CONCHOID OF NICOMEDES AND LIMACON OF PASCAL AS ELECTRODE OF STATIC FIELD AND AS WAVEGUIDE OF HIGH FREQUENCY WAVE
|
|
- Emmeline Higgins
- 5 years ago
- Views:
Transcription
1 Progress In Eectromagnetics Research, PIER 30, 73 84, 001 CONCHOID OF NICOMEDES AND LIMACON OF PASCAL AS ELECTRODE OF STATIC FIELD AND AS WAVEGUIDE OF HIGH FREQUENCY WAVE W. Lin and Z. Yu University of Eectronic Science and Technoogy of China Chengdu, , China E. K. N. Yuang and K. M. Luk City University of Hong Kong Hong Kong, China Abstract Groove guides have sharp corners in the guide proper so are not ow oss and high power carrying wave guides. Proposed has been a groove rectanguar waveguide with rounded interna anges [1]. However, the atter is not practica in having the infinite arms. Let us try the Conchoid of Nicomedes and Limacon of Pasca to see how can they be boundaries of static fieds or high frequency waves. Aso has been proposed the pentagon waveguide [] for the high power transmission in high frequency. Yet we fee that we have very imited forms of waveguides, so we study the existing curves to see whether we can find some better guides such as rounded corner rectanguar waveguide and others by starting from the boundary curves of ρ = a/ cos ϕ + and ρ = a cos ϕ Conchoid of Nicomedes. Limacon of Pasca 3. The Theory 4. Appications 5. Concuding Remarks References
2 74 Lin et a. curves Xmax Ymax y 0 M M1 Mx,y) x=a Figure 1. Locus of Mx, y). 1. CONCHOID OF NICOMEDES We now see how to construct the Conchoid of Nicomedes, given a poe 0 and a straight ine, the base ine x = a, Fig. 1, and a ine segment, > 0, we can write down how to describe M1 and M ρ = a cos ϕ + 1a) x = a + cos ϕ 1b) y = a tan ϕ + sin ϕ 1c) cos ϕ = x x + y 1d) So that x a) z = x ) We can write from 1a), by the mutipication of e jϕ z = x + jy = e jϕ + a ejϕ cos ϕ now we construct a new equation containing ω = u + jν,byrepacing ϕ by bω, b is a constant to be fixed ejbw z = e jbw + a cosbw) and then take jb = 1 for simpicity we have then e w z = e w + a coshw) = eω + a1 + tanhw) 3)
3 Conchoid of Nicomedes and Limacon of Pasca as eectrode 75 where bw = ju + ϕ,sofor u =0,ϕ= ν. Equation ) describes curve in Fig. and Fig. 3, which can be the boundaries of waveguide of groove type and cose type of good quaity. The ta arms of the former wi go up rapidy to meet each other at x = a.. LIMACON OF PASCAL Now in Fig. 4, we have a poe o and a base curve of a circe x a/) + y = a /4 instead of a base ine, so the points M1 and M obey the equation in poar coordinates ρ = a cos ϕ + 4a) so that x = a + a cos ϕ + cos ϕ 4b) y = a sin ϕ + sin ϕ 4c) now 4b) gives x a = a cos ϕ + cos ϕ so together with 4c) we have x a ) + y = a + a cos ϕ + 4 that is x + y ax = a cos ϕ + ) =ρ by 4a), we finay have then x + y ax ) = x + y ) 5) as the equation of curves in Fig. 5, which are then boundaries of a new famiy of rectanguar waveguides of rounded inner corner, so the cut-off waveength of the first mode wi be approximatey λ c =4y max 6)
4 76 Lin et a. Figure. Groove guide the ta arms of which do not open up to infinity but asymptoticay meets the base ine x = a. We may take a =1,they may be eectrode of static fied or guide of surface waves. The equation of the curves are x = a+ cos ϕ and y = a tan ϕ+ sin ϕ. Figure 3. The cose waveguides, when changed from 1.5 to 10 and a =1. The equations of the curves are x = a + cos ϕ, y = a tan ϕ + sin ϕ, x max = a) 1/3 ; y max =a 1/3 a 1/3 ): /3 + a /3 ) 1/.
5 Conchoid of Nicomedes and Limacon of Pasca as eectrode 77 y M1x,y) o ϕ M p B a x Figure 4. Base Circe, OB = a,locus of M1 and M. Figure 5. The Limacon Curves ρ = a cos ϕ+) a =1,=0.8; 1; 1.5; ; 3. There are two oops, when < a the inner oop shrinks with increasing. When a, there is one oop. We can have ridge waveguide and cose empty guide. x = a cos ϕ + cos ϕ, y = a sin ϕ cos ϕ + sin ϕ. where the x max,y max ) are the maximums on the Fig. 5 and are on the curve ρ = a sin ϕ cos ϕ
6 78 Lin et a. so together with 5) we find x Max = a +1 + y Max = x max +1 From 4a), repace ϕ by bw, w = u + jν, make jb =1,wefind, by timing by e jϕ first, z = e w acoshw + ) = a + a ) ew ew + 6a) therefore we find, as needed in 5) x + y = e u acoshu cos ν + + jasinhu sin ν = e u { coshu + cos ν)a + +acoshu cos ν } 6b) and z a = e u a ) ew + a = e u 4 eu + + ae u cos ν 6c) and z a ) a a 4 eu e u = e u sinhu + + ae u cos ν = x +y ax 6d) Now from 6a) we find a ew + e w z a ) =0 So we can sove for e w, e w = 1 a ± +a z a ) ) and then we have, w = u + jv, e u = 1 a ± a +a z a ) 7a)
7 Conchoid of Nicomedes and Limacon of Pasca as eectrode 79 and we have transformed 5) to 7), with u as a parameter, et m + jn = a +a z a ), then so finay where a e u = ± m + m + n [ a e u m n ] =4 m 7b) m + n = ± +ax) +4a y m = 1 { ax ± } +ax) +4a y n = 1 { ax ± } +ax) +4a y of course the curve 5) in the x-y pane of curve in Fig. 5 can be transform into curve in w = u + jν pane by means of 6b) and 6d) ) a e u 4 sinhu + + ae u cos ν = [coshu + cos ν] a + +acoshu cos ν ) 8a) so that dz dw = e u { a e u +ae u cos ν + } 8b) and for the inner region of a guide, u<0, dz dw e u,sowemay find the soution of the guide probem by starting from this approximation. 3. THE THEORY Now eq. ) is the we known concoid of Nicomedes from which we construct a compex equation 3): z in term of ω,so we can interpret ω = u + jν as a compex potentia, the rea part u and the imaginary part ν of which are both soution of the Lapace s equation, by the
8 80 Lin et a. Cauchy Riemahn reations between u and ν. When u = 0, ω = jν, and 3) goes back to eq. 1), so 1) is the zero potentia boundary of the static fied defined by 3). Aso we can interpret 3) as a coordinate transformation, transforming the points on the z-pane onto points on the w-pane, so that the two dimensiona Hemohotz equation ψ x + ψ y + k c ψ =0 9) is changed into that in the w-pane as ψ u + ψ ν + k dz c dw =0 10) Whie in the case where u is a static potentia, the magnitude dw dz is the fied intensity E, E = dw dz = 1 11) h From 3), we have dw dz = eω + = e ω + 1 cos u cos ν + jsinhu) sin ν ) {coshu) cosν) jsinhu sin ν} cos u + cos ν) So that 6) becomes, making a = 1 without oss of generaity h cos u cos ν +1 = eu cos ν + cosh4u + cos 4ν + cos u cos ν sin sinhu cos ν +1 ν eu cos u cos ν + cos 4u + cos 4ν + 4 Now from 3) we find, et a =1, x = e u cos ν + eu + cos ν coshu + cos ν ) y = sin ν e u cos ν + coshu + cos ν 1) 13) 14a) 14b)
9 Conchoid of Nicomedes and Limacon of Pasca as eectrode 81 to observe how do u and ν vary aong the rea axia y = 0,wesee 1) y =0,ν=0,x= e u +1+tanhu 0 for u x = e ω +1+tanhu =0for u =0; ) y =0,ν= π, x 0asu,x as u 3) y =0,e u cos ν = then 14a) gives coshu + cos ν e u 1 x = 14c) coshu + cos ν which gives x = 0when u = 0ify = 0ony, whie 14a) gives x = cos ν +1. Sowehave to go back to 3) to investigate w as function of z, bychanging 3) into the foowing equation, etting a =1: e 3w z) + e ω + e ω + e ω z/ =0 15) we can sove e ω from this equation, first by soving ξ 3 + pξ + q =0 16) e ω = ξ 1 z 3 17) p =1 1 ) z 3 3 q = z 1 ) z + z 3 7 the discriminate is q ) 3 q = + 3 { ) [ z = z ) ) ]} z = 1 { z } η ) 3 where ) z ) z η =
10 8 Lin et a. Here η is given by prescribed z and, usuay, prescribed is z = x+jy. Then 16) is soved, and soved is e ω in 17), so that we find the static fied E, 11) and the scae factor h, 13), enabing us to study the fied ψ in 10). From 13), e u = e w = ξ 1 z = F x, y), so that ) is transform into an equation with u of 3) as a parameter. Of course ) can be transformed into an equation of u and ν, a curve in u-ν-pane. To do so, we find from 3) x a = e u asinhu cos ν + cos u + cos ν ) z = e u 4acoshu cos ν + coshu + cos ν and write ) in the from x a) z =x a + a) =x a) +ax a)+a so that we can sove for x a) : a ± a z x a) = z 1 18a) 18b) and to remove the ± sign, we have to write a a z ) x a z ) = z ) ) 1 1 then putting in 15a) and 15b) we have an equation in u and ν competey, for curves in u-ν pane or w-pane of ) of x and y variabes. Now from 3) we find Making use of dz dw = ew +1 tanh w) d dw tanhw = tanh w 1 ) = cos ν +jcoshu sin ν coshu + cos ν),
11 Conchoid of Nicomedes and Limacon of Pasca as eectrode 83 so that it is neary true that dz dw e u and is free of ν so is separabe in variabe u and ν. 4. APPLICATIONS A pot eq. ), with >0asaparameter, x = a is the asymptote in a the cases of different.sothe vertica arms wi be cosed for arge y, not open up to infinity. Each curve of Fig. can be made to act as an eectrode when charged and/or to act as waveguides, in three difference ways: a) The part in the eft haf pane, shown aso the maximum y and its positions can be some forms of ridge wave guide to give wide operating frequency ranges; b) On the right are ta waveguides good for high microwave power and high frequency transmissions which ca for high cearance for high microwave fied and for ow waveength; c) The whoe structure is a guiding structure to a surface wave, i.e., the whoe is artificia dieectric wave guide. If the configuration of Fig. is charged with a charge Q per meter the fied intensity can be found by 14), at the surface u =0,itcan be cacuated from 14), { E = ε dz 1 dw = Q 1+ ) } 4 1 π x ) x 1 We pot Fig. 5 simiary as another famiy of smooth wave guides, so we have very arge choices of guide figure in appications. Detai studies wi foow in coming papers. 5. CONCLUDING REMARKS Nowadays we have trianguar, rectanguar, pentagon, circuar, eiptica and paraboic waveguides. In this paper we propose two famiies of waveguides whose boundaries are described by simpe equations and ρ = a/ cos ϕ +. ρ = a cos ϕ + respectivey. They are shown in Fig., Fig. 3 and Fig. 5. They ook ike reguar rectanguar waveguide with smooth, rounded inner corners, so they
12 84 Lin et a. shoud be more efficient in transmission of high frequency power. Further study, numericay and experimentay wi be carried out and wi be reported shorty. ACKNOWLEDGMENT This work is supported by Chinese NSF under the grant and the State Key Lab. on Miimeter Wave, South East University, Nanjing, Jiangsu, China. REFERENCES 1. Lin, W., E. K.-N. Yung, and K.-M. Luk, Groove rectanguar waveguide with rounded interna anges, MWSYM, , Lin, W., H. Zhou, Z. Yu, E. K.-N. Yung, and K.-M. Luk Theoretica and experimenta study of a new famiy high-power waveguides, Journa of Eectromagnetic waves and Appications, Vo. 13 No. 8, , Vygodsky, M., Mathematica Handbook, 4th ed., Mir. Pubishers, Moscus, 1984 or Bayer, CRC Stan. Math Tabe, 8th Ed.
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationJackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More informationSupporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers
Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,
More informationDYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE
3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationTwo Kinds of Parabolic Equation algorithms in the Computational Electromagnetics
Avaiabe onine at www.sciencedirect.com Procedia Engineering 9 (0) 45 49 0 Internationa Workshop on Information and Eectronics Engineering (IWIEE) Two Kinds of Paraboic Equation agorithms in the Computationa
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationTHE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Sevie, Spain, -6 June 04 THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS M. Wysocki a,b*, M. Szpieg a, P. Heström a and F. Ohsson c a Swerea SICOMP
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationA Simple and Efficient Algorithm of 3-D Single-Source Localization with Uniform Cross Array Bing Xue 1 2 a) * Guangyou Fang 1 2 b and Yicai Ji 1 2 c)
A Simpe Efficient Agorithm of 3-D Singe-Source Locaization with Uniform Cross Array Bing Xue a * Guangyou Fang b Yicai Ji c Key Laboratory of Eectromagnetic Radiation Sensing Technoogy, Institute of Eectronics,
More informationFFTs in Graphics and Vision. Spherical Convolution and Axial Symmetry Detection
FFTs in Graphics and Vision Spherica Convoution and Axia Symmetry Detection Outine Math Review Symmetry Genera Convoution Spherica Convoution Axia Symmetry Detection Math Review Symmetry: Given a unitary
More informationBohr s atomic model. 1 Ze 2 = mv2. n 2 Z
Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More information1. Measurements and error calculus
EV 1 Measurements and error cacuus 11 Introduction The goa of this aboratory course is to introduce the notions of carrying out an experiment, acquiring and writing up the data, and finay anayzing the
More informationProblem Set 6: Solutions
University of Aabama Department of Physics and Astronomy PH 102 / LeCair Summer II 2010 Probem Set 6: Soutions 1. A conducting rectanguar oop of mass M, resistance R, and dimensions w by fas from rest
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationComputational Investigation of Arc Behavior in an Auto-Expansion Circuit Breaker with Gases of SF 6 and N 2 **
Computationa Investigation of Arc Behavior in an Auto-Expansion Circuit Breaker with Gases of SF 6 and N 2 ** Jining Zhang Abstract An eectric arc behavior in a fu-scae 245 kv autoexpansion circuit breaker
More informationWeek 5 Lectures, Math 6451, Tanveer
Week 5 Lectures, Math 651, Tanveer 1 Separation of variabe method The method of separation of variabe is a suitabe technique for determining soutions to inear PDEs, usuay with constant coefficients, when
More informationT.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA
ON THE SYMMETRY OF THE POWER INE CHANNE T.C. Banwe, S. Gai {bct, sgai}@research.tecordia.com Tecordia Technoogies, Inc., 445 South Street, Morristown, NJ 07960, USA Abstract The indoor power ine network
More information(Refer Slide Time: 2:34) L C V
Microwave Integrated Circuits Professor Jayanta Mukherjee Department of Eectrica Engineering Indian Intitute of Technoogy Bombay Modue 1 Lecture No 2 Refection Coefficient, SWR, Smith Chart. Heo wecome
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationFormulas for Angular-Momentum Barrier Factors Version II
BNL PREPRINT BNL-QGS-06-101 brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY 11973 March 19, 2015 abstract A
More informationNumerical solution of one dimensional contaminant transport equation with variable coefficient (temporal) by using Haar wavelet
Goba Journa of Pure and Appied Mathematics. ISSN 973-1768 Voume 1, Number (16), pp. 183-19 Research India Pubications http://www.ripubication.com Numerica soution of one dimensiona contaminant transport
More informationPhysics 505 Fall Homework Assignment #4 Solutions
Physics 505 Fa 2005 Homework Assignment #4 Soutions Textbook probems: Ch. 3: 3.4, 3.6, 3.9, 3.0 3.4 The surface of a hoow conducting sphere of inner radius a is divided into an even number of equa segments
More informationForces of Friction. through a viscous medium, there will be a resistance to the motion. and its environment
Forces of Friction When an object is in motion on a surface or through a viscous medium, there wi be a resistance to the motion This is due to the interactions between the object and its environment This
More informationSolution Set Seven. 1 Goldstein Components of Torque Along Principal Axes Components of Torque Along Cartesian Axes...
: Soution Set Seven Northwestern University, Cassica Mechanics Cassica Mechanics, Third Ed.- Godstein November 8, 25 Contents Godstein 5.8. 2. Components of Torque Aong Principa Axes.......................
More informationAn approximate method for solving the inverse scattering problem with fixed-energy data
J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999
More informationParaxial Beam, Gaussian Basics
Paraxia Beam, Gaussian Basics ECE 5368/6358 han q e - copyrighted Use soey for students registered for UH ECE 6358/5368 during courses - DO NOT distributed (copyrighted materias). Introduction. Paraxia
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationAn Algorithm for Pruning Redundant Modules in Min-Max Modular Network
An Agorithm for Pruning Redundant Modues in Min-Max Moduar Network Hui-Cheng Lian and Bao-Liang Lu Department of Computer Science and Engineering, Shanghai Jiao Tong University 1954 Hua Shan Rd., Shanghai
More informationOn Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 x 2 ) and Their Byproducts
Commun. Theor. Phys. 66 (216) 369 373 Vo. 66, No. 4, October 1, 216 On Integras Invoving Universa Associated Legendre Poynomias and Powers of the Factor (1 x 2 ) and Their Byproducts Dong-Sheng Sun ( 孙东升
More informationStatistical Learning Theory: A Primer
Internationa Journa of Computer Vision 38(), 9 3, 2000 c 2000 uwer Academic Pubishers. Manufactured in The Netherands. Statistica Learning Theory: A Primer THEODOROS EVGENIOU, MASSIMILIANO PONTIL AND TOMASO
More informationEXACT CLOSED FORM FORMULA FOR SELF INDUC- TANCE OF CONDUCTOR OF RECTANGULAR CROSS SECTION
Progress In Eectromagnetics Research M, Vo. 26, 225 236, 22 EXACT COSED FORM FORMUA FOR SEF INDUC- TANCE OF CONDUCTOR OF RECTANGUAR CROSS SECTION Z. Piatek, * and B. Baron 2 Czestochowa University of Technoogy,
More informationPaper presented at the Workshop on Space Charge Physics in High Intensity Hadron Rings, sponsored by Brookhaven National Laboratory, May 4-7,1998
Paper presented at the Workshop on Space Charge Physics in High ntensity Hadron Rings, sponsored by Brookhaven Nationa Laboratory, May 4-7,998 Noninear Sef Consistent High Resoution Beam Hao Agorithm in
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More information(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
(This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna
More informationMultiple Beam Interference
MutipeBeamInterference.nb James C. Wyant 1 Mutipe Beam Interference 1. Airy's Formua We wi first derive Airy's formua for the case of no absorption. ü 1.1 Basic refectance and transmittance Refected ight
More informationSupplementary Material: An energy-speed-accuracy relation in complex networks for biological discrimination
Suppementary Materia: An energy-speed-accuracy reation in compex networks for bioogica discrimination Feix Wong, 1,2 Arie Amir, 1 and Jeremy Gunawardena 2, 1 Schoo of Engineering and Appied Sciences, Harvard
More informationLecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
More informationVolume 13, MAIN ARTICLES
Voume 13, 2009 1 MAIN ARTICLES THE BASIC BVPs OF THE THEORY OF ELASTIC BINARY MIXTURES FOR A HALF-PLANE WITH CURVILINEAR CUTS Bitsadze L. I. Vekua Institute of Appied Mathematics of Iv. Javakhishvii Tbiisi
More informationMeasurement of acceleration due to gravity (g) by a compound pendulum
Measurement of acceeration due to gravity (g) by a compound penduum Aim: (i) To determine the acceeration due to gravity (g) by means of a compound penduum. (ii) To determine radius of gyration about an
More informationMassive complex scalar field in the Kerr Sen geometry: Exact solution of wave equation and Hawking radiation
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 3 MARCH 2003 Massive compex scaar fied in the Kerr Sen geometry: Exact soution of wave equation and Hawking radiation S. Q. Wu a) Interdiscipinary Center
More informationEffect of transport ratio on source term in determination of surface emission coefficient
Internationa Journa of heoretica & Appied Sciences, (): 74-78(9) ISSN : 975-78 Effect of transport ratio on source term in determination of surface emission coefficient Sanjeev Kumar and Apna Mishra epartment
More informationGaussian Curvature in a p-orbital, Hydrogen-like Atoms
Advanced Studies in Theoretica Physics Vo. 9, 015, no. 6, 81-85 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.5115 Gaussian Curvature in a p-orbita, Hydrogen-ike Atoms Sandro-Jose Berrio-Guzman
More information8 Digifl'.11 Cth:uits and devices
8 Digif'. Cth:uits and devices 8. Introduction In anaog eectronics, votage is a continuous variabe. This is usefu because most physica quantities we encounter are continuous: sound eves, ight intensity,
More informationApplied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationLECTURE 10. The world of pendula
LECTURE 0 The word of pendua For the next few ectures we are going to ook at the word of the pane penduum (Figure 0.). In a previous probem set we showed that we coud use the Euer- Lagrange method to derive
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More informationNonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles
ISSN 002-3640, JETP Letters, 20, Vo. 94, No., pp. 5. Peiades Pubishing, Inc., 20. Origina Russian Text V.I. Matveev, D.N. Makarov, 20, pubished in Pis ma v Zhurna Eksperimenta noi i Teoreticheskoi Fiziki,
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More information2.1. Cantilever The Hooke's law
.1. Cantiever.1.1 The Hooke's aw The cantiever is the most common sensor of the force interaction in atomic force microscopy. The atomic force microscope acquires any information about a surface because
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationIntroduction. Figure 1 W8LC Line Array, box and horn element. Highlighted section modelled.
imuation of the acoustic fied produced by cavities using the Boundary Eement Rayeigh Integra Method () and its appication to a horn oudspeaer. tephen Kirup East Lancashire Institute, Due treet, Bacburn,
More informationCompletion. is dense in H. If V is complete, then U(V) = H.
Competion Theorem 1 (Competion) If ( V V ) is any inner product space then there exists a Hibert space ( H H ) and a map U : V H such that (i) U is 1 1 (ii) U is inear (iii) UxUy H xy V for a xy V (iv)
More informationTM Electromagnetic Scattering from 2D Multilayered Dielectric Bodies Numerical Solution
TM Eectromagnetic Scattering from D Mutiayered Dieectric Bodies Numerica Soution F. Seydou,, R. Duraiswami, N.A. Gumerov & T. Seppänen. Department of Eectrica and Information Engineering University of
More informationPHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I
6 n terms of moment of inertia, equation (7.8) can be written as The vector form of the above equation is...(7.9 a)...(7.9 b) The anguar acceeration produced is aong the direction of appied externa torque.
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationExtensions of Laplace Type Problems in the Euclidean Space
Internationa Mathematica Forum, Vo. 9, 214, no. 26, 1253-1259 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/imf.214.46112 Extensions of Lapace Type Probems in the Eucidean Space Giuseppe Caristi
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationA. Distribution of the test statistic
A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch
More informationOn the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems
Internationa Journa of Bifurcation and Chaos Vo. 25 No. 10 2015 1550131 10 pages c Word Scientific Pubishing Company DOI: 10.112/S02181271550131X On the Number of Limit Cyces for Discontinuous Generaized
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More informationOn a geometrical approach in contact mechanics
Institut für Mechanik On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof Universität Karsruhe, Institut für Mechanik Institut für Mechanik Kaiserstr. 12, Geb. 20.30 76128
More informationAPPENDIX C FLEXING OF LENGTH BARS
Fexing of ength bars 83 APPENDIX C FLEXING OF LENGTH BARS C.1 FLEXING OF A LENGTH BAR DUE TO ITS OWN WEIGHT Any object ying in a horizonta pane wi sag under its own weight uness it is infinitey stiff or
More informationCluster modelling. Collisions. Stellar Dynamics & Structure of Galaxies handout #2. Just as a self-gravitating collection of objects.
Stear Dynamics & Structure of Gaaxies handout # Custer modeing Just as a sef-gravitating coection of objects. Coisions Do we have to worry about coisions? Gobuar custers ook densest, so obtain a rough
More informationAkaike Information Criterion for ANOVA Model with a Simple Order Restriction
Akaike Information Criterion for ANOVA Mode with a Simpe Order Restriction Yu Inatsu * Department of Mathematics, Graduate Schoo of Science, Hiroshima University ABSTRACT In this paper, we consider Akaike
More informationHigh Efficiency Development of a Reciprocating Compressor by Clarification of Loss Generation in Bearings
Purdue University Purdue e-pubs Internationa Compressor Engineering Conference Schoo of Mechanica Engineering 2010 High Efficiency Deveopment of a Reciprocating Compressor by Carification of Loss Generation
More informationModule 22: Simple Harmonic Oscillation and Torque
Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationResearch Article Solution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singularly Perturbed Method
Science and Technoogy of Nucear Instaations Voume 213, Artice ID 261327, 6 pages http://dx.doi.org/1.1155/213/261327 Research Artice Soution of Point Reactor Neutron Kinetics Equations with Temperature
More informationSpherical perfect lens: Solutions of Maxwell s equations for spherical geometry
PHYSICAL REVIEW B 69, 55 004 Spherica perfect ens: Soutions of Maxwe s equations for spherica geometry S. Anantha Ramakrishna Department of Physics, Indian Institute of Technoogy, Kanpur 0806, India J.
More informationKeywords: Rayleigh scattering, Mie scattering, Aerosols, Lidar, Lidar equation
CEReS Atmospheric Report, Vo., pp.9- (007 Moecuar and aeroso scattering process in reation to idar observations Hiroaki Kue Center for Environmenta Remote Sensing Chiba University -33 Yayoi-cho, Inage-ku,
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationTracking Control of Multiple Mobile Robots
Proceedings of the 2001 IEEE Internationa Conference on Robotics & Automation Seou, Korea May 21-26, 2001 Tracking Contro of Mutipe Mobie Robots A Case Study of Inter-Robot Coision-Free Probem Jurachart
More informationLecture 8 February 18, 2010
Sources of Eectromagnetic Fieds Lecture 8 February 18, 2010 We now start to discuss radiation in free space. We wi reorder the materia of Chapter 9, bringing sections 6 7 up front. We wi aso cover some
More informationMath 1600 Lecture 5, Section 2, 15 Sep 2014
1 of 6 Math 1600 Lecture 5, Section 2, 15 Sep 2014 Announcements: Continue reading Section 1.3 and aso the Exporation on cross products for next cass. Work through recommended homework questions. Quiz
More informationTunnel Geological Prediction Radar Alternating Electromagnetic Field Propagation Attenuation in Lossy Inhomogeneous Medium
Tunne Geoogica Prediction Radar Aternating Eectromagnetic Fied Propagation Attenuation in Lossy Inhomogeneous Medium Si Yang Chen,a, Yan Peng Zhu,b, Zhong Li,c, Tian Yu Zhang Coage of civi engineering,lanhou
More informationApproximate description of the two-dimensional director field in a liquid crystal display
JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 9 1 MAY 21 Approximate description of the two-dimensiona director fied in a iquid crysta dispay G. Panasyuk, a) D. W. Aender, J. Key Liquid Crysta Institute
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Voume 9, 23 http://acousticasociety.org/ ICA 23 Montrea Montrea, Canada 2-7 June 23 Architectura Acoustics Session 4pAAa: Room Acoustics Computer Simuation II 4pAAa9.
More informationPhysics 566: Quantum Optics Quantization of the Electromagnetic Field
Physics 566: Quantum Optics Quantization of the Eectromagnetic Fied Maxwe's Equations and Gauge invariance In ecture we earned how to quantize a one dimensiona scaar fied corresponding to vibrations on
More informationLecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential
Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia 16.1 4.4 3.6 9.9 Meson Theory of Nucear potentia 4.5 3.11 9.10 I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider
More informationCoupling of LWR and phase transition models at boundary
Couping of LW and phase transition modes at boundary Mauro Garaveo Dipartimento di Matematica e Appicazioni, Università di Miano Bicocca, via. Cozzi 53, 20125 Miano Itay. Benedetto Piccoi Department of
More informationD. Prémel, J.M. Decitre and G. Pichenot. CEA, LIST, F Gif-sur-Yvette, France
SIMULATION OF EDDY CURRENT INSPECTION INCLUDING MAGNETIC FIELD SENSOR SUCH AS A GIANT MAGNETO-RESISTANCE OVER PLANAR STRATIFIED MEDIA COMPONENTS WITH EMBEDDED FLAWS D. Préme, J.M. Decitre and G. Pichenot
More informationDual Integral Equations and Singular Integral. Equations for Helmholtz Equation
Int.. Contemp. Math. Sciences, Vo. 4, 9, no. 34, 1695-1699 Dua Integra Equations and Singuar Integra Equations for Hemhotz Equation Naser A. Hoshan Department of Mathematics TafiaTechnica University P.O.
More informationCombining reaction kinetics to the multi-phase Gibbs energy calculation
7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More information