Chapter 15: Superposition and Interference of Waves

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 15: Superposition and Interference of Waves"

Transcription

1 Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when harmonic waves combine o form more complicaed waves

2 The sum (or difference of wo soluions of he wave equaion is again a soluion of he wave equaion > superposiion principle Lineariy of he wave equaion b v a v x b x a The soluions o he wave equaion are linear True for small displacemens where he resoring force remains linear ( If a(x, and b(x, are soluions o he wave equaion, hen so is, (, (, ( x b x a x Wha does his mean? v x ( ( b a v x b a Check: a v x a b v x b is rue if and

3 Inerference The superposiion principle allows us o add waves of differen ampliudes, wavelenghs, and frequencies, each of hem moving in differen direcions. > The resul: Inerference Mahemaically: he sums are algebraic sums Examples: (Red curves: algebraic sum of he blue and green curves

4 Simple example for desrucive inerference: ( x, ( x, Imporan rigonomeric relaionships for general cases of inerference: sin( a cos( a b b sin( kx sin( kx ω ω π sin( kx Simple example for consrucive inerference: Example: ( x, ( x, sin( kx sin( kx ω ω ω sin( acos( b cos( asin( b cos( acos( b sin( asin( b ( x, ( x, sin( kx sin( kx sin( kx ω? ω ω ϕ [sin( kx ω sin( kx ω ϕ] a a b

5 > if sin( if kx π ϕ ω ϕ sin( cos( cos( sin( sin( ϕ ω ϕ ω ϕ ω kx kx kx use: sin( cos( sin( cos( ( ] sin( cos( cos( sin( [sin( ] sin( [sin( ϕ ω ω ϕ ϕ ω ϕ ω ω ϕ ω ω kx kx kx kx kx kx kx plug in: Coherence: Phase is sable in ime.

6 Sanding Waves Through Inerference Adding wo opposiely raveling waves o ge a sanding wave lef( x, sin( ω kx x, sin( ω kx righ ( ( x, righ ( x, lef( x, ( x, sin( ω kx sin( ω kx [ ] sin( ωcos( kx sin( kxcos( ω sin( ωcos( kx sin( kxcos( ω sin( ωcos( kx ( x, sin( ωcos( kx noe ha ω and k are same for raveling waves and resuling sanding wave

7 Superposiion of raveling waves

8

9 Inerference of inciden and refleced waves To ge node a wall, refleced wave mus have opposie sign as inciden wave (bu same magniude a wall

10 Beas Beas occur when wo waves of nearly he same frequency are superposed. Example: Two combs wih differen ooh spacing Where he wo combs superpose, a new paern emerges (here called Moire paern. A similar hing happens when wo waves superpose.

11 Two sine waves wih nearly same wavelenghs superpose o a wave ha has an envelope ha is again a sine wave bu wih longer wavelengh. > beas > How o calculae he wavelengh of he beas?

12 To simplify analysis, fix posiion x and look a wave as funcion of ime. (Could equivalenly fix ime and look a wave as a funcion of posiion Les define wo new frequencies: (x, sin( ω kx sin( ω fix posiion x δω (x, sin( ω kx sin( ω δω ω ω and Ω ω ( ω ω ω Ω (x, δω δω δω sin Ω sin( Ω cos cos( sin Ω (x, ( x, ( x, Ω ( ω sin( ω sin( ω ω kx kx and sin( ω sin( ω δω ω ω ω δω δω δω sin Ω sin( cos cos( sin Ω Ω δω (x, (x, sin( Ω cos δω ( x, ( x, sin( Ωcos( average frequency: Ω (fas bea frequency: ω bea δω (slow

13 f bea ωbea π ω ω π Noe: since he ear is only sensiive o he ampliude, i canno ell he difference beween cos( δω and cos( δω he perceived beaing frequency is f pulse fbea f f ( f f pulse frequency

14 Demo: Bea frequency ω ω Can hear average frequency And pulse frequency f f Ω π pulse fbea f f (fas Wha happens o he pulse frequency when he wo frequencies ge close o each oher? Wha happens o he pulse frequency when he wo frequencies move apar?

15 Spaial Inerference Phenomena (fix ime by aking snapsho and look a x-dependence

16 In phase: Consrucive inerference n x nλ, ±, ±, Phase difference: n π Ou of phase: Desrucive Inerference x ( n λ n, ±, ±, Phase difference: ( n π

17 LL -L minima when L(n/λ, maxima when Lnλ, n, ±, ±, Recall: λ π cos( x λ cos( x λ λ π cos( x cos( x λ π λ cos( x sin( x 4 4

18 Locaions of maxima and minima: L dsin θ for maxima: n λ L n λ sin θ d for minima: n λ L n λ sinθ d where n, ±, ±, ± 3

19 Demo: Inerference from wo speakers S S Raise your hand when an acousic minimum passes you.

20 Collision of pulses

21 Wha happens here?? Noe ha he cener poin a he do remains moionless. Connecion o reflecion of pulse on fixed end

22 Reflecion from a fixed end Think of his as a real pulse moving o he righ and an imaginary pulse moving o he lef. Before reflecion: only real pulse visible ( imaginary one is somewhere beyond wall Afer reflecion: previously imaginary pulse is real now and moves o lef Wha happens during he reflecion? superposiion of lef and righ moving pulses (such ha rope end remains fixed, see previous slide

23 Reflecion from a verically free end Wha happens here?

24 Reflecion due o change in µ (noe ension is consan! v? Ligher o heavier medium

25 heavier o ligher medium

26 Power is conserved (power inpower ou µω Av µω Av µω Av v v v and v v i i i r r r i r ω ω ω i r

27 Fourier decomposiion of waves

28 f( A sin(nω φ n n ' n g(x A sin(nkx θ n This is analogous o decomposing a vecor ino orhogonal direcion componens (x, y, and, excep here we are decomposing a funcion ino orhogonal frequency componens n n

29 8 ( cos(nω π n oddn (.8 cos( ω cos(3ω cos(5ω

30 Real Waves

31 f f f ( bea Ch 5 Useful Equaions Inerference due o wo sources L d sinθ if wo sources are in phase ( φ : nλ for maxima: L nλ sinθ d n λ for minima: L n λ sinθ d where n, ±, ±, ± 3 If wo sources are ou of phase ( φ π n λ for maxima: L n λ sinθ d nλ for minima: L nλ sinθ d µω i Av i i µω r Av r r µω Av v v v and v v ω ω ω i r i r

2. Waves in Elastic Media, Mechanical Waves

2. Waves in Elastic Media, Mechanical Waves 2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in

More information

Interference, Diffraction and Polarization

Interference, Diffraction and Polarization L.1 - Simple nerference Chaper L nerference, Diffracion and Polarizaion A sinusoidal wave raveling in one dimension has he form: Blinn College - Physics 2426 - Terry Honan A coshk x w L where in he case

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Chapter 14: Wave Motion

Chapter 14: Wave Motion Chaper 14: Wave Moion Tpes of mechanical waves Mechanical waves are disurbances ha ravel hrough some maerial or subsance called medium for he waves. ravel hrough he medium b displacing paricles in he medium

More information

Intro to Fourier Series

Intro to Fourier Series Inro o Fourier Series Vecor decomposiion Even and Odd funcions Fourier Series definiion and examples Copyrigh 27 by M.H. Perro All righs reserved. M.H. Perro 27 Inro o Fourier Series, Slide 1 Review of

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009.

Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009. Name: Teacher: DO NOT OPEN THE EXMINTION PPER UNTIL YOU RE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINL EXMINTION June 2009 Value: 100% General Insrucions This examinaion consiss of wo pars. Boh pars

More information

Application of kinematic equation:

Application of kinematic equation: HELP: See me (office hours). There will be a HW help session on Monda nigh from 7-8 in Nicholson 109. Tuoring a #10 of Nicholson Hall. Applicaion of kinemaic equaion: a = cons. v= v0 + a = + v + 0 0 a

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Rotational Inertia of a Point Mass

Rotational Inertia of a Point Mass Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE. Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

More information

Week #9 - The Integral Section 5.1

Week #9 - The Integral Section 5.1 Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

LAB 6: SIMPLE HARMONIC MOTION

LAB 6: SIMPLE HARMONIC MOTION 1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

( ) in the following way. ( ) < 2

( ) in the following way. ( ) < 2 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

More information

4.2 Trigonometric Functions; The Unit Circle

4.2 Trigonometric Functions; The Unit Circle 4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

More information

PHYS245 Lab: RC circuits

PHYS245 Lab: RC circuits PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

Physics 107 HOMEWORK ASSIGNMENT #2

Physics 107 HOMEWORK ASSIGNMENT #2 Phsics 7 HOMEWORK ASSIGNMENT # Cunell & Johnson, 7 h ediion Chaper : Problem 5 Chaper : Problems 44, 54, 56 Chaper 3: Problem 38 *5 Muliple-Concep Example 9 deals wih he conceps ha are imporan in his problem.

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

AP1 Kinematics (A) (C) (B) (D) Answer: C

AP1 Kinematics (A) (C) (B) (D) Answer: C 1. A ball is hrown verically upward from he ground. Which pair of graphs bes describes he moion of he ball as a funcion of ime while i is in he air? Neglec air resisance. y a v a (A) (C) y a v a (B) (D)

More information

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance. 5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

m m m m m correct

m m m m m correct Version 055 Miderm 1 OConnor (05141) 1 This prin-ou should have 36 quesions. Muliple-choice quesions ma coninue on he ne column or pae find all choices before answerin. V1:1, V:1, V3:3, V4:, V5:1. 001

More information

MOTION ALONG A STRAIGHT LINE

MOTION ALONG A STRAIGHT LINE Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

More information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

The Reality Ride. Introduction. Key Concept: Each decision we make has a consequence. The decisions we make today directly affect the future.

The Reality Ride. Introduction. Key Concept: Each decision we make has a consequence. The decisions we make today directly affect the future. The Realiy Ride Inroducion Key Concep: Each decision we make has a consequence. The decisions we make oday direcly affec he fuure. y - au worh i Wha consequences Vocabulary Realiy Goals Challenges Choices

More information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

t t t Numerically, this is an extension of the basic definition of the average for a discrete

t t t Numerically, this is an extension of the basic definition of the average for a discrete Average and alues of a Periodic Waveform: (Nofziger, 8) Begin by defining he average value of any ime-varying funcion over a ime inerval as he inegral of he funcion over his ime inerval, divided by : f

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

Fourier Series and Fourier Transform

Fourier Series and Fourier Transform Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007

More information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F

4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

Velocity & Acceleration Analysis

Velocity & Acceleration Analysis Velociy & Acceleraion Analysis Secion 4 Velociy analysis deermines how fas pars of a machine are moving. Linear Velociy (v) Sraigh line, insananeous speed of a poin. ds s v = d Linear velociy is a vecor.

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Lenz's Law. Definition from the book:

Lenz's Law. Definition from the book: Lenz's Law Definiion from he book: The induced emf resuling from a changing magneic flux has a polariy ha leads o an induced curren whose direcion is such ha he induced magneic field opposes he original

More information

A Curriculum Module for AP Calculus BC Curriculum Module

A Curriculum Module for AP Calculus BC Curriculum Module Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

More information

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t, Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

More information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

6.003: Signals and Systems

6.003: Signals and Systems 6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20 2 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers. 3 Fourier Series

More information

11. Properties of alternating currents of LCR-electric circuits

11. Properties of alternating currents of LCR-electric circuits WS. Properies of alernaing currens of L-elecric circuis. Inroducion So-called passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science

More information

Damped Harmonic Motion Closing Doors and Bumpy Rides

Damped Harmonic Motion Closing Doors and Bumpy Rides Prerequisies and Goal Damped Harmonic Moion Closing Doors and Bumpy Rides Andrew Forreser May 4, 21 Assuming you are familiar wih simple harmonic moion, is equaion of moion, and is soluions, we will now

More information

Chapter4: Superposition and Interference

Chapter4: Superposition and Interference Chapter4: Superposition and Interference Sections Superposition Principle Superposition of Sinusoidal Waves Interference of Sound Waves Standing Waves Beats: Interference in Time Nonsinusoidal Wave Patterns

More information

Discussion Examples Chapter 10: Rotational Kinematics and Energy

Discussion Examples Chapter 10: Rotational Kinematics and Energy Discussion Examples Chaper : Roaional Kinemaics and Energy 9. The Crab Nebula One o he mos sudied objecs in he nigh sky is he Crab nebula, he remains o a supernova explosion observed by he Chinese in 54.

More information

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

More information

Digital Data Acquisition

Digital Data Acquisition ME231 Measuremens Laboraory Spring 1999 Digial Daa Acquisiion Edmundo Corona c The laer par of he 20h cenury winessed he birh of he compuer revoluion. The developmen of digial compuer echnology has had

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

Signal Rectification

Signal Rectification 9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

More information

Motion Along a Straight Line

Motion Along a Straight Line Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

Chapter 8 Copyright Henning Umland All Rights Reserved

Chapter 8 Copyright Henning Umland All Rights Reserved Chaper 8 Copyrigh 1997-2004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon

More information

Brown University PHYS 0060 INDUCTANCE

Brown University PHYS 0060 INDUCTANCE Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide INTODUCTION INDUCTANCE Anyone who has ever grabbed an auomobile spark-plug wire a he wrong place, wih he engine running, has an appreciaion

More information

Fourier Series Approximation of a Square Wave

Fourier Series Approximation of a Square Wave OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Interference: Two Spherical Sources

Interference: Two Spherical Sources Interference: Two Spherical Sources Superposition Interference Waves ADD: Constructive Interference. Waves SUBTRACT: Destructive Interference. In Phase Out of Phase Superposition Traveling waves move through

More information

Circuit Types. () i( t) ( )

Circuit Types. () i( t) ( ) Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

More information

Brown University PHYS 0060 Physics Department LAB B Measuring the Earth s Magnetic Field

Brown University PHYS 0060 Physics Department LAB B Measuring the Earth s Magnetic Field Measuring he Earh s Magneic Field As is well known he Earh has a small magneic field on he order of 0.5 Gauss. I is an ineresing and challenging lab exercise o ry and measure i. elow we lis 5 mehods wih

More information

A Mathematical Description of MOSFET Behavior

A Mathematical Description of MOSFET Behavior 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

More information

University Physics 226N/231N Old Dominion University Wave Motion, Interference, Reflection (Chapter 14)

University Physics 226N/231N Old Dominion University Wave Motion, Interference, Reflection (Chapter 14) University Physics 226N/231N Old Dominion University Wave Motion, Interference, Reflection (Chapter 14) Dr. Todd Satogata (ODU/Jefferson Lab) http://www.toddsatogata.net/2012-odu Monday November 26, 2012

More information

Lecture 2: Interference

Lecture 2: Interference Lecture 2: Interference λ S 1 d S 2 Lecture 2, p.1 Today Interference of sound waves Two-slit interference Lecture 2, p.2 Review: Wave Summary ( ) ( ) The formula y x,t = Acos kx ωt describes a harmonic

More information

A NOTE ON UNIT SYSTEMS

A NOTE ON UNIT SYSTEMS Tom Aage Jelmer NTNU eparmen of Peroleum Engineering and Applied Geophysics Inroducory remarks A NOTE ON UNIT SYSTEMS So far, all equaions have been expressed in a consisen uni sysem. The SI uni sysem

More information

Physics 53. Wave Motion 1

Physics 53. Wave Motion 1 Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can

More information

24. Frequency-Shift Keying (FSK) Modulation and Quadrature Amplitude Modulation (QAM)

24. Frequency-Shift Keying (FSK) Modulation and Quadrature Amplitude Modulation (QAM) Frequency-Shif Keying (FSK) Modulaion and Quadraure mpliude Modulaion (QM) on Mac 4. Frequency-Shif Keying (FSK) Modulaion and Quadraure mpliude Modulaion (QM) Frequency-Shif Keying (BFSK) [-3] binary

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

Transient Analysis of First Order RC and RL circuits

Transient Analysis of First Order RC and RL circuits Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage

More information

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t ) Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y

More information

Section A: Forces and Motion

Section A: Forces and Motion I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and

More information