SOME REMARKS ON THE TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE
|
|
- Julie Barrett
- 6 years ago
- Views:
Transcription
1 R904 Philips Res. Repts 30, 232*-239*, 1975 Issue in honour of C. J. Bouwkamp SOME REMARKS ON THE TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE by Josef MEIXNER *) and Schiu SCHE Rheinisch-Westfälische Technische Hochschule Aachen Aachen, B.R.D. (Received January 14, 1975) Bouwkamp's thesis 1) is a cornerstone in the theory of spheroidal functions, in their numerical treatment and their application to a specific problem which was of great interest at that time. As a matter of fact, he treated the diffraction of scalar waves through a circular aperture from long wavelengths down to such wavelengths where the Kirchhoff-Rayleigh approximations are already quite good. At that time the numerical computations were quite cumbersome. With the advent of the present fast computers the situation has much changed. Not only can one easily carry through the calculations with an increased number of significant digits, one can also go to much smaller wavelengths as compared to the radius of the aperture. Our goal in this note is to extend Bouwkamp's results to wavelengths which are one half of his, but also to compare the exact solution, which is a series in spheroidal wave functions, with the two Kirchhoff-Rayleigh approximations, also expanded in spheroidal wave functions. We restrict ourselves, however, to the case of a hard screen with a circular aperture. The other case of an infinitely soft screen with a circular aperture can be treated in a similar fashion. The problem to be treated is the diffraction of a scalar plane wave by a rigid plane with a circular aperture A. The diffracting screen S is in the plane z = 0, the centre of the aperture is at x = y = 0, its radius is a. The plane wave exp (-ikz) is impinging from the half-space z ~ 0 normally to the plane z = O. The transmitted wave in z ;;:::: 0 can be written in terms of spheroidal functions as co "P(g, 'fj, (/)) = L C 2L S2L(4)(-i~; L=O iy) PS2L('fj; -y2) (1).) This paper is dedicated to C. J. Bouwkamp, from whose thesis the first author not only learned a little of the Dutch language but also got the confidence that it is worth while to study the spheroidal functions.
2 TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 233* with where Q2Lty) is given by (12). The spheroidal coordinates ç, 'fj, cp are defined by Furthermore x = a (ç2 1)1/ 2 (1 - 'fj2)1/2 cos cp, Y = a (ç2 + 1)1/2 (1 - 'fj2)1/2 sin cp, z = aç'fj. where k is the wavenumber, l the wavelength. The boundary conditions are 'b1p -=0 ons, 'bz (2) y = ka = 2najl (4) (3) 1p = 1 in A. (5) The problem is Bouwkamp's problem (lb) 1). The notation ofthe spheroidal functions is taken from ref. 2. We also introduce the Kirchhoff-Rayleigh approximate solution 1pK2, for Bouwkamp's case K2. It satisfies the conditions 'b1pk2 'b1pk2 -- = 0 on S, - = -i k in A. (6) öz' 'bz It can be expanded in spheroidal functions like in (1), but the coefficients C2L are to be replaced by where b 2L(y2) I (y2) C2LK2 = y2 (4L + 1) 0 2L 2 ' (7) ps2l(l; -y ) 1 12L(y2) = J'YJ PS2L('fJ; _y2) d'yj. o In the derivation of this expansion the Wronskian (ref. 2, p. 294) and (11) are used. Bouwkamp has compared the solutions 1p and 1pK2 by expanding them in spherical wave functions and by comparing the numerical values of the coefficients for y = 5 and y = 10. Quite interesting results are obtained if, instead, one compares the coefficients C2L and C2L K2 in the expansion (1) and in the corresponding expansion for 1pK2. This will be done for y = 10 and y = 20. The coefficients bn 2L (y2) in the expansion (8) PS2L('fJ; _y2) <Xl = L: bn 2L (y2) P2N('YJ), N=O (9)
3 234* JOSEF MEIXNER AND SCHIU seas where P2N('fJ) = Legendre polynomial of degree 2N, with the normalization condition (10) N=O have been calculated for l' = 10 and l' = 20. The eigenvalues of the differential equation for PS2L('fJ; -y2) have been taken from the tables 3). Thus the calculation, which otherwise used Bouwkamp's method of continued fractions, could be considerably simplified. These tables which also contain S2L W (-io; i1'), j = 1, 2, provided a useful check of our numerical results through the two relations (11) (12) Table I contains values of PS2L(0; _1'2), ps2l(i; _1'2), h02l(1'2) and 12L(1'2) for l' = 10 and l' = 20. In table II we give the values of the coefficients C2L> C2L K2 and their contributions to the transmission coefficient, 2 1 a K2 = _ Ic K212 2L r2 4L + 1 2L (13) For the same values of l' the transmission coefficients D 2 and D K 2 are obtained by summing these contributions over L. For the rigorous solution we obtain Dz(10) = , (14) D 2 (20) = , while the asymptotic formula 4) yields 1 cos (2r - n/4) 7 sin (21' - n/4) D2(r) = /2 5/ /2 7/2 + l' n v n l' 1 sin (41' - n/2) + + O( -9/ 2 ) 161'4-64 n 1'4 l' Dz(10),.., , Dz(20),.., (15) (16)
4 TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 235* TABLE I Numerical values of the spheroidal functions PS2L(0; _1'2), ps2l(i; -y2), the first term of its expansion in Legendre polynomials and the integral 12L(1'2) 2L ps2l(o; -100) ps2l(i; -100) bo 2L (100) 12L(100) , O , , , , ' ~ ~ L ps2l(o; -400) ps2l(i; -400) bo 2L (4oo) 12L(400) ' 'Ü , , ~ , ~ , ~ , \
5 TABLE 11 I~ Numerical values of the expansion coefficients C 2L and C 2L K2 and of the partial amplitudes 0'2L and 0'2L K2 2L I Re C 2L (lo) Im C 2L (IO) C 2L K2 (10) 0'2L(10) 0'2L K2 (10) _10-12 R:j _10-14 R:j 7 _10-11 ~ 14 R:j -2 _10-18 R:j _10-20 R:j 2 _ L Re C 2L (20) Im C2L(20) C K2 2L (20) 0'2L(20) 0'2L K2 (20) R:j -1 _ R:j 2 _ ê '"o ~ R:j 9 _ R:j 5 _ R:j 2 _10-16 R:j 3 _10: R:j -5 _ R:j 3 _10-21 R:j 4 _10-15 '-< 0 i:tj '" "1 is: i:tj ~ := ~ '"o
6 TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 237* The transmission coefficient of the Kirchhoff approximation is D K 2(10) = , D K2 (20) = It is obtained by summing the individual contributions from the rigorous expression DK2(y) = 1_ J 1 (2y), Y (17) in table II, but also (18) where J 1 (2y) is the Bessel function of the first order. The calculations have been carried through on the CD 6400 computer of the "Rechenzentrum der Rheinisch-Westfälischen Technischen Hochschule". All digits given in the tables except for the last digit are considered to be safe. The surprising result is that DK2, although pretty close to D2 in numerical value, has an asymptotic expression which departs from 1 by a term of order y-3/2 in contrast to (15) while, on the other hand, there is such close agreement between the contributions (Jo and O'OK2. One expects that both can be expanded asymptotically in descending powers of y. The close numerical agreement at least to 10-4 for y = 10 and to 10-9 for y = 20 would, however, indicate, that the two asymptotic expansions agree in quite a number of leading terms. Therefore we have studied the situation by comparing the expression with e2l K2 for large positive values of y. The quantity e 2L ' is obtained from e2l in (2) by using (19) and noting that for ~ = 0, y = 20 and L = 0, 1 the imaginary part is about times the real part or less. For y = 40 this would be true even for L = 0, 1,...,6 (see ref. 3). We therefore examine the asymptotic expansion of e 2L '(y) 1 [ps2l(1; _y2)y K(y) := e 2L K2(y) = y2 b02l(y2) 12L(y2) (21) for large positive y and moderate values of L. An asymptotic expansion of the functions PS2L(17; _y2) has been given in ref. 5. Although the normalization (10) drops out in (21), we give the expansions of the individual terms in (21) with regard to the normalized functions PS2L(17; -y2). They are, with p = 2L + 1,
7 238* JOSEF MEIXNER AND SCHIU SCHE (22) (23) (_I)L I2L(Y) '" 2 )1/2 [ 3p r«. ( Y (4L + 1) 1-4y 8p2 + 5 P ] (26 p2 + 57) y2 128 y3 (24) Introducing these expansions into (21) yields K(y) = 1 + O(y-4), as y_oo. (25) Our conjecture is that the non-occurrence of terms of the orders y-l, y-2, y-3 is not accidental, and that all higher powers of y-1 also drop out. But no general proof is available at the present time. This does not mean that C2L' and C2L K2 are equal, rather is it expected that the O(y-4) can be replaced by O(yn exp (-y» with some value of n, which may depend on L. Bouwkamp has also considered the other Kirchhoff-Rayleigh approximation Kl which has the boundary conditions 'ljjkl = 0 on S, 'ljjkl = 1 in A. (26) It can also be easily expanded in spheroidal functions 00 V)K1= :L C2L+1K1 S2L+1(4)(-i~; iy) PS2L+1(1]; _y2) (27) L=O with 0 ~ ~ < 00, 0 ~ 1] ~ 1, where b12l+ 1(y2) 1 C2L+IK1=-ty2(4L+3) Jps2L+l(1];-y2)d1]. (28) _ PS2L+ 1(1 ; _y2) 0 At first sight it does not seem possible to relate simply the individual terms in the expansions (7) and (28). However, at large distance from the aperture, ~» 1, we have Moreover, for large positive values of y we have, apart from terms of order yn exp (-y) with some number nel) (29)
8 TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 239* The square root of (4L + 1)/(4L + 3) enters these expressions due to the different normalization factor of ps2l and PS2L+1. Thus in the far field corresponding terms in both Kirchhoff expansions Kl and K2 become equal to any order y-m (m = 1,2,3,... ) as y -)0 00. But how fast this approach to equality is, depends on the value of L. We have chosen y = 10 and the numerical results of Bouwkamp 1) in order to calculate the contributions of the partial spheroidal waves to the total transmission coefficient for the case Kl. Their consecutive values are ; ; ; ; of which four decimals are considered to be correct. These values should be compared with the 0"2L(10) and the 0"2L K2 (10) in table 11. REFERENCES 1) C. J. Bouwkamp, Thesis Groningen, J. B. Wolters Uitgevers-maatschappij N.V., Groningen-Batavia, 1941 (English translation in IEEE Trans. Antennas and Propagation AP-18, , 1970). 2) J. Meixner and F. W. Schaefke, Mathieu'sche Funktionen und Sphäroidfunktionen, Springer, Berlin, ) S. Hanish, R. V. Baier, V. L. Van Buren and B. J. King, Tables of radial spheroidal wave functions, Vol. 4, N.R.L. Report 7091, Naval Research Laboratory, Washington O.C., ) D. S. Jones, Proc. Camb. Phil. Soc. 61, , ) J. Meixner, Z. angew. Math. Mech. 28, , 1948.
ACCURATE CALCULATION OF PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST KIND AND THEIR FIRST DERIVATIVES
QUARTERLY OF APPLIED MATHEMATICS VOLUME LX, NUMBER 3 SEPTEMBER 2002, PAGES 589-599 ACCURATE CALCULATION OF PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST KIND AND THEIR FIRST DERIVATIVES By ARNIE L.
More informationFourier Optics - Exam #1 Review
Fourier Optics - Exam #1 Review Ch. 2 2-D Linear Systems A. Fourier Transforms, theorems. - handout --> your note sheet B. Linear Systems C. Applications of above - sampled data and the DFT (supplement
More informationElectromagnetic Theory for Microwaves and Optoelectronics
Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Translated by authors With 259 Figures Springer Contents 1 Basic Electromagnetic Theory 1 1.1 Maxwell's Equations 1 1.1.1
More informationThe Oscillating Water Column Wave-energy Device
J. Inst. Maths Applies (1978) 22, 423-433 The Oscillating Water Column Wave-energy Device D. V. EVANS Department of Mathematics, University of Bristol, Bristol [Received 19 September 1977 in revised form
More informationSemi-analytical computation of acoustic scattering by spheroids and disks
Semi-analytical computation of acoustic scattering by spheroids and disks Ross Adelman, a),b) Nail A. Gumerov, c) and Ramani Duraiswami b),c) Institute for Advanced Computer Studies, University of Maryland,
More informationModeling Focused Beam Propagation in a Scattering Medium. Janaka Ranasinghesagara
Modeling Focused Beam Propagation in a Scattering Medium Janaka Ranasinghesagara Lecture Outline Introduction Maxwell s equations and wave equation Plane wave and focused beam propagation in free space
More informationEFFECTIVE CONDUCTIVITY, DIELECTRIC CONSTANT AND PERMEABILITY OF A DILUTE SUSPENSION*)
R893 I Philips Res. Repts 30, 83*-90*, 1~75 Issue in honour of C. J. Bouwkamp EFFECTIVE CONDUCTIVITY, DIELECTRIC CONSTANT AND PERMEABILITY OF A DILUTE SUSPENSION*) by Joseph B. KELLER Courant Institute
More informationNoise in enclosed spaces. Phil Joseph
Noise in enclosed spaces Phil Joseph MODES OF A CLOSED PIPE A 1 A x = 0 x = L Consider a pipe with a rigid termination at x = 0 and x = L. The particle velocity must be zero at both ends. Acoustic resonances
More informationInterference, Diffraction and Fourier Theory. ATI 2014 Lecture 02! Keller and Kenworthy
Interference, Diffraction and Fourier Theory ATI 2014 Lecture 02! Keller and Kenworthy The three major branches of optics Geometrical Optics Light travels as straight rays Physical Optics Light can be
More informationContents. 1 Basic Equations 1. Acknowledgment. 1.1 The Maxwell Equations Constitutive Relations 11
Preface Foreword Acknowledgment xvi xviii xix 1 Basic Equations 1 1.1 The Maxwell Equations 1 1.1.1 Boundary Conditions at Interfaces 4 1.1.2 Energy Conservation and Poynting s Theorem 9 1.2 Constitutive
More informationSum Rules and Physical Bounds in Electromagnetics
Sum Rules and Physical Bounds in Electromagnetics Mats Gustafsson Department of Electrical and Information Technology Lund University, Sweden Complex analysis and passivity with applications, SSF summer
More information21. Propagation of Gaussian beams
1. Propagation of Gaussian beams How to propagate a Gaussian beam Rayleigh range and confocal parameter Transmission through a circular aperture Focusing a Gaussian beam Depth of field Gaussian beams and
More informationLecture 16 February 25, 2016
MTH 262/CME 372: pplied Fourier nalysis and Winter 2016 Elements of Modern Signal Processing Lecture 16 February 25, 2016 Prof. Emmanuel Candes Scribe: Carlos. Sing-Long, Edited by E. Bates 1 Outline genda:
More informationarxiv:hep-th/ v1 14 Oct 1992
ITD 92/93 11 Level-Spacing Distributions and the Airy Kernel Craig A. Tracy Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA arxiv:hep-th/9210074v1
More informationElectromagnetic Scattering from an Anisotropic Uniaxial-coated Conducting Sphere
Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August 22-26 43 Electromagnetic Scattering from an Anisotropic Uniaxial-coated Conducting Sphere You-Lin Geng 1,2, Xin-Bao Wu 3, and
More informationVector diffraction theory of refraction of light by a spherical surface
S. Guha and G. D. Gillen Vol. 4, No. 1/January 007/J. Opt. Soc. Am. B 1 Vector diffraction theory of refraction of light by a spherical surface Shekhar Guha and Glen D. Gillen* Materials and Manufacturing
More informationCommunicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths
Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths David A. B. Miller A rigorous method for finding the best-connected orthogonal communication
More informationPHASE VELOCITY AND ATTENUATION OF SH WAVES IN A FIBER-
PHASE VELOCITY AND ATTENUATION OF SH WAVES IN A FIBER- REINFORCED COMPOSITE Ruey-Bin Yang and Ajit K. Mal Department of Mechanical, Aerospace and Nuclear Engineering University of California, Los Angeles,
More informationSound Transmission in an Extended Tube Resonator
2016 Published in 4th International Symposium on Innovative Technologies in Engineering and Science 3-5 November 2016 (ISITES2016 Alanya/Antalya - Turkey) Sound Transmission in an Extended Tube Resonator
More informationBounds for Eigenvalues of Tridiagonal Symmetric Matrices Computed. by the LR Method. By Gene H. Golub
Bounds for Eigenvalues of Tridiagonal Symmetric Matrices Computed by the LR Method By Gene H. Golub 1. Introduction. In recent years, a number of methods have been proposed for finding the eigenvalues
More informationTV Breakaway Fail-Safe Lanyard Release Plug Military (D38999/29 & D38999/30)
y il- y l l iliy (/9 & /0) O O O -..... 6.. O ix i l ll iz y o l yi oiio / 9. O / i --, i, i- oo. i lol il l li, oi ili i 6@0 z iiio i., o l y, 00 i ooio i oli i l li, 00 o x l y, 0@0 z iiio i.,. &. ll
More informationSCATIERING BY A PERIODIC ARRAY OF COWNEAR CRACKS. Y. Mikata
SCATIERING BY A PERIODIC ARRAY OF COWNEAR CRACKS Y. Mikata Center of Excellence for Advanced Materials Department of Applied Mechanics and Engineering Sciences University of California at San Diego, B-010
More informationElectromagnetic Theory for Microwaves and Optoelectronics
Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Second Edition With 280 Figures and 13 Tables 4u Springer Basic Electromagnetic Theory 1 1.1 Maxwell's Equations 1 1.1.1
More informationModeling microlenses by use of vectorial field rays and diffraction integrals
Modeling microlenses by use of vectorial field rays and diffraction integrals Miguel A. Alvarez-Cabanillas, Fang Xu, and Yeshaiahu Fainman A nonparaxial vector-field method is used to describe the behavior
More informationGenerating Bessel beams by use of localized modes
992 J. Opt. Soc. Am. A/ Vol. 22, No. 5/ May 2005 W. B. Williams and J. B. Pendry Generating Bessel beams by use of localized modes W. B. Williams and J. B. Pendry Condensed Matter Theory Group, The Blackett
More informationTRAN S F O R M E R S TRA N SMI S S I O N. SECTION AB Issue 2, March, March,1958, by American Telephone and Telegraph Company
B B, ch, 9 B h f h c h l f f Bll lh, c chcl l l k l, h, h ch f h f ll ll l f h lh h c ll k f Bll lh, c ck ll ch,9, c lh lh B B c x l l f f f 9 B l f c l f f l 9 f B c, h f c x ch 9 B B h f f fc Bll c f
More informationTHE IMPEDANCE SCATTERING PROBLEM FOR A POINT-SOURCE FIELD. THE SMALL RESISTIVE SPHERE
THE IMPEDANCE SCATTERING PROBLEM FOR A POINT-SOURCE FIELD. THE SMALL RESISTIVE SPHERE By GEORGE DASSIOS (Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, Greece)
More informationA family of closed form expressions for the scalar field of strongly focused
Scalar field of non-paraxial Gaussian beams Z. Ulanowski and I. K. Ludlow Department of Physical Sciences University of Hertfordshire Hatfield Herts AL1 9AB UK. A family of closed form expressions for
More informationCopyright 1966, by the author(s). All rights reserved.
Copyright 1966, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are
More informationMie theory for light scattering by a spherical particle in an absorbing medium
Mie theory for light scattering by a spherical particle in an absorbing medium Qiang Fu and Wenbo Sun Analytic equations are developed for the single-scattering properties of a spherical particle embedded
More informationA far-field based T-matrix method for three dimensional acoustic scattering
ANZIAM J. 50 (CTAC2008) pp.c121 C136, 2008 C121 A far-field based T-matrix method for three dimensional acoustic scattering M. Ganesh 1 S. C. Hawkins 2 (Received 14 August 2008; revised 4 October 2008)
More informationTheory of the Magnetotelluric Method
Geophys. J. R. asir. SOC. (1966) 11, 313-381. Theory of the Magnetotelluric Method for a Spherical Conductor * 4 S. P. Srivastava* (Received 1964 September 24. Revised 1965 August 19) Summary Impedance
More informationFrequency Dependence of Ultrasonic Wave Scattering from Cracks
Proceedings of the ARPA/AFML Review of Progress in Quantitative NDE, July 1977 June 1978 Interdisciplinary Program for Quantitative Flaw Definition Annual Reports 1-1979 Frequency Dependence of Ultrasonic
More informationMETHODS OF THEORETICAL PHYSICS
METHODS OF THEORETICAL PHYSICS Philip M. Morse PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Herman Feshbach PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY PART II: CHAPTERS 9
More informationMorphology-Dependent Resonances of an Infinitely Long Circular Cylinder Illuminated by a Diagonally Incident Plane Wave or a Focused Gaussian Beam
Cleveland State University EngagedScholarship@CSU Physics Faculty Publications Physics Department 3-1-1997 Morphology-Dependent Resonances of an Infinitely Long Circular Cylinder Illuminated by a Diagonally
More informationComparative study of scattering by hard core and absorptive potential
6 Comparative study of scattering by hard core and absorptive potential Quantum scattering in three dimension by a hard sphere and complex potential are important in collision theory to study the nuclear
More informationTime part of the equation can be separated by substituting independent equation
Lecture 9 Schrödinger Equation in 3D and Angular Momentum Operator In this section we will construct 3D Schrödinger equation and we give some simple examples. In this course we will consider problems where
More informationOffset Spheroidal Mirrors for Gaussian Beam Optics in ZEMAX
Offset Spheroidal Mirrors for Gaussian Beam Optics in ZEMAX Antony A. Stark and Urs Graf Smithsonian Astrophysical Observatory, University of Cologne aas@cfa.harvard.edu 1 October 2013 This memorandum
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More informationChapter 1 Introduction
Introduction This monograph is concerned with two aspects of optical engineering: (1) the analysis of radiation diffraction valid for all wavelengths of the incident radiation, and (2) the analysis of
More informationBabinet s Principle for Electromagnetic Fields
Babinet s Principle for Electromagnetic Fields 1 Problem Zhong Ming Tan and Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (January 19, 2012) In optics, Babinet s
More informationA R T A - A P P L I C A T I O N N O T E
Loudspeaker Free-Field Response This AP shows a simple method for the estimation of the loudspeaker free field response from a set of measurements made in normal reverberant rooms. Content 1. Near-Field,
More informationA wavenumber approach to characterizing the diffuse field conditions in reverberation rooms
PROCEEDINGS of the 22 nd International Congress on Acoustics Isotropy and Diffuseness in Room Acoustics: Paper ICA2016-578 A wavenumber approach to characterizing the diffuse field conditions in reverberation
More information252 P. ERDÖS [December sequence of integers then for some m, g(m) >_ 1. Theorem 1 would follow from u,(n) = 0(n/(logn) 1/2 ). THEOREM 2. u 2 <<(n) < c
Reprinted from ISRAEL JOURNAL OF MATHEMATICS Vol. 2, No. 4, December 1964 Define ON THE MULTIPLICATIVE REPRESENTATION OF INTEGERS BY P. ERDÖS Dedicated to my friend A. D. Wallace on the occasion of his
More informationA FINITE BASIS THEOREM FOR PACKING BOXES WITH BRICKS*)
R912 Philips Res. Repts 30,337*-343*,1975 Issue in honour of C. J. Bouwkamp A FINITE BASIS THEOREM FOR PACKING BOXES WITH BRICKS*) by N. G. de BRUIJN Technological University Eindhoven Eindhoven, The Netherlands
More informationPhys102 Lecture Diffraction of Light
Phys102 Lecture 31-33 Diffraction of Light Key Points Diffraction by a Single Slit Diffraction in the Double-Slit Experiment Limits of Resolution Diffraction Grating and Spectroscopy Polarization References
More informationOn spherical-wave scattering by a spherical scatterer and related near-field inverse problems
IMA Journal of Applied Mathematics (2001) 66, 539 549 On spherical-wave scattering by a spherical scatterer and related near-field inverse problems C. ATHANASIADIS Department of Mathematics, University
More informationSurface tension driven oscillatory instability in a rotating fluid layer
J. Fluid Mech. (1969), wol. 39, part 1, pp. 49-55 Printed in Great Britain 49 Surface tension driven oscillatory instability in a rotating fluid layer By G. A. McCONAGHYT AND B. A. FINLAYSON University
More information18 BESSEL FUNCTIONS FOR LARGE ARGUMENTS. By M. Goldstein and R. M. Thaler
18 BESSEL FUNCTIONS FOR LARGE ARGUMENTS Bessel Functions for Large Arguments By M. Goldstein R. M. Thaler Calculations of Bessel Functions of real order argument for large values of the argument can be
More informationProgress In Electromagnetics Research M, Vol. 21, 33 45, 2011
Progress In Electromagnetics Research M, Vol. 21, 33 45, 211 INTERFEROMETRIC ISAR THREE-DIMENSIONAL IMAGING USING ONE ANTENNA C. L. Liu *, X. Z. Gao, W. D. Jiang, and X. Li College of Electronic Science
More information! #! % && ( ) ) +++,. # /0 % 1 /21/ 3 && & 44&, &&7 4/ 00
! #! % && ( ) ) +++,. # /0 % 1 /21/ 3 &&4 2 05 6. 4& 44&, &&7 4/ 00 8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 2, FEBRUARY 2008 345 Moment Method Analysis of an Archimedean Spiral Printed
More informationOPSE FINAL EXAM Fall 2016 YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT.
CLOSED BOOK. Equation Sheet is provided. YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT. ALL NUMERICAL ANSERS MUST HAVE UNITS INDICATED. (Except dimensionless units like
More informationDiffraction by a Half-Plane LL. G. CHAMBERS. {Received 29th March Read 5th May 1950.)
Diffraction by a Half-Plane By LL. G. CHAMBERS {Received 29th March 1950. Read 5th May 1950.) 1. Introduction. The diffraction of a simple harmonic wave train by a straightedged semi-infinite screen was
More informationACCURACY ESTIMATION OF CROSS POLAR RADIATION PREDICTION OF OPEN-ENDED THIN- WALL CIRCULAR WAVEGUIDE BY APPROXIMATE METHODS
International Conference on Antenna Theory and Techniques, 6-9 October, 009, Lviv, Uraine pp. 8-86 ACCURACY ESTIMATION OF CROSS POLAR RADIATION PREDICTION OF OPEN-ENDED THIN- WALL CIRCULAR WAVEGUIDE BY
More informationTHERMAL CONVECTION IN A HORIZONTAL FLUID LAYER WITH INTERNAL HEAT SOURCES
THERMAL CONVECTION IN A HORIZONTAL FLUID LAYER WITH INTERNAL HEAT SOURCES by Morten Tveitereid Department of Mechanics University of Oslo, Norway Abstract This paper is concerned with thermal convection
More informationSound radiation from a loudspeaker, from a spherical pole cap, and from a piston in an infinite baffle1
Sound radiation from a loudspeaker, from a spherical pole cap, and from a piston in an infinite baffle1 Ronald M. Aarts, Philips Research Europe HTC 36 (WO-02), NL-5656 AE Eindhoven, The Netherlands, Also
More informationBy C. W. Nelson. 1. Introduction. In an earlier paper by C. B. Ling and the present author [1], values of the four integrals, h I f _wk dw 2k Ç' xkdx
New Tables of Howland's and Related Integrals By C. W. Nelson 1. Introduction. In an earlier paper by C. B. Ling and the present author [1], values of the four integrals, (1) () h I f _wk dw k Ç' xkdx
More informationBinary Puzzles as an Erasure Decoding Problem
Binary Puzzles as an Erasure Decoding Problem Putranto Hadi Utomo Ruud Pellikaan Eindhoven University of Technology Dept. of Math. and Computer Science PO Box 513. 5600 MB Eindhoven p.h.utomo@tue.nl g.r.pellikaan@tue.nl
More informationxi is asymptotically equivalent to multiplication by Xbxi/bs, where S SOME REMARKS CONCERNING SCHRODINGER'S WA VE EQ UA TION
(6 eks (VO + vl +... ) ~(1) VOL. 19, 1933 MA THEMA TICS: G. D. BIRKHOFF 339 is necessary due to the weakness that only the approximate frequency distribution ml is known except at the value I = '/2 n.
More informationChapter 6 SCALAR DIFFRACTION THEORY
Chapter 6 SCALAR DIFFRACTION THEORY [Reading assignment: Hect 0..4-0..6,0..8,.3.3] Scalar Electromagnetic theory: monochromatic wave P : position t : time : optical frequency u(p, t) represents the E or
More informationLecture notes 5: Diffraction
Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through
More informationModeling Focused Beam Propagation in scattering media. Janaka Ranasinghesagara, Ph.D.
Modeling Focused Beam Propagation in scattering media Janaka Ranasinghesagara, Ph.D. Teaching Objectives The need for computational models of focused beam propagation in scattering media Introduction to
More informationLearn how to use Desmos
Learn how to use Desmos Maclaurin and Taylor series 1 Go to www.desmos.com. Create an account (click on bottom near top right of screen) Change the grid settings (click on the spanner) to 1 x 3, 1 y 12
More informationCHAPTER 3 CYLINDRICAL WAVE PROPAGATION
77 CHAPTER 3 CYLINDRICAL WAVE PROPAGATION 3.1 INTRODUCTION The phase and amplitude of light propagating from cylindrical surface varies in space (with time) in an entirely different fashion compared to
More informationSCATTERING FROM PERFECTLY MAGNETIC CON- DUCTING SURFACES: THE EXTENDED THEORY OF BOUNDARY DIFFRACTION WAVE APPROACH
Progress In Electromagnetics Research M, Vol. 7, 13 133, 009 SCATTERING FROM PERFECTLY MAGNETIC CON- DUCTING SURFACES: THE EXTENDED THEORY OF BOUNDARY DIFFRACTION WAVE APPROACH U. Yalçın Department of
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationPhysics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010
Physics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude
More informationAnalytical Study of Electromagnetic Wave Diffraction Through a Circular Aperture with Fringes on a Perfect Conducting Screen
International Journal of High Energy Physics 016; 3(5): 33-40 http://wwwsciencepublishinggroupcom/j/ijhep doi: 1011648/jijhep016030511 ISSN: 376-7405 (Print); ISSN: 376-7448 (Online) Analytical Study of
More informationSpectrum of spatial frequency of terahertz vortex Bessel beams formed using phase plates with spiral zones
Spectrum of spatial frequency of terahertz vortex Bessel beams formed using phase plates with spiral zones Zhabin V.N., Novosibirsk State University Volodkin B.O., Samara State Aerospace University Knyazev
More informationAN EFFICIENT INTEGRAL TRANSFORM TECHNIQUE OF A SINGULAR WIRE ANTENNA KERNEL. S.-O. Park
AN EFFICIENT INTEGRAL TRANSFORM TECHNIQUE OF A SINGULAR WIRE ANTENNA KERNEL S.-O. Park Department of Electronics Engineering Information and Communications University 58-4 Hwaam-dong, Yusung-gu Taejon,
More informationOn the Equation of the Parabolic Cylinder Functions.
On the Equation of the Parabolic Cylinder Functions. By AKCH. MILNE, Research Student, Edinburgh University Mathematical Laboratory. (Bead 9th January 1914- Beceived 29th January 191 Jj). 1. Introductory.
More informationChapter 1 Computer Arithmetic
Numerical Analysis (Math 9372) 2017-2016 Chapter 1 Computer Arithmetic 1.1 Introduction Numerical analysis is a way to solve mathematical problems by special procedures which use arithmetic operations
More informationA rigorous Theory of the Diffraction of electromagnetic Waves on a perfectly conducting Disk
A rigorous Theory of the Diffraction of electromagnetic Waves on a perfectly conducting Disk By Joseph Meixner Of the Institute for Theoretical Physics of RWTH Aachen University Published July 15, 1948
More informationAzimuthally Propagating Modes in an Axially-Truncated, Circular, Coaxial Waveguide
Azimuthally Propagating Modes in an Axially-Truncated, Circular, Coaxial Waveguide Clifton C. Courtney Voss Scientific Albuquerque, NM There is considerable interest in transition structures that efficiently
More informationCHALLENGES TO THE SWARM MISSION: ON DIFFERENT INTERNAL SHA MAGNETIC FIELD MODELS OF THE EARTH IN DEPENDENCE ON SATELLITE ALTITUDES
CHALLENGES TO THE SWARM MISSION: ON DIFFERENT INTERNAL SHA MAGNETIC FIELD MODELS OF THE EARTH IN DEPENDENCE ON SATELLITE ALTITUDES Wigor A. Webers Helmholtz- Zentrum Potsdam, Deutsches GeoForschungsZentrum,
More information1. (3) Write Gauss Law in differential form. Explain the physical meaning.
Electrodynamics I Midterm Exam - Part A - Closed Book KSU 204/0/23 Name Instructions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try to tell about the physics involved,
More information2. Linear recursions, different cases We discern amongst 5 different cases with respect to the behaviour of the series C(x) (see (. 3)). Let R be the
MATHEMATICS SOME LINEAR AND SOME QUADRATIC RECURSION FORMULAS. I BY N. G. DE BRUIJN AND P. ERDÖS (Communicated by Prow. H. D. KLOOSTERMAN at the meeting ow Novemver 24, 95). Introduction We shall mainly
More informationLecture 2: Weak Interactions and BEC
Lecture 2: Weak Interactions and BEC Previous lecture: Ideal gas model gives a fair intuition for occurrence of BEC but is unphysical (infinite compressibility, shape of condensate...) Order parameter
More informationModelling of Ridge Waveguide Bends for Sensor Applications
Modelling of Ridge Waveguide Bends for Sensor Applications Wilfrid Pascher FernUniversität, Hagen, Germany n 1 n2 n 3 R Modelling of Ridge Waveguide Bends for Sensor Applications Wilfrid Pascher FernUniversität,
More informationOH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9
OH BOY! O h Boy!, was or igin a lly cr eat ed in F r en ch an d was a m a jor s u cc ess on t h e Fr en ch st a ge f or young au di enc es. It h a s b een s een by ap pr ox i ma t ely 175,000 sp ect at
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherical Bessel Functions We would like to solve the free Schrödinger equation [ h d l(l + ) r R(r) = h k R(r). () m r dr r m R(r) is the radial wave function ψ( x)
More informationA Local-Global Principle for Diophantine Equations
A Local-Global Principle for Diophantine Equations (Extended Abstract) Richard J. Lipton and Nisheeth Vishnoi {rjl,nkv}@cc.gatech.edu Georgia Institute of Technology, Atlanta, GA 30332, USA. Abstract.
More informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationA super-resolution method based on signal fragmentation
Contributed paper OPTO-ELECTRONICS REVIEW 11(4), 339 344 (2003) A super-resolution method based on signal fragmentation M.J. MATCZAK *1 and J. KORNIAK 2 1 Institute of Physics, University of Rzeszów, 16a
More informationUniversity of Saskatchewan Department of Electrical Engineering
University of Saskatchewan Department of Electrical Engineering December 9,2004 EE30 1 Electricity, Magnetism and Fields Final Examination Professor Robert E. Johanson Welcome to the EE301 Final. This
More informationANALYSIS OF THE COUPLING GAP BETWEEN MICROSTRIP ANTENNAS
Engineering Journal of University of Qatar, Vol. 10, 1997, p. 123-132 ANALYSIS OF THE COUPLING GAP BETWEEN MICROSTRIP ANTENNAS Moataza A. Hindy and Abdela Aziz A. Mitkees Electrical Engineering Department,
More informationDIFFRACTION OF LIGHT BY ULTRASONIC WAVES. BY RAM RATAN AGGARWAL (Delhi)
DIFFRACTION OF LIGHT BY ULTRASONIC WAVES (Deduction of the Different Theories from the Generalised Theory of Raman and Nath) BY RAM RATAN AGGARWAL (Delhi) Received March 1, 1910 (Communicated by Sir C.
More informationMATH 341 MIDTERM 2. (a) [5 pts] Demonstrate that A and B are row equivalent by providing a sequence of row operations leading from A to B.
11/01/2011 Bormashenko MATH 341 MIDTERM 2 Show your work for all the problems. Good luck! (1) Let A and B be defined as follows: 1 1 2 A =, B = 1 2 3 0 2 ] 2 1 3 4 Name: (a) 5 pts] Demonstrate that A and
More informationSurface Waves and Free Oscillations. Surface Waves and Free Oscillations
Surface waves in in an an elastic half spaces: Rayleigh waves -Potentials - Free surface boundary conditions - Solutions propagating along the surface, decaying with depth - Lamb s problem Surface waves
More informationP1 (cos 0) by Jo([ ] sin 0/2). While these approximations are very good. closed set that is not closed.
62 PHYSICS: R. SERBER PROC. N. A. S. the collection { aj + E an }. Then every set is compact, but not every set is closed n =,m (e.g.! X - a is not closed). Thus X has Property a but contains a compactly
More informationIf the wavelength is larger than the aperture, the wave will spread out at a large angle. [Picture P445] . Distance l S
Chapter 10 Diffraction 10.1 Preliminary Considerations Diffraction is a deviation of light from rectilinear propagation. t occurs whenever a portion of a wavefront is obstructed. Hecht; 11/8/010; 10-1
More informationSEAFLOOR MAPPING MODELLING UNDERWATER PROPAGATION RAY ACOUSTICS
3 Underwater propagation 3. Ray acoustics 3.. Relevant mathematics We first consider a plane wave as depicted in figure. As shown in the figure wave fronts are planes. The arrow perpendicular to the wave
More informationCalculation of Cylindrical Functions using Correction of Asymptotic Expansions
Universal Journal of Applied Mathematics & Computation (4), 5-46 www.papersciences.com Calculation of Cylindrical Functions using Correction of Asymptotic Epansions G.B. Muravskii Faculty of Civil and
More informationPapers On Sturm-Liouville Theory
Papers On Sturm-Liouville Theory References [1] C. Fulton, Parametrizations of Titchmarsh s m()- Functions in the limit circle case, Trans. Amer. Math. Soc. 229, (1977), 51-63. [2] C. Fulton, Parametrizations
More informationON THE NUMBER OF NIVEN NUMBERS UP TO
ON THE NUMBER OF NIVEN NUMBERS UP TO Jean-Marie DeKoninck 1 D partement de Math matiques et de statistique, University Laval, Quebec G1K 7P4, Canada e-mail: jmdk@mat.ulaval.ca Nicolas Doyon Departement
More informationHolography and Optical Vortices
WJP, PHY381 (2011) Wabash Journal of Physics v3.3, p.1 Holography and Optical Vortices Z. J. Rohrbach, J. M. Soller, and M. J. Madsen Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated:
More informationANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION
R898 Philips Res. Repts 30, 161*-170*,1975 Issue in honour of C. J. Bouwkamp ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION by J. BOERSMA Technological University Eindhoven Eindhoven, The Netherlands (Received
More informationChapter 2 Acoustical Background
Chapter 2 Acoustical Background Abstract The mathematical background for functions defined on the unit sphere was presented in Chap. 1. Spherical harmonics played an important role in presenting and manipulating
More informationPolarized light propagation and scattering in random media
Polarized light propagation and scattering in random media Arnold D. Kim a, Sermsak Jaruwatanadilok b, Akira Ishimaru b, and Yasuo Kuga b a Department of Mathematics, Stanford University, Stanford, CA
More informationLARGE AMPLITUDE OSCILLATIONS OF A TUBE OF INCOMPRESSIBLE ELASTIC MATERIAL*
71 LARGE AMPLITUDE OSCILLATIONS OF A TUBE OF INCOMPRESSIBLE ELASTIC MATERIAL* BT JAMES K. KNOWLES California Institute of Technology 1. Introduction. In recent years a number of problems have been solved
More information