Wave Phenomena Physics 15c

Similar documents
The Mathematics of Harmonic Oscillators

Introduction. Voice Coil Motors. Introduction - Voice Coil Velocimeter Electromechanical Systems. F = Bli

FL/VAL ~RA1::1. Professor INTERVI of. Professor It Fr recru. sor Social,, first of all, was. Sys SDC? Yes, as a. was a. assumee.

( ) ( ) ( ) 0. Conservation of Energy & Poynting Theorem. From Maxwell s equations we have. M t. From above it can be shown (HW)

Summary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns

Chapter 7 Stead St y- ate Errors

L.3922 M.C. L.3922 M.C. L.2996 M.C. L.3909 M.C. L.5632 M.C. L M.C. L.5632 M.C. L M.C. DRIVE STAR NORTH STAR NORTH NORTH DRIVE

Relation between Fourier Series and Transform

Consider a system of 2 simultaneous first order linear equations

Chapter 5 Transient Analysis

Jonathan Turner Exam 2-10/28/03

INF5820 MT 26 OCT 2012

Introduction to Inertial Dynamics

Section 3: Antiderivatives of Formulas

Chapter 4 A First Analysis of F edback edbac

Pupil / Class Record We can assume a word has been learned when it has been either tested or used correctly at least three times.

Erlkönig. t t.! t t. t t t tj "tt. tj t tj ttt!t t. e t Jt e t t t e t Jt

Advanced Queueing Theory. M/G/1 Queueing Systems

1. Accident preve. 3. First aid kit ess 4. ABCs of life do. 6. Practice a Build a pasta sk

Chapter 11: Matter-Photon Interactions and Cavity Quantum Electrodynamics

Ğ ğ ğ Ğ ğ Öğ ç ğ ö öğ ğ ŞÇ ğ ğ

1 Finite Automata and Regular Expressions

Quantum Harmonic Oscillator

ADORO TE DEVOTE (Godhead Here in Hiding) te, stus bat mas, la te. in so non mor Je nunc. la in. tis. ne, su a. tum. tas: tur: tas: or: ni, ne, o:

Derivation of the differential equation of motion

Math 266, Practice Midterm Exam 2

Inverse Fourier Transform. Properties of Continuous time Fourier Transform. Review. Linearity. Reading Assignment Oppenheim Sec pp.289.

PHY2053 Summer C 2013 Exam 1 Solutions

RUTH. land_of_israel: the *country *which God gave to his people in the *Old_Testament. [*map # 2]

M A. L I O E T O W A R D N O N E A. N I D O H A R I T Y F O R A L L. " An Old Timor's DesorSptlon of HI* Camp Outfit. THE DEATH OF M R L A. R. WEEKS.

INTERQUARTILE RANGE. I can calculate variabilityinterquartile Range and Mean. Absolute Deviation

MODEL SOLUTIONS TO IIT JEE ADVANCED 2014

Root behavior in fall and spring planted roses...

Single Correct Type. cos z + k, then the value of k equals. dx = 2 dz. (a) 1 (b) 0 (c)1 (d) 2 (code-v2t3paq10) l (c) ( l ) x.

EE Control Systems LECTURE 11

Appendix. In the absence of default risk, the benefit of the tax shield due to debt financing by the firm is 1 C E C

w x a f f s t p q 4 r u v 5 i l h m o k j d g DT Competition, 1.8/1.6 Stainless, Black S, M, L, XL Matte Raw/Yellow

Grid Game Policing Poses Problem

Copyright A.Milenin, 2017, AGH University of Science and Technology

SYMMETRICAL COMPONENTS

- Double consonant - Wordsearch 3

Engineering Circuit Analysis 8th Edition Chapter Nine Exercise Solutions

J = 1 J = 1 0 J J =1 J = Bout. Bin (1) Ey = 4E0 cos(kz (2) (2) (3) (4) (5) (3) cos(kz (1) ωt +pπ/2) (2) (6) (4) (3) iωt (3) (5) ωt = π E(1) E = [E e

Rotations.

4.8 Improper Integrals

Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =

Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall

Linear System Review. Linear System Review. Descriptions of Linear Systems: 2008 Spring ME854 - GGZ Page 1

II The Z Transform. Topics to be covered. 1. Introduction. 2. The Z transform. 3. Z transforms of elementary functions

Handout on. Crystal Symmetries and Energy Bands

Convergence Theorems for Two Iterative Methods. A stationary iterative method for solving the linear system: (1.1)

Chapter 8: Propagating Quantum States of Radiation

THE LOWELL LEDGER. INDEPENDENT-NOT NEUTRAL.

Ch 1.2: Solutions of Some Differential Equations

An action with positive kinetic energy term for general relativity. T. Mei

Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction

T h e C S E T I P r o j e c t

a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...

PH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations.

n r t d n :4 T P bl D n, l d t z d th tr t. r pd l

More Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations

More on FT. Lecture 10 4CT.5 3CT.3-5,7,8. BME 333 Biomedical Signals and Systems - J.Schesser

Special Curves of 4D Galilean Space

Campsites are approximately 20 wide x 50 deep. Campsites must be paid for when the site is staked.

Convergence tests for the cluster DFT calculations

Title. Author(s)Ito, Yasuhisa; Igarashi, Hajime. CitationIEEE Transactions on Magnetics, 49(5): Issue Date Doc URL. Rights.

Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues

P441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba

Trader Horn at Strand This Week

Canonical Quantizing of Spinor Fields: Anti-Commutation Relations

0 t b r 6, 20 t l nf r nt f th l t th t v t f th th lv, ntr t n t th l l l nd d p rt nt th t f ttr t n th p nt t th r f l nd d tr b t n. R v n n th r

Analysis of Laser-Driven Particle Acceleration from Planar Transparent Boundaries *

Lecture 12: Introduction to nonlinear optics II.

46 D b r 4, 20 : p t n f r n b P l h tr p, pl t z r f r n. nd n th t n t d f t n th tr ht r t b f l n t, nd th ff r n b ttl t th r p rf l pp n nt n th

Fractions. Mathletics Instant Workbooks. Simplify. Copyright

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

ERAOAL COERECE UCOE CURRE RED ECHOLOGY O 0 l n ll n l Gnlly g l lw hv g l % % xly n g v n n hv g v l h l Bg: R Dg Hgh h x g l lly l lly h ly n HDR n h

ELEG 413 Lecture #6. Mark Mirotznik, Ph.D. Professor The University of Delaware

Generalized Half Linear Canonical Transform And Its Properties

" W I T H M A L I C E T O W A - P t D N O I S T E A - I S T D O H A n i T Y F O R. A L L. " A TENDERFOOT. an awful storm." At this juncture,

Review: Transformations. Transformations - Viewing. Transformations - Modeling. world CAMERA OBJECT WORLD CSE 681 CSE 681 CSE 681 CSE 681

x, x, e are not periodic. Properties of periodic function: 1. For any integer n,

Pricing Arithmetic Average Reset Options With Control Variates

Forms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:

Major: All Engineering Majors. Authors: Autar Kaw, Luke Snyder

CONSTACYCLIC CODES OF LENGTH OVER A FINITE FIELD

Walk Like a Mathematician Learning Task:

Instructions for Section 1

Exponential Stability Analysis of a System Comprised of a Robot and its Associated Safety Mechanism

Floating Point Number System -(1.3)

Floating Point Number System -(1.3)

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS

Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano

Introduction to Laplace Transforms October 25, 2017

1K21 LED GR N +33V 604R VR? 1K0 -33V -33V 0R0 MUTE SWTH? JA? T1 T2 RL? +33V 100R A17 CB? 1N N RB? 2K0 QBI? OU T JE182 4K75 RB? 1N914 D?

o C *$ go ! b», S AT? g (i * ^ fc fa fa U - S 8 += C fl o.2h 2 fl 'fl O ' 0> fl l-h cvo *, &! 5 a o3 a; O g 02 QJ 01 fls g! r«'-fl O fl s- ccco

Types of forces. Types of Forces

Higher Order Binaries with Time Dependent Coefficients and Two Factors - Model for Defaultable Bond with Discrete Default Information

NEW FLOODWAY (CLOMR) TE TE PIN: GREENS OF ROCK HILL, LLC DB: 12209, PG: ' S67 46'18"E APPROX. FLOODWAY NEW BASE FLOOD (CLOMR)


Transcription:

Wv hnon hyscs 5c cur 4 Coupl Oscllors! H& con 4.

Wh W D s T " u forc oscllon " olv h quon of oon wh frcon n foun h sy-s soluon " Oscllon bcos lr nr h rsonnc frquncy " hs chns fro 0 π/ π s h frquncy ncrss hrouh h rsonnc " Wor on by h forc s consu by h frcon " Enry consupon s lr nr h rsonnc

Gols for Toy " Coupl Oscllors " How pr of hronc oscllors bhv whn hy r connc wh ch ohr. " Wll us so lnr lbr " rpr for rl wvs

Coupl Oscllors " Two ncl pnulus r connc by sprn " Consr sll oscllon θ θ " prn nson s F " Rsorn forc fro rvy F sn θ F Equon of oon θ θ F F F

Equon of oon " us solv wo ffrnl quons sulnously " Bru forc " yry " nr lbr θ θ F F F

Bru Forc 0 4 4 X 0 4 X X X

Bru Forc 0 0 4 X X X X X, ± ± For ch soluon of X, w Thn w pu no X Wh pn

yry " Th wo quons r syrc " By n & subrcn h, you " W cn solv h quons for n oos l spl hronc oscllors

yry " oluons r: " How o hy loo l? B A

rlll Oscllon A B " B 0.! 0 " Two pnulus r ovn n prlll " Th sprn os nohn " nurl frquncy of fr pnulu whou coupln

yrc Oscllon A B " A 0! 0 " Two pnulus r ovn syrclly " Th sprn s pn/ shrun by wc h ovn of ch pnulu " s rn by boh h pnulus n h sprn

orl os " Th wo osclln prns r cll h norl os " Boh r spl hronc oscllon " Consn frquncy & plu " Two norl os for wo coupl oscllors " Two pnulus hv wo nl conons ch n / " Two norl os hv wo prrs ch cos bsn

Gnrl oluon " Onc you now h norl os, h nrl soluon s lnr cobnon of h. " s fn soluon h ssfs n nl conon: 0 0 0, 0 0 0, 0!! B A B A B A B A

pcfc oluon " T h rl pr: " Us cos cos cos cos sn sn cos cos cos cos cos cos β α β α β α β α β α β α sn sn cos cos

pcfc oluon.

pcfc oluon cos cos sn sn " lo shows h cs whr n r vry clos << " Th sprn consn s sll w coupln bwn h wo pnulus sll

Bs " Two oscllons of slhly ffrn frquncs prouc bs oulon of plu " Coupl oscllors chn hr plus by bs " B frquncy ffrnc of wo frquncs " Ths s us n unn pno, ur, c " Wll co bc whn w scuss roup vlocy

Fnn orl os " Is hr sysc wy of fnn norl os? " yry s usful. Bu os no lwys wor " Wh f h wo pnulus r ffrn? " Wh f w coupl hr pnulus? " W wn rcp h s urn o wor " You n lnr lbr " If you now lnr lbr, hs s n sy pplcon " If you on, sy. Ths won hur

Ch h: nr Albr ± ± ± ± 3 3 b b b b 3 3 b b b b 3 33 3 3 3 3 3 33 3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Rwrn Equon of oon " W r loon for norl o " Boh n oscll on s frquncy " Throw hs no

oon for orl o " r K s vcor! consn s slf " W cll n nvcor of K " Th consn s cll h nvlu " In nrl, n n n r hs n nvcors " Brrn unforun concnc v o h " W pc o fn wo norl os hr K How?

Fnn Envlus " Equon 0 cn b ssf f 0. " Th rnn n hs cs s " Ths vs h wo soluons 0 K 0 K 0,

Fnn Envcors " For " For 0, 0 0

Bc o orl os " W foun h nvlus n nvcors. " Th norl os r vn by

Wh D W rn? " nr lbr vs you rcp for fnn norl os by solvn n nvlu probl " W sw how wor for our spl probl " I urns h h norl os s " Thr wll b n norl os f w coupl n pnulus " W cn now n our probl nown h h norl-o, consn frquncy soluons s " Ths s nouh nforon o l us c h n probl ny ny ny coupl oscllors

ny Coupl nulus " Connc pnulus wh sprns " Dsplcn of h n-h pnulu s n n,, " Equon of oon: n n n n n n

ss-prn Trnssson n " Assu h s vry lon! Inor / " Th srn jus p h ss fro flln " Equvln o ss-sprn rnssson ln n H& 4. n n n n n n n n n n n

Gon Connuous " ow w vry lr, whl n h ss n h sprn sllr n sllr " I srs o loo l sprn wh srbu ss " Goo ol for chncl wvs such s soun

ury " u coupl oscllors " Gnrl soluon s n lnr cobnon of norl os prns of oscllon wh consn frquncs " urprsn prn shows up Bs " nr lbr urns h h norl os s " Envlus! orl frquncs " Envcors! orl os " W r ry o n coupl oscllors no ss-sprn rnssson ln " Rl connuous wvs n!