Wave Phenomena Physics 15c
|
|
- Jasmine Chapman
- 6 years ago
- Views:
Transcription
1 Wave Phenomena Physics 5c Lecture Fourier Analysis (H&L Sections 3. 4) (Georgi Chapter )
2 What We Did Last ime Studied reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical impedance is defined by F = ± Zv For transverse/longitudinal waves: Useful in analyzing reflection Studied standing waves Created by reflecting sinusoidal waves Oscillation pattern has nodes and antinodes Musical instruments use standing waves to produce their distinct sound Z = [ or K] ρ l
3 Goals For oday Define Fourier integral Fourier series is defined for repetitive functions Discreet values of frequencies contribute = + n n + n n n= Extend the definition to include non-repetitive functions Sum becomes an integral Discuss pulses and wave packets Sending information using waves Signal speed and bandwidth ( ω ω ) f() t a a cos t b sin t Connection with Quantum Mechanics
4 Looking Back In Lecture #5, we solved the wave equation ξ( xt, ) = c w ξ( xt, ) t x ω i( kx± ωt) c Normal-mode solutions ξ( xt, ) = ξe w = k Using Fourier series, we can make any arbitrary waveform with linear combination of the normal modes Example: forward-going repetitive waves ( ) ξ( xt, ) = f( x ct) = a cos( kx ω t) + bsin( kx ω t) w n n n n n n n= Non-repetitive waves also OK if we make his makes ω continuous ω = k A little math work needed n = π n c ω n w n
5 Fourier Series For repetitive function f(t) a = f ( t) dt = + n n + n n n= ( ω ω ) f() t a a cos t b sin t an f t tdt = ()cosω n n = ω = π n b f()sin t ωntdt Express cosω n t and sinω n t with complex exponentials an ibn iωnt an + ibn iωnt ancosωnt+ bnsin ωnt = e + e n am + ibm iωmt an + ibn iωnt = e + e m= n= ( ) n= = m = n a m = a n b m = b n ω m = ω n n
6 Fourier Series Sum includes n = an + ibn Define Fn and How do we calculate F n? iω nt Fn = ()cos ()sin () f t ω ntdt + i f t ω ntdt = f t e d t F = f() t dt = It s useful later if I shift the integration range here iω nt Fn = f() t e dt Now we take it to the continuous limit F = a same i n f() t = Fne ω n= OK because f(t) is repetitive t
7 Fourier Integral i nt f() t Fne ω = iω nt Fn = f() t e dt n= Make iω nt Fn in ωt f() t = lim Fn e = lim e ω ω n= = lim Fe n π = iωt iωt F( ω) e dω d ω n= F(ω) is the Fourier integral of f(t) iωt f() t = F( ω) e dω ω = n ω π n π iωt F( ω) lim Fn = f( t) e dt π π iωt F( ω) = f( t) e dt π
8 Fourier Integral iωt f() t = F( ω) e dω Fourier integral F(ω) is A decomposition of f(t) into different frequencies An alternative, complete representation of f(t) One can convert f(t) into F(ω) and vice versa f(t) is in the time domain F(ω) is in the frequency domain iωt F( ω) = f( t) e dt π F(ω) and f(t) are two equally-good representations of a same function
9 Warning Different conventions exist in Fourier integrals iωt f() t F( ω) e d = () = ( ) π iωt f t F ω e d iωt f() t F( ω) e d Watch out when you read other textbooks ω and ω iωt F( ω) = f( t) e dt π and = ω and π F( ω) = f( t) e iωt dt iωt F( ω) = f( t) e dt π
10 Square Pulse t < f() t = t > iωt iωt ω F( ω) = f( t) e dt e dt sin π = π = πω Consider a short pulse with unit area Fourier F(ω) is a bunch of little ripples around ω = Height is /π Area is / F() = π π ω
11 Pulse Width ω F( ω) = sin π width πω he shorter the pulse, the wider the F(ω) Pulse of duration (width in t) (width in ω) = π = const his is a general feature of Fourier transformation Example: Gaussian function f() t = π e t F( ω) = e π ω
12 Sending Information Consider sending information using waves Voice in the air Voice converted into EM signals on a phone cable Video signals through a V cable You can t do it with pure sine waves cos(kx ωt) It just goes on Completely predictable No information You need waves that change patterns with time What you really need are pulses Pulse width determines the speed Pulses must be separated by at least
13 Amplitude Modulation Audio signals range from to khz oo low for efficient radio transmission Use a better frequency and modulate amplitude Carrier wave Audio signal Modulated waves are no longer pure sine waves What is the frequency composition? Amplitude-modulated waves
14 Wave Packet Consider carrier waves modulated by a pulse his makes a short train of waves A wave packet f() t = ω e t < = /( khz) for audio signals Fourier integral is i t t > f() t iω ( t iωt ω ω) F( ω) = e e dt = sin π π( ω ω)
15 Wave Packet ( ω ω) F( ω) = sin πω ( ω ) Similar to the square pulse Width is π/ Centered at ω = ω o send pulses every second, your signal must have a minimum spread of π/ in ω, which corresponds to / in frequency his is called the bandwidth of your radio station his limits how close the frequencies of radio stations can be You need khz for HiFi audio ω It s more like 5 khz in commercial AM stations π ω
16 Bandwidth Speed of information transfer = # of pulses / second Determined by the pulse width in the time domain ranslated into bandwidth in the frequency domain We say bandwidth to mean speed of communication Broadband means fast communication Each medium has its maximum bandwidth You can split it into smaller bandwidth channels Radio wave frequencies Regulated by the government Cable V 75 MHz / 6 MHz = 5 channels You want to minimize the bandwidth of each channel elephones carry only between 4 and 34 Hz
17 Delta Function ake the square pulse again Make it narrower by he height grows / We get an infinitely narrow pulse with unit area Dirac s delta function δ(t) δ () t t = = t and For any function f(t) δ () tdt= f() t δ () t dt = f() f t δ t t dt = f t () ( ) ( )
18 Delta Function What is the Fourier integral of δ(t)? iωt F( ω) = δ( t) e dt = π π δ(t) contains all frequencies equally iωt δ () t = e dω Another way of defining δ(t) π You can get this also by making in ω F( ω) = sin πω
19 Pure Sine Waves Consider pure sine waves with angular frequency ω f() t = i t e ω F ω = e e dt e dt δω ω π = π = iωt i( ) t ( ) i ω t ω ω ( ) f() t F( ω) t ω ω
20 How hings Fit ogether Waveform t domain t width ω domain ω width Sinusoidal uniform infinite δ(ω ω) δ pulse δ(t) uniform infinite Finite pulse and everything else f(t) F(ω) / Pure sine waves and δ pulses are the two extreme cases of all waves Everything falls in between Widths in t and ω are inversely proportional to each other Wait Did I prove it?
21 Arbitrary Signal Width Now we consider a signal with an arbitrary shape f () t F( ω) Fourier Let s define the average time and the average frequency t = t f() t f() t dt dt ω = ω F( ω) dω F( ω) dω Because (energy density) (amplitude) Now we define the r.m.s. widths in t and ω ( t) = ( t t ) ( ω) = ( ω ω ) r.m.s. = root mean square
22 Arbitrary Signal Width ( ) ( ) ( ) () t = t t = f() t dt t t f t dt ( ) ( ) ( ) F( ) ω = ω ω = F( ω) dω ω ω ω dω What can we do with this mess?? We can express F(ω) with f(t) as F ω dω f t f se dωdtds 4π * = f () t f ( s) δ ( t s) dtds π = f() t dt π * iω ( t s) ( ) = ( ) ( ) = π ( ) ( ) i ω F ω f t e t dt
23 Arbitrary Signal Width Next we take iωt f() t = F( ω) e dω d [ f () t ] i F ( ) e iωt d with t dt = ω ω ω i t d ω ωf( ω) e dω = i f ( t) dt We can use this to construct Differentiate * ( ) ( ) = ( ) ( )( ) ( ) ( ) d = i ω f() t dt π dt [ ] ω ω F ω dω ω ω F ω ω ω F ω δ ω ω dωdω * i( ω ω ) t = ( ω ω ) F( ω) ( ω ω ) F ( ω ) e dωdω d π t
24 Arbitrary Signal Width Now we have ( t) = ( ) t t f() t dt f() t Here comes the trick: we calculate the integral I ( κ ) It s a positive number divided by a positive number κ is a real number dt ( ω ) = d i ω f() t dt dt f() t d t t iκ i ω f() t dt dt = > f() t dt dt
25 Arbitrary Signal Width I ( κ) ( t) κ ( ω) κ = + + he integral in the denominator becomes κ d d dt dt ( ) + ( ) * * t t f () t f () t f () t t t f () t dt Integrate the first term in parts d d dt dt * * ( t t ) f() t + κ ( t t ) f() t f () t + f() t ( t t ) f () t dt κ = because the pulse has a finite extent d * d * ( t t ) f() t iκ i ω f () t + iκ i ω f() t ( t t ) f () t dt dt dt f() t d * d * ( tf () t ) f () t + f () t tf () t dt = κ f () t dt dt dt dt
26 Arbitrary Signal Width We ve come a long way Now we got I ( ) ( t) ( ) κ = + κ ω κ > If a quadratic function of κ is always positive, ( ) ( ) D= 4 t ω < finally! t ω > For any signal, the product of the r.m.s. widths t and ω in the time and frequency domain is greater than /
27 Space and Wavenumber We have studied Fourier transformation in time t and frequency ω We can also do it in space x and wavenumber k Everything works the same way ikx f( x) = ikx F( k) e dk Fk ( ) = f ( xe ) dx π In particular, x k > for any signal traveling in space Why is it important?
28 Uncertainty Principle In Quantum Mechanics, particles are wave packets Unlike a classical particle, wave packet has a length he position cannot be determined more accurately than x Momentum is related to the wavenumber by p = hk his means h = x p= h x k > h h π Planck s constant = J s Heisenberg s Uncertainty Principle
29 Summary Defined Fourier integral iωt f() t = iωt F( ω) e dω F( ω) = f( t) e dt π f(t) and F(ω) represent a function in time/frequency domains Analyzed pulses and wave packets ime resolution t and bandwidth ω related by Proved for arbitrary waveform Rate of information transmission bandwidth Dirac s δ(t) a limiting case of infinitely fast pulse t ω > Connection with Heisenberg s Uncertainty Principle in QM
Wave Phenomena Physics 15c. Lecture 10 Fourier Transform
Wave Phenomena Physics 15c Lecture 10 Fourier ransform What We Did Last ime Reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical impedance is defined by For transverse/longitudinal
More informationWave Phenomena Physics 15c. Lecture 11 Dispersion
Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed
More informationWave Phenomena Physics 15c
Wave Pheomea Physics 5c Lecture Fourier Aalysis (H&L Sectios 3. 4) (Georgi Chapter ) Admiistravia! Midterm average 68! You did well i geeral! May got the easy parts wrog, e.g. Problem (a) ad 3(a)! erm
More informationLOPE3202: Communication Systems 10/18/2017 2
By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.
More informationHow many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation?
How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation? (A) 0 (B) 1 (C) 2 (D) more than 2 (E) it depends or don t know How many of
More informationNotes on Fourier Series and Integrals Fourier Series
Notes on Fourier Series and Integrals Fourier Series et f(x) be a piecewise linear function on [, ] (This means that f(x) may possess a finite number of finite discontinuities on the interval). Then f(x)
More informationI WAVES (ENGEL & REID, 13.2, 13.3 AND 12.6)
I WAVES (ENGEL & REID, 13., 13.3 AND 1.6) I.1 Introduction A significant part of the lecture From Quantum to Matter is devoted to developing the basic concepts of quantum mechanics. This is not possible
More informationLECTURE 4 WAVE PACKETS
LECTURE 4 WAVE PACKETS. Comparison between QM and Classical Electrons Classical physics (particle) Quantum mechanics (wave) electron is a point particle electron is wavelike motion described by * * F ma
More information= k, (2) p = h λ. x o = f1/2 o a. +vt (4)
Traveling Functions, Traveling Waves, and the Uncertainty Principle R.M. Suter Department of Physics, Carnegie Mellon University Experimental observations have indicated that all quanta have a wave-like
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics Kevin S. Huang Contents 1 The Wave Function 1 1.1 The Schrodinger Equation............................ 1 1. Probability.................................... 1.3 Normalization...................................
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More informationFourier Series and Integrals
Fourier Series and Integrals Fourier Series et f(x) beapiece-wiselinearfunctionon[, ] (Thismeansthatf(x) maypossessa finite number of finite discontinuities on the interval). Then f(x) canbeexpandedina
More informationFourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko
Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, 20 Prof. Young-Chai Ko koyc@korea.ac.kr Summary Fourier transform Properties Fourier transform of special function Fourier
More informationTHE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3
THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3 Any periodic function f(t) can be written as a Fourier Series a 0 2 + a n cos( nωt) + b n sin n
More informationWave Phenomena Physics 15c. Lecture 9 Wave Reflection Standing Waves
Wave Phenomena Physics 15c Lecture 9 Wave Reflection Standing Waves What We Did Last Time Energy and momentum in LC transmission lines Transfer rates for normal modes: and The energy is carried by the
More informationSignals and Fourier Analysis
Chapter 10 Signals and Fourier Analysis Traveling waves with a definite frequency carry energy but no information They are just there, always have been and always will be To send information, we must send
More informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationSpectral Broadening Mechanisms
Spectral Broadening Mechanisms Lorentzian broadening (Homogeneous) Gaussian broadening (Inhomogeneous, Inertial) Doppler broadening (special case for gas phase) The Fourier Transform NC State University
More informationBASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?
1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,
More informationProblem 1. Part a. Part b. Wayne Witzke ProblemSet #4 PHY ! iθ7 7! Complex numbers, circular, and hyperbolic functions. = r (cos θ + i sin θ)
Problem Part a Complex numbers, circular, and hyperbolic functions. Use the power series of e x, sin (θ), cos (θ) to prove Euler s identity e iθ cos θ + i sin θ. The power series of e x, sin (θ), cos (θ)
More informatione iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that
Phys 53 Fourier Transforms In this handout, I will go through the derivations of some of the results I gave in class (Lecture 4, /). I won t reintroduce the concepts though, so if you haven t seen the
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationExplanations of animations
Explanations of animations This directory has a number of animations in MPEG4 format showing the time evolutions of various starting wave functions for the particle-in-a-box, the free particle, and the
More information4E : The Quantum Universe. Lecture 9, April 13 Vivek Sharma
4E : The Quantum Universe Lecture 9, April 13 Vivek Sharma modphys@hepmail.ucsd.edu Just What is Waving in Matter Waves? For waves in an ocean, it s the water that waves For sound waves, it s the molecules
More informationINTRODUCTION TO QUANTUM MECHANICS PART-II MAKING PREDICTIONS in QUANTUM
Lecture Notes PH4/5 ECE 598 A. La Rosa INTRODUCTION TO QUANTUM MECHANICS PART-II MAKING PREDICTIONS in QUANTUM MECHANICS and the HEISENBERG s PRINCIPLE CHAPTER-4 WAVEPACKETS DESCRIPTION OF A FREE-PARTICLE
More informationCHAPTER-5 WAVEPACKETS DESCRIBING the MOTION of a FREE PARTICLE in the context of the WAVE-PARTICLE DUALITY hypothesis
Lecture Notes PH 4/5 ECE 598 A. La Rosa INTRODUCTION TO QUANTUM MECHANICS CHAPTER-5 WAVEPACKETS DESCRIBING the MOTION of a FREE PARTICLE in the context of the WAVE-PARTICLE DUALITY hypothesis 5. Spectral
More informationLinear second-order differential equations with constant coefficients and nonzero right-hand side
Linear second-order differential equations with constant coefficients and nonzero right-hand side We return to the damped, driven simple harmonic oscillator d 2 y dy + 2b dt2 dt + ω2 0y = F sin ωt We note
More informationα(t) = ω 2 θ (t) κ I ω = g L L g T = 2π mgh rot com I rot
α(t) = ω 2 θ (t) ω = κ I ω = g L T = 2π L g ω = mgh rot com I rot T = 2π I rot mgh rot com Chapter 16: Waves Mechanical Waves Waves and particles Vibration = waves - Sound - medium vibrates - Surface ocean
More informationCHAPTER-5 WAVEPACKETS DESCRIBING the MOTION of a FREE PARTICLE in the context of the WAVE-PARTICLE DUALITY hypothesis
Lecture Notes PH 4/5 ECE 598 A. La Rosa INTRODUCTION TO QUANTUM MECHANICS CHAPTER-5 WAVEPACKETS DESCRIBING the MOTION of a FREE PARTICLE in the context of the WAVE-PARTICLE DUALITY hypothesis 5. Spectral
More informationmultiply both sides of eq. by a and projection overlap
Fourier Series n x n x f xa ancos bncos n n periodic with period x consider n, sin x x x March. 3, 7 Any function with period can be represented with a Fourier series Examples (sawtooth) (square wave)
More informationSpectral Broadening Mechanisms. Broadening mechanisms. Lineshape functions. Spectral lifetime broadening
Spectral Broadening echanisms Lorentzian broadening (Homogeneous) Gaussian broadening (Inhomogeneous, Inertial) Doppler broadening (special case for gas phase) The Fourier Transform NC State University
More informationChapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves
Chapter 16 Waves Types of waves Mechanical waves exist only within a material medium. e.g. water waves, sound waves, etc. Electromagnetic waves require no material medium to exist. e.g. light, radio, microwaves,
More informatione iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that
Phys 531 Fourier Transforms In this handout, I will go through the derivations of some of the results I gave in class (Lecture 14, 1/11). I won t reintroduce the concepts though, so you ll want to refer
More informationLecture 4: Oscillators to Waves
Matthew Schwartz Lecture 4: Oscillators to Waves Review two masses Last time we studied how two coupled masses on springs move If we take κ=k for simplicity, the two normal modes correspond to ( ) ( )
More informationGeneral Physics I. Lecture 14: Sinusoidal Waves. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 14: Sinusoidal Waves Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Motivation When analyzing a linear medium that is, one in which the restoring force
More informationChapter 16 Mechanical Waves
Chapter 6 Mechanical Waves A wave is a disturbance that travels, or propagates, without the transport of matter. Examples: sound/ultrasonic wave, EM waves, and earthquake wave. Mechanical waves, such as
More informationWave Phenomena Physics 15c. Lecture 8 LC Transmission Line Wave Reflection
Wave Phenomena Physics 15c Lecture 8 LC Transmission Line Wave Reflection Midterm Exam #1 Midterm #1 has been graded Class average = 80.4 Standard deviation = 14.6 Your exam will be returned in the section
More informationFourier Series. Fourier Transform
Math Methods I Lia Vas Fourier Series. Fourier ransform Fourier Series. Recall that a function differentiable any number of times at x = a can be represented as a power series n= a n (x a) n where the
More informationX(t)e 2πi nt t dt + 1 T
HOMEWORK 31 I) Use the Fourier-Euler formulae to show that, if X(t) is T -periodic function which admits a Fourier series decomposition X(t) = n= c n exp (πi n ) T t, then (1) if X(t) is even c n are all
More informationBasics of Radiation Fields
Basics of Radiation Fields Initial questions: How could you estimate the distance to a radio source in our galaxy if you don t have a parallax? We are now going to shift gears a bit. In order to understand
More informationWAVE PACKETS & SUPERPOSITION
1 WAVE PACKETS & SUPERPOSITION Reading: Main 9. PH41 Fourier notes 1 x k Non dispersive wave equation x ψ (x,t) = 1 v t ψ (x,t) So far, we know that this equation results from application of Newton s law
More informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationGroup & Phase Velocities (2A)
(2A) 1-D Copyright (c) 2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationPhysics 1C. Lecture 13B
Physics 1C Lecture 13B Speed of Sound! Example values (m/s): Description of a Sound Wave! A sound wave may be considered either a displacement wave or a pressure wave! The displacement of a small element
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 15c Lecture 15 lectromagnetic Waves (H&L Sections 9.5 7) What We Did Last Time! Studied spherical waves! Wave equation of isotropic waves! Solution e! Intensity decreases with! Doppler
More informationSpectral Analysis of Random Processes
Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all
More informationLecture 34. Fourier Transforms
Lecture 34 Fourier Transforms In this section, we introduce the Fourier transform, a method of analyzing the frequency content of functions that are no longer τ-periodic, but which are defined over the
More informationChapter 16 - Waves. I m surfing the giant life wave. -William Shatner. David J. Starling Penn State Hazleton PHYS 213. Chapter 16 - Waves
I m surfing the giant life wave. -William Shatner David J. Starling Penn State Hazleton PHYS 213 There are three main types of waves in physics: (a) Mechanical waves: described by Newton s laws and propagate
More informationA3. Frequency Representation of Continuous Time and Discrete Time Signals
A3. Frequency Representation of Continuous Time and Discrete Time Signals Objectives Define the magnitude and phase plots of continuous time sinusoidal signals Extend the magnitude and phase plots to discrete
More informationNo Lecture on Wed. But, there is a lecture on Thursday, at your normal recitation time, so please be sure to come!
Announcements Quiz 6 tomorrow Driscoll Auditorium Covers: Chapter 15 (lecture and homework, look at Questions, Checkpoint, and Summary) Chapter 16 (Lecture material covered, associated Checkpoints and
More informationMatter Waves. Chapter 5
Matter Waves Chapter 5 De Broglie pilot waves Electromagnetic waves are associated with quanta - particles called photons. Turning this fact on its head, Louis de Broglie guessed : Matter particles have
More informationNotes 07 largely plagiarized by %khc
Notes 07 largely plagiarized by %khc Warning This set of notes covers the Fourier transform. However, i probably won t talk about everything here in section; instead i will highlight important properties
More informationProblem Sheet 1 Examples of Random Processes
RANDOM'PROCESSES'AND'TIME'SERIES'ANALYSIS.'PART'II:'RANDOM'PROCESSES' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Problem'Sheets' Problem Sheet 1 Examples of Random Processes 1. Give
More informationFourier Series & The Fourier Transform
Fourier Series & The Fourier Transform What is the Fourier Transform? Anharmonic Waves Fourier Cosine Series for even functions Fourier Sine Series for odd functions The continuous limit: the Fourier transform
More informationUnstable Oscillations!
Unstable Oscillations X( t ) = [ A 0 + A( t ) ] sin( ω t + Φ 0 + Φ( t ) ) Amplitude modulation: A( t ) Phase modulation: Φ( t ) S(ω) S(ω) Special case: C(ω) Unstable oscillation has a broader periodogram
More informationChapter 16 Waves in One Dimension
Chapter 16 Waves in One Dimension Slide 16-1 Reading Quiz 16.05 f = c Slide 16-2 Reading Quiz 16.06 Slide 16-3 Reading Quiz 16.07 Heavier portion looks like a fixed end, pulse is inverted on reflection.
More informationSeries FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 24 g.s.mcdonald@salford.ac.uk 1. Theory
More informationQuantum Physics Lecture 3
Quantum Physics Lecture 3 If light (waves) are particle-like, are particles wave-like? Electron diffraction - Davisson & Germer Experiment Particle in a box -Quantisation of energy Wave Particle?? Wave
More informationChapter 14: Wave Motion Tuesday April 7 th
Chapter 14: Wave Motion Tuesday April 7 th Wave superposition Spatial interference Temporal interference (beating) Standing waves and resonance Sources of musical sound Doppler effect Sonic boom Examples,
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationLinear and Nonlinear Oscillators (Lecture 2)
Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical
More informationPHYS 3313 Section 001 Lecture #16
PHYS 3313 Section 001 Lecture #16 Monday, Mar. 24, 2014 De Broglie Waves Bohr s Quantization Conditions Electron Scattering Wave Packets and Packet Envelops Superposition of Waves Electron Double Slit
More informationThe Generation of Ultrashort Laser Pulses II
The Generation of Ultrashort Laser Pulses II The phase condition Trains of pulses the Shah function Laser modes and mode locking 1 There are 3 conditions for steady-state laser operation. Amplitude condition
More informationThe formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)
More informationWave nature of particles
Wave nature of particles We have thus far developed a model of atomic structure based on the particle nature of matter: Atoms have a dense nucleus of positive charge with electrons orbiting the nucleus
More informationLecture 30. Chapter 21 Examine two wave superposition (-ωt and +ωt) Examine two wave superposition (-ω 1 t and -ω 2 t)
To do : Lecture 30 Chapter 21 Examine two wave superposition (-ωt and +ωt) Examine two wave superposition (-ω 1 t and -ω 2 t) Review for final (Location: CHEM 1351, 7:45 am ) Tomorrow: Review session,
More informationENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University
ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
More informationDamped Harmonic Oscillator
Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or
More informationExplanations of quantum animations Sohrab Ismail-Beigi April 22, 2009
Explanations of quantum animations Sohrab Ismail-Beigi April 22, 2009 I ve produced a set of animations showing the time evolution of various wave functions in various potentials according to the Schrödinger
More informationChapter 4. The wave like properties of particle
Chapter 4 The wave like properties of particle Louis de Broglie 1892 1987 French physicist Originally studied history Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons
More informationReview Quantitative Aspects of Networking. Decibels, Power, and Waves John Marsh
Review Quantitative spects of Networking Decibels, ower, and Waves John Marsh Outline Review of quantitative aspects of networking Metric system Numbers with Units Math review exponents and logs Decibel
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More information26. The Fourier Transform in optics
26. The Fourier Transform in optics What is the Fourier Transform? Anharmonic waves The spectrum of a light wave Fourier transform of an exponential The Dirac delta function The Fourier transform of e
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More information( ) f (k) = FT (R(x)) = R(k)
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)
More informationA second look at waves
A second loo at waves ravelling waves A first loo at Amplitude Modulation (AM) Stationary and reflected waves Lossy waves: dispersion & evanescence I thin this is the MOS IMPORAN of my eight lectures,
More information1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by
result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM)
More information221B Lecture Notes on Resonances in Classical Mechanics
1B Lecture Notes on Resonances in Classical Mechanics 1 Harmonic Oscillators Harmonic oscillators appear in many different contexts in classical mechanics. Examples include: spring, pendulum (with a small
More informationECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:
ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off
More informationChapter 38 Quantum Mechanics
Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle
More informationElectrodynamics HW Problems 06 EM Waves
Electrodynamics HW Problems 06 EM Waves 1. Energy in a wave on a string 2. Traveling wave on a string 3. Standing wave 4. Spherical traveling wave 5. Traveling EM wave 6. 3- D electromagnetic plane wave
More informationPhysics 217 Problem Set 1 Due: Friday, Aug 29th, 2008
Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationJean Morlet and the Continuous Wavelet Transform
Jean Brian Russell and Jiajun Han Hampson-Russell, A CGG GeoSoftware Company, Calgary, Alberta, brian.russell@cgg.com ABSTRACT Jean Morlet was a French geophysicist who used an intuitive approach, based
More information1D Wave PDE. Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) Richard Sear.
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) November 12, 2018 Wave equation in one dimension This lecture Wave PDE in 1D Method of Separation of
More informationWave Phenomena Physics 15c. Masahiro Morii
Wave Phenomena Physics 15c Masahiro Morii Teaching Staff! Masahiro Morii gives the lectures.! Tuesday and Thursday @ 1:00 :30.! Thomas Hayes supervises the lab.! 3 hours/week. Date & time to be decided.!
More informationa k cos kω 0 t + b k sin kω 0 t (1) k=1
MOAC worksheet Fourier series, Fourier transform, & Sampling Working through the following exercises you will glean a quick overview/review of a few essential ideas that you will need in the moac course.
More informationPhysics 443, Solutions to PS 1 1
Physics 443, Solutions to PS. Griffiths.9 For Φ(x, t A exp[ a( mx + it], we need that + h Φ(x, t dx. Using the known result of a Gaussian intergral + exp[ ax ]dx /a, we find that: am A h. ( The Schrödinger
More informationTime-resolved vs. frequency- resolved spectroscopy
Time-resolved vs. frequency- resolved spectroscopy 1 Response to the impulsation Response to the impulsation Using the equation: Linear response to the δ function: It implies that the response function
More information16 SUPERPOSITION & STANDING WAVES
Chapter 6 SUPERPOSITION & STANDING WAVES 6. Superposition of waves Principle of superposition: When two or more waves overlap, the resultant wave is the algebraic sum of the individual waves. Illustration:
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationTransverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string)
1 Part 5: Waves 5.1: Harmonic Waves Wave a disturbance in a medium that propagates Transverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string) Longitudinal
More informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More informationFourier Series. Spectral Analysis of Periodic Signals
Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at
More informationBME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY
1 Lucas Parra, CCNY BME 50500: Image and Signal Processing in Biomedicine Lecture 2: Discrete Fourier Transform Lucas C. Parra Biomedical Engineering Department CCNY http://bme.ccny.cuny.edu/faculty/parra/teaching/signal-and-image/
More information1. Types of Waves. There are three main types of waves:
Chapter 16 WAVES I 1. Types of Waves There are three main types of waves: https://youtu.be/kvc7obkzq9u?t=3m49s 1. Mechanical waves: These are the most familiar waves. Examples include water waves, sound
More information