porous media, geometry and image process- ing.
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• The Emergence of Chaos in Quantum Mechanics
S S symmetry Article The Emergence of Chaos in Quantum Mechanics Emilio Fiordilino Dipartimento di Fisica e Chimica—Emilio Segrè, Università degli Studi di Palermo, Via Archiraﬁ 36, 90123 Palermo, Italy; emilio.ﬁ[email protected] Received: 4 April 2020; Accepted: 7 May 2020; Published: 8 May 2020 Abstract: Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difﬁcult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser ﬁeld assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular ﬁlling of the Poincaré map and exponential separation of the trajectories of the Bloch vector s in the Bloch sphere. Keywords: nonlinear Schrödinger equation; chaos; high order harmonic generation 1. Introduction The theory of chaos gained the role of a new paradigm of Science for explaining a large variety of phenomena by introducing the concept of unpredictability within the perimeter of Classical Physics. Chaos occurs when the trajectory of a particle is sensitive to the initial conditions and is produced by a nonlinear equation of motion [1]. As examples we quote the equation for the Dufﬁng oscillator (1918) 3 mx¨ + gx˙ + rx = F cos(wLt) (1) or its elaborated form 3 mx¨ + gx˙ + kx + rx = F cos(wLt) (2) that are among the most studied equation of mathematical physics [1,2] and, as far as we know, cannot be analytically solved.
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• Wave Nature of Fractal Lacunarity: a Higher Resolution Scale to Capture Subtle Changes in ECG Signals
Wave Nature of Fractal Lacunarity: A Higher Resolution Scale to Capture Subtle Changes in ECG Signals AN Esgiar, PK Chakravorty University of Al Tahadi, Sirte, Libya Abstract ECG signal is a statistically fractal set (unlike any systematic fractal set) and hence its lacunarity measure is Lacunarity is a feature of a fractal set. Its wave a function of the box size and it can be proved that the nature was predicted theoretically by Bloomenfeld and lacunarity (based on even moments of mass) plot should Mandelbrot. In this paper, we present our result conﬁrming be generally convex upwards and lacunarity (based on odd the existence of wave nature of lacunarity for ECG signals moments of mass) should be generally concave upwards as which are statistical fractal sets. We have applied this was presented earlier[1]. In this paper, ﬁrst we conﬁrm the wave nature of lacunarity to capture the subtle changes prediction of Bloomenfeld and Mandelbrot regarding the in ECG signals in a very short time span of about one wave nature of Lacunarity. This wave nature of lacunarity minute to predict occurrence of termination of atrial is used to discern the changes occurring in the ECG signals ﬁbrillation for a subject, whose ECG is being monitored in a very short time span. continuously. This method can be used to detect the All the fractal features have been estimated by a new box minute subtle changes of ECG signals which are caused counting algorithm. We have named it True Box Counting by other physiological events as well and needs further method.
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• Soap Bubbles: Part 2 ______
Andrew Glassner’s Notebook http://www.glassner.com Soap Bubbles: Part 2 ________________________________ oap bubbles are fragile, beautiful phenomena. the points I want to make, so I’ll just ignore them. Andrew SThey’re fun to make and play with, and their geom- To get an endless succession of waves in the tank, Glassner etry is as clean and elegant as anything in nature, which plunge the cylinder in repeatedly at just the right rate. makes them particularly suited to computer graphics. Each time the pipe drops into the water, it creates a In the last issue I discussed the nature of soap ﬁlms. wave, which starts to spread outward. If you measure These thin sheets of soapy water have less surface ten- the distance from the crest of one wave to the crest of sion than water itself, so they can billow out into curved the next one—as they move down the length of the surfaces like bubbles. In this column I’ll talk about where tank—you’ll ﬁnd that that distance doesn’t change as a bubble’s beautiful colors come from, the geometry of the wave moves out. This distance is called the wave- bubble clusters, and how to make bubbles in a 3D mod- length. As we watch the wave move, it’s like a distur- eler. I’ll use some of the material discussed in previous bance moving through the water. columns: reflection, refraction, and Snell’s Law If we set a piece of foam on top of the water as the wave (January/February 1998 and March/April 1998) and goes by, we’ll see the foam rise up and down.
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• Wave Chaos for the Helmholtz Equation Olivier Legrand, Fabrice Mortessagne
Wave chaos for the Helmholtz equation Olivier Legrand, Fabrice Mortessagne To cite this version: Olivier Legrand, Fabrice Mortessagne. Wave chaos for the Helmholtz equation. Matthew Wright & Richard Weaver. New Directions in Linear Acoustics and Vibration, Cambridge University Press, pp.24-41, 2010, 978-0-521-88508-9. hal-00847896 HAL Id: hal-00847896 https://hal.archives-ouvertes.fr/hal-00847896 Submitted on 24 Jul 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Wave chaos for the Helmholtz equation O. Legrand and F. Mortessagne Laboratoire de Physique de la Mati`ere Condens´ee CNRS - UMR 6622 Universit´ede Nice-Sophia Antipolis Parc Valrose F-06108 Nice Cedex 2 FRANCE I. INTRODUCTION The study of wave propagation in complicated structures can be achieved in the high frequency (or small wavelength) limit by considering the dynamics of rays. The complexity of wave media can be either due to the presence of inhomogeneities (scattering centers) of the wave velocity, or to the geometry of boundaries enclosing a homogeneous medium. It is the latter case that was originally addressed by the ﬁeld of Quantum Chaos to describe solutions of the Schr¨odinger equation when the classical limit displays chaos.
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