Waves, the Wave Equation, and Phase Velocity. We ll start with optics. The one-dimensional wave equation. What is a wave? Optional optics texts: f(x)

Size: px
Start display at page:

Download "Waves, the Wave Equation, and Phase Velocity. We ll start with optics. The one-dimensional wave equation. What is a wave? Optional optics texts: f(x)"

Transcription

1 We ll start with optics Optional optics texts: Waves, the Wave Equation, and Phase Velocity What is a wave? f(x) f(x-) f(x-) f(x-3) Eugene Hecht, Optics, 4th ed. J.F. James, A Student's Guide to Fourier Transforms Forward [f(x-vt)] and backward [f(x+vt)] propagating waves The one-dimensional wave equation Harmonic waves Optics played a major role in all the physics revolutions of the 0th century, so we ll do some. Wavelength, frequency, period, etc. And waves and the Fourier transform play major roles in all of science, so we ll do that, too. Complex numbers A wave is anything that moves. If we let a = v t, where v is positive and t is time, then the displacement will increase with time. v will be the velocity of the wave. 3 x Plane waves and laser beams We ll derive the wave equation from Maxwell s equations. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f: f(x) f(x-) f(x-) f(x-3) f x So f(x - v t) represents a rightward, or forward, propagating wave. Similarly, f(x + v t) represents a leftward, or backward, propagating wave. The one-dimensional wave equation What is a wave? To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number. 0 Phase velocity 0 3 x " f v t = 0 Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.

2 The solution to the one-dimensional wave equation The wave equation has the simple solution: Proof that f (x ± vt) solves the wave equation u u Write f (x ± vt) as f (u), where u = x ± vt. So = and v x t = ± f f u Now, use the chain rule: = f f u = x u x t u t f ( x, t) = f ( x ± v t) f f So = f f f f = and = ± v x u x u t u f f t u = v Substituting into the wave equation: where f (u) can be any twice-differentiable function. f f f " f # $ = $ v 0 % & = x v t u v ' u ( The D wave equation for light waves A simpler equation for a harmonic wave: E E x t # µ" = 0 where E is the light electric field Use the trigonometric identity: E(x,t) = A cos[(kx t) "] We ll use cosine- and sine-wave solutions: E( x, t) = B cos[ k( x ± v t)] + C sin[ k( x ± v t)] cos(z y) = cos(z) cos(y) + sin(z) sin(y) where z = k x t and y = " to obtain: or where: E( x, t) = B cos( kx ± t) + C sin( kx ± t) k kx ± ( kv) t = v = µ" The speed of light in vacuum, usually called c, is 3 x 0 0 cm/s. E(x,t) = A cos(kx t) cos(") + A sin(kx t) sin(") which is the same result as before, as long as: E( x, t) = B cos( kx " t) + C sin( kx " t) A cos(") = B and A sin(") = C For simplicity, we ll just use the forwardpropagating wave.

3 Definitions: Amplitude and Absolute phase E(x,t) = A cos[(k x t ) " ] A = Amplitude " = Absolute phase (or initial phase) Definitions Spatial quantities: " Temporal quantities: The Phase Velocity Human wave How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v = # / $ Since % = /$ : v = # v In terms of the k-vector, k = " / #, and the angular frequency, = " / $, this is: v = / k A typical human wave has a phase velocity of about 0 seats per second.

4 The Phase of a Wave The phase is everything inside the cosine. E(x,t) = A cos(&), where & = k x t " Complex numbers Consider a point, P = (x,y), on a D Cartesian grid. In terms of the phase, And & = &(x,y,z,t) and is not a constant, like " = $& /$t k = $& /$x $& /$t v = $& /$x This formula is useful when the wave is really complicated. Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number. So, instead of using an ordered pair, (x,y), we write: where i = (-) / P = x + i y = A cos(#) + i A sin(#) Euler's Formula exp(i&) = cos(&) + i sin(&) so the point, P = A cos(#) + i A sin(#), can be written: P = A exp(i&) Proof of Euler's Formula exp(i&) = cos(&) + i sin(&) 3 x x x Use Taylor Series: f ( x) = f (0) + f '(0) + f ''(0) + f '''(0) x x x x exp( x) = x x x x cos( x) = x x x x x sin( x) = where A = Amplitude & = Phase If we substitute x = i& into exp(x), then: 3 4 i i exp( i ) = + " " # $ # $ = % " i... 4 & + % " + 3 & ' ( ' ( = cos( ) + i sin( )

5 Complex number theorems If exp( i ) = cos( ) + i sin( ) exp( i ) = # exp( i / ) = i exp(- i" ) = cos( ") # i sin( ") cos( ") = [ exp( i" ) + exp( # i" )] sin( ") = [ exp( i" ) # exp( # i" )] i A exp( i" ) $ A exp( i" ) = A A exp i( " + " ) A exp( i" ) / A exp( i" ) = A / A exp i( " #" ) More complex number theorems Any complex number, z, can be written: z = Re{ z } + i Im{ z } So Re{ z } = / ( z + z* ) and Im{ z } = /i ( z z* ) where z* is the complex conjugate of z ( i % i ) The "magnitude," z, of a complex number is: z = z z* = Re{ z } + Im{ z } To convert z into polar form, A exp(i&): A = Re{ z } + Im{ z } tan(#) = Im{ z } / Re{ z } We can also differentiate exp(ikx) as if the argument were real. Proof : d dx d exp( ikx) = ik exp( ikx) dx cos( kx) + i sin( kx) = k sin( kx) + ik cos( kx) " # = ik $ sin( kx ) + cos( kx ) & i % ' But / i = i, so : = ik i sin( kx) + cos( kx) Waves using complex numbers Recall that the electric field of a wave can be written: E(x,t) = A cos(kx t ") Since exp(i&) = cos(&) + i sin(&), E(x,t) can also be written: or E(x,t) = Re { A exp[i(kx t ")] } E(x,t) = / A exp[i(kx t ")] + c.c. where "+ c.c." means "plus the complex conjugate of everything before the plus sign." We often write these expressions without the, Re, or +c.c.

6 Waves using complex amplitudes We can let the amplitude be complex: where we've separated the constant stuff from the rapidly changing stuff. The resulting complex amplitude is: So: (, ) = exp ( # #" ) E x t A $ & i kx t %' { } (, ) = { Aexp ( i" )} exp ( x # ) E x t # $ & i k t %' E0 = Aexp( " i ) # (note the " ~ ") (, ) = exp ( " ) E x t E0 i kx t How do you know if E 0 is real or complex? As written, this entire field is complex Complex numbers simplify waves Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency. This isn't so obvious using trigonometric functions, but it's easy with complex exponentials: Etot ( x, t) = E exp i( kx " t) + E exp i( kx " t) + E3 exp i( kx " t) = ( E + E + E3)exp i( kx " t) where all initial phases are lumped into E, E, and E 3. Sometimes people use the "~", but not always. So always assume it's complex. The 3D wave equation for the electric field and its solution A wave can propagate in any direction in space. So we must allow the space derivative to be 3D: µ " E " t # E $ = 0 E 0 exp[ i ( k " r # t )] " is called a plane wave. A plane wave s contours of maximum phase, called wave-fronts or phase-fronts, are planes. They extend over all space. or E E E E x y z t + + # µ" = 0 which has the solution: E( x, y, z, t) = E0 exp[ i( k " r # t)] " " where k ( kx, k y, kz) r ( x, y, z) and k r " k x + k y + k z x y z k k + k + k x y z Wave-fronts are helpful for drawing pictures of interfering waves. A plane wave's wave-fronts are equally spaced, a wavelength apart. They're perpendicular to the propagation direction. A wave's wavefronts sweep along at the speed of light. Usually, we just draw lines; it s easier.

7 Laser beams vs. Plane waves A plane wave has flat wave-fronts throughout all space. It also has infinite energy. It doesn t exist in reality. A laser beam is more localized. We can approximate a laser beam as a plane wave vs. z times a Gaussian in x and y: " # x + y E( x, y, z, t) = E0 exp % $ exp[ i( kz $ t) ] ' ( w & z Localized wave-fronts y x w Laser beam spot on wall

Waves, the Wave Equation, and Phase Velocity

Waves, the Wave Equation, and Phase Velocity Waves, the Wave Equation, and Phase Velocity What is a wave? The one-dimensional wave equation Wavelength, frequency, period, etc. Phase velocity Complex numbers and exponentials Plane waves, laser beams,

More information

2. Waves and the Wave Equation

2. Waves and the Wave Equation 2. Waves and the Wave Equation What is a wave? Forward vs. backward propagating waves The one-dimensional wave equation Phase velocity Reminders about complex numbers The complex amplitude of a wave What

More information

18. Standing Waves, Beats, and Group Velocity

18. Standing Waves, Beats, and Group Velocity 18. Standing Waves, Beats, and Group Velocity Superposition again Standing waves: the sum of two oppositely traveling waves Beats: the sum of two different frequencies Group velocity: the speed of information

More information

Lecture 1. Rejish Nath. Optics, IDC202

Lecture 1. Rejish Nath. Optics, IDC202 Lecture 1. Rejish Nath Optics, IDC202 Contents 1. Waves: The wave equation 2. Harmonic Waves 3. Plane waves 4. Spherical Waves Literature: 1. Optics, (Eugene Hecht and A. R. Ganesan) 2. Optical Physics,

More information

= k, (2) p = h λ. x o = f1/2 o a. +vt (4)

= 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 information

28. Fourier transforms in optics, part 3

28. Fourier transforms in optics, part 3 28. Fourier transforms in optics, part 3 Magnitude and phase some examples amplitude and phase of light waves what is the spectral phase, anyway? The Scale Theorem Defining the duration of a pulse the

More information

THE PHYSICS OF WAVES CHAPTER 1. Problem 1.1 Show that Ψ(x, t) = (x vt) 2. is a traveling wave.

THE PHYSICS OF WAVES CHAPTER 1. Problem 1.1 Show that Ψ(x, t) = (x vt) 2. is a traveling wave. CHAPTER 1 THE PHYSICS OF WAVES Problem 1.1 Show that Ψ(x, t) = (x vt) is a traveling wave. Show thatψ(x, t) is a wave by substitutioninto Equation 1.1. Proceed as in Example 1.1. On line version uses Ψ(x,

More information

PHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015.

PHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. PHYS 408, Optics Problem Set 1 - Spring 2016 Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. 1. An electric field in vacuum has the wave equation, Let us consider the solution, 2 E 1 c 2 2 E =

More information

Constructive vs. destructive interference; Coherent vs. incoherent interference

Constructive vs. destructive interference; Coherent vs. incoherent interference Constructive vs. destructive interference; Coherent vs. incoherent interference Waves that combine in phase add up to relatively high irradiance. = Constructive interference (coherent) Waves that combine

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 12.510 Introduction

More information

MITOCW 8. Electromagnetic Waves in a Vacuum

MITOCW 8. Electromagnetic Waves in a Vacuum MITOCW 8. Electromagnetic Waves in a Vacuum The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources

More information

Chapter 9. Electromagnetic Waves

Chapter 9. Electromagnetic Waves Chapter 9. Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation What is a "wave?" Let's start with the simple case: fixed shape, constant speed: How would you represent such a string

More information

in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD

in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD 2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light

More information

Physics 142 Wave Optics 1 Page 1. Wave Optics 1. For every complex problem there is one solution that is simple, neat, and wrong. H.L.

Physics 142 Wave Optics 1 Page 1. Wave Optics 1. For every complex problem there is one solution that is simple, neat, and wrong. H.L. Physics 142 Wave Optics 1 Page 1 Wave Optics 1 For every complex problem there is one solution that is simple, neat, and wrong. H.L. Mencken Interference and diffraction of waves The essential characteristic

More information

Course Updates. 2) This week: Electromagnetic Waves +

Course Updates.  2) This week: Electromagnetic Waves + Course Updates http://www.phys.hawaii.edu/~varner/phys272-spr1/physics272.html Reminders: 1) Assignment #11 due Wednesday 2) This week: Electromagnetic Waves + 3) In the home stretch [review schedule]

More information

Electromagnetic Waves

Electromagnetic Waves Electromagnetic Waves As the chart shows, the electromagnetic spectrum covers an extremely wide range of wavelengths and frequencies. Though the names indicate that these waves have a number of sources,

More information

1D Wave PDE. Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) Richard Sear.

1D 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 information

Class 30: Outline. Hour 1: Traveling & Standing Waves. Hour 2: Electromagnetic (EM) Waves P30-

Class 30: Outline. Hour 1: Traveling & Standing Waves. Hour 2: Electromagnetic (EM) Waves P30- Class 30: Outline Hour 1: Traveling & Standing Waves Hour : Electromagnetic (EM) Waves P30-1 Last Time: Traveling Waves P30- Amplitude (y 0 ) Traveling Sine Wave Now consider f(x) = y = y 0 sin(kx): π

More information

1 (2n)! (-1)n (θ) 2n

1 (2n)! (-1)n (θ) 2n Complex Numbers and Algebra The real numbers are complete for the operations addition, subtraction, multiplication, and division, or more suggestively, for the operations of addition and multiplication

More information

Lecture 4: Oscillators to Waves

Lecture 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 information

Complex Number Review:

Complex Number Review: Complex Numbers and Waves Review 1 of 14 Complex Number Review: Wave functions Ψ are in general complex functions. So it's worth a quick review of complex numbers, since we'll be dealing with this all

More information

Chapter 2 Solutions. Chapter 2 Solutions 1. 4( z t) ( z t) z t

Chapter 2 Solutions. Chapter 2 Solutions 1. 4( z t) ( z t) z t Chapter Solutions 1 Chapter Solutions.1. 1 z t ( z t) z z ( z t) t t It s a twice differentiable function of ( z t ), where is in the negative z direction. 1 y t ( y, t) ( y4 t) ( y 4 t) y y 8( y 4 t)

More information

3 What You Should Know About Complex Numbers

3 What You Should Know About Complex Numbers 3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make

More information

Differential equations. Background mathematics review

Differential equations. Background mathematics review Differential equations Background mathematics review David Miller Differential equations First-order differential equations Background mathematics review David Miller Differential equations in one variable

More information

Lab 2: Mach Zender Interferometer Overview

Lab 2: Mach Zender Interferometer Overview Lab : Mach Zender Interferometer Overview Goals:. Study factors that govern the interference between two light waves with identical amplitudes and frequencies. Relative phase. Relative polarization. Learn

More information

Plane waves and spatial frequency. A plane wave

Plane waves and spatial frequency. A plane wave Plane waves and spatial frequency A plane wave Complex representation E(,) z t = E cos( ωt kz) = E cos( ωt kz) o Ezt (,) = Ee = Ee j( ωt kz) j( ωt kz) o = 1 2 A B t + + + [ cos(2 ω α β ) cos( α β )] {

More information

Periodic functions: simple harmonic oscillator

Periodic functions: simple harmonic oscillator Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω2 0y = 0 Solution can be written in various ways: y(t) = Ae iω 0t y(t) = A cos

More information

3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series

3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechanics for Scientists and Engineers David Miller Background mathematics 7 Differential equations Differential equations in one variable A differential equation involves derivatives of some function

More information

Plane waves and spatial frequency. A plane wave

Plane waves and spatial frequency. A plane wave Plane waves and spatial frequency A plane wave Complex representation E(,) zt Ecos( tkz) E cos( tkz) o Ezt (,) Ee Ee j( tkz) j( tkz) o 1 cos(2 ) cos( ) 2 A B t Re atbt () () ABcos(2 t ) Complex representation

More information

Chapter 16 - Waves. I m surfing the giant life wave. -William Shatner. David J. Starling Penn State Hazleton PHYS 213. Chapter 16 - Waves

Chapter 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 information

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves Physics 331 Lecture 9 February 13, 019 LAST TIME: Finished mirrors and aberrations, more on plane waves Recall, Represents a plane wave having a propagation vector k that propagates in any direction with

More information

Electromagnetic waves in free space

Electromagnetic waves in free space Waveguide notes 018 Electromagnetic waves in free space We start with Maxwell s equations for an LIH medum in the case that the source terms are both zero. = =0 =0 = = Take the curl of Faraday s law, then

More information

Fourier Approach to Wave Propagation

Fourier Approach to Wave Propagation Phys 531 Lecture 15 13 October 005 Fourier Approach to Wave Propagation Last time, reviewed Fourier transform Write any function of space/time = sum of harmonic functions e i(k r ωt) Actual waves: harmonic

More information

Chapter 16 Waves in One Dimension

Chapter 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 information

Light as a Transverse Wave.

Light as a Transverse Wave. Waves and Superposition (Keating Chapter 21) The ray model for light (i.e. light travels in straight lines) can be used to explain a lot of phenomena (like basic object and image formation and even aberrations)

More information

2 u 1-D: 3-D: x + 2 u

2 u 1-D: 3-D: x + 2 u c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function

More information

Complex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:

Complex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers: Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together

More information

MET 4301 LECTURE SEP17

MET 4301 LECTURE SEP17 ------------------------------------------------------------------------------------------------------------------------------------ Objectives: To review essential mathematics for Meteorological Dynamics

More information

Traveling Waves: Energy Transport

Traveling Waves: Energy Transport Traveling Waves: Energ Transport wave is a traveling disturbance that transports energ but not matter. Intensit: I P power rea Intensit I power per unit area (measured in Watts/m 2 ) Intensit is proportional

More information

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis

SEISMIC 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 information

MATH 308 COURSE SUMMARY

MATH 308 COURSE SUMMARY MATH 308 COURSE SUMMARY Approximately a third of the exam cover the material from the first two midterms, that is, chapter 6 and the first six sections of chapter 7. The rest of the exam will cover the

More information

A new Technique for Simulating Semiconductor Laser Resonators

A new Technique for Simulating Semiconductor Laser Resonators A new Technique for Simulating Semiconductor Laser Resonators Britta Heubeck Christoph Pflaum Department of System Simulation (LSS) University of Erlangen-Nuremberg, Germany NUSOD conference, September

More information

Phase difference plays an important role in interference. Recalling the phases in (3.32) and (3.33), the phase difference, φ, is

Phase difference plays an important role in interference. Recalling the phases in (3.32) and (3.33), the phase difference, φ, is Phase Difference Phase difference plays an important role in interference. Recalling the phases in (3.3) and (3.33), the phase difference, φ, is φ = (kx ωt + φ 0 ) (kx 1 ωt + φ 10 ) = k (x x 1 ) + (φ 0

More information

Coordinate systems and vectors in three spatial dimensions

Coordinate systems and vectors in three spatial dimensions PHYS2796 Introduction to Modern Physics (Spring 2015) Notes on Mathematics Prerequisites Jim Napolitano, Department of Physics, Temple University January 7, 2015 This is a brief summary of material on

More information

Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration

Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration Lana Sheridan De Anza College May 17, 2018 Last time pulse propagation the wave equation Overview solutions to the wave

More information

01 Harmonic Oscillations

01 Harmonic Oscillations Utah State University DigitalCommons@USU Foundations of Wave Phenomena Library Digital Monographs 8-2014 01 Harmonic Oscillations Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu

More information

Electromagnetic fields and waves

Electromagnetic fields and waves Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell

More information

OPAC102. The Acoustic Wave Equation

OPAC102. The Acoustic Wave Equation OPAC102 The Acoustic Wave Equation Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating

More information

An Overly Simplified and Brief Review of Differential Equation Solution Methods. 1. Some Common Exact Solution Methods for Differential Equations

An Overly Simplified and Brief Review of Differential Equation Solution Methods. 1. Some Common Exact Solution Methods for Differential Equations An Overly Simplified and Brief Review of Differential Equation Solution Methods We will be dealing with initial or boundary value problems. A typical initial value problem has the form y y 0 y(0) 1 A typical

More information

10 Motivation for Fourier series

10 Motivation for Fourier series 10 Motivation for Fourier series Donkeys and scholars in the middle! French soldiers, forming defensive squares in Egypt, during Napoleon s campaign The central topic of our course is the so-called Fourier

More information

Wave Mechanics in One Dimension

Wave Mechanics in One Dimension Wave Mechanics in One Dimension Wave-Particle Duality The wave-like nature of light had been experimentally demonstrated by Thomas Young in 1820, by observing interference through both thin slit diffraction

More information

5.1 Second order linear differential equations, and vector space theory connections.

5.1 Second order linear differential equations, and vector space theory connections. Math 2250-004 Wed Mar 1 5.1 Second order linear differential equations, and vector space theory connections. Definition: A vector space is a collection of objects together with and "addition" operation

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

More information

Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; differentiation; integration; vectors.

Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; differentiation; integration; vectors. Revision Checklist Unit C4: Core Mathematics 4 Unit description Assessment information Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; differentiation; integration;

More information

09 The Wave Equation in 3 Dimensions

09 The Wave Equation in 3 Dimensions Utah State University DigitalCommons@USU Foundations of Wave Phenomena Physics, Department of --2004 09 The Wave Equation in 3 Dimensions Charles G. Torre Department of Physics, Utah State University,

More information

Measuring the Universal Gravitational Constant, G

Measuring the Universal Gravitational Constant, G Measuring the Universal Gravitational Constant, G Introduction: The universal law of gravitation states that everything in the universe is attracted to everything else. It seems reasonable that everything

More information

Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator

Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound

More information

Lecture 1 Notes: 06 / 27. The first part of this class will primarily cover oscillating systems (harmonic oscillators and waves).

Lecture 1 Notes: 06 / 27. The first part of this class will primarily cover oscillating systems (harmonic oscillators and waves). Lecture 1 Notes: 06 / 27 The first part of this class will primarily cover oscillating systems (harmonic oscillators and waves). These systems are very common in nature - a system displaced from equilibrium

More information

Quantum Physics Lecture 8

Quantum Physics Lecture 8 Quantum Physics Lecture 8 Applications of Steady state Schroedinger Equation Box of more than one dimension Harmonic oscillator Particle meeting a potential step Waves/particles in a box of >1 dimension

More information

Lab 1: Damped, Driven Harmonic Oscillator

Lab 1: Damped, Driven Harmonic Oscillator 1 Introduction Lab 1: Damped, Driven Harmonic Oscillator The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic

More information

= x iy. Also the. DEFINITION #1. If z=x+iy, then the complex conjugate of z is given by magnitude or absolute value of z is z =.

= x iy. Also the. DEFINITION #1. If z=x+iy, then the complex conjugate of z is given by magnitude or absolute value of z is z =. Handout # 4 COMPLEX FUNCTIONS OF A COMPLEX VARIABLE Prof. Moseley Chap. 3 To solve z 2 + 1 = 0 we "invent" the number i with the defining property i 2 = ) 1. We then define the set of complex numbers as

More information

Maxwell s Equations and Electromagnetic Waves W13D2

Maxwell s Equations and Electromagnetic Waves W13D2 Maxwell s Equations and Electromagnetic Waves W13D2 1 Announcements Week 13 Prepset due online Friday 8:30 am Sunday Tutoring 1-5 pm in 26-152 PS 10 due Week 14 Friday at 9 pm in boxes outside 26-152 2

More information

Lab 1: damped, driven harmonic oscillator

Lab 1: damped, driven harmonic oscillator Lab 1: damped, driven harmonic oscillator 1 Introduction The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic

More information

(Total 1 mark) IB Questionbank Physics 1

(Total 1 mark) IB Questionbank Physics 1 1. A transverse wave travels from left to right. The diagram below shows how, at a particular instant of time, the displacement of particles in the medium varies with position. Which arrow represents the

More information

Concepts from linear theory Extra Lecture

Concepts from linear theory Extra Lecture Concepts from linear theory Extra Lecture Ship waves from WW II battleships and a toy boat. Kelvin s (1887) method of stationary phase predicts both. Concepts from linear theory A. Linearize the nonlinear

More information

Chapter 15. Mechanical Waves

Chapter 15. Mechanical Waves Chapter 15 Mechanical Waves A wave is any disturbance from an equilibrium condition, which travels or propagates with time from one region of space to another. A harmonic wave is a periodic wave in which

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS SPECIALIST MATHEMATICS (Year 11 and 12) UNIT A A1: Combinatorics Permutations: problems involving permutations use the multiplication principle and factorial notation permutations and restrictions with

More information

Reinterpreting the Solutions of Maxwell s Equations in Vacuum

Reinterpreting the Solutions of Maxwell s Equations in Vacuum 1 Reinterpreting the Solutions of Maxwell s Equations in Vacuum V.A.I. Menon, M.Rajendran, V.P.N.Nampoori International School of Photonics, Cochin Univ. of Science and tech., Kochi 680022, India. The

More information

Transverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string)

Transverse 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 information

Superposition of waves (review) PHYS 258 Fourier optics SJSU Spring 2010 Eradat

Superposition of waves (review) PHYS 258 Fourier optics SJSU Spring 2010 Eradat Superposition of waves (review PHYS 58 Fourier optics SJSU Spring Eradat Superposition of waves Superposition of waves is the common conceptual basis for some optical phenomena such as: Polarization Interference

More information

FILTERING IN THE FREQUENCY DOMAIN

FILTERING IN THE FREQUENCY DOMAIN 1 FILTERING IN THE FREQUENCY DOMAIN Lecture 4 Spatial Vs Frequency domain 2 Spatial Domain (I) Normal image space Changes in pixel positions correspond to changes in the scene Distances in I correspond

More information

1 Fundamentals of laser energy absorption

1 Fundamentals of laser energy absorption 1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms

More information

Examples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.

Examples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case. s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:

More information

single uniform density, but has a step change in density at x = 0, with the string essentially y(x, t) =A sin(!t k 1 x), (5.1)

single uniform density, but has a step change in density at x = 0, with the string essentially y(x, t) =A sin(!t k 1 x), (5.1) Chapter 5 Waves II 5.1 Reflection & Transmission of waves et us now consider what happens to a wave travelling along a string which no longer has a single uniform density, but has a step change in density

More information

The de Broglie hypothesis for the wave nature of matter was to simply re-write the equations describing light's dual nature: Elight = hf

The de Broglie hypothesis for the wave nature of matter was to simply re-write the equations describing light's dual nature: Elight = hf Modern Physics (PHY 3305) Lecture Notes Modern Physics (PHY 3305) Lecture Notes Describing Nature as Waves (Ch. 4.3-4.4) SteveSekula, 10 February 010 (created 13 December 009) Review tags: lecture We concluded

More information

(TRAVELLING) 1D WAVES. 1. Transversal & Longitudinal Waves

(TRAVELLING) 1D WAVES. 1. Transversal & Longitudinal Waves (TRAVELLING) 1D WAVES 1. Transversal & Longitudinal Waves Objectives After studying this chapter you should be able to: Derive 1D wave equation for transversal and longitudinal Relate propagation speed

More information

Chapter 9. Electromagnetic waves

Chapter 9. Electromagnetic waves Chapter 9. lectromagnetic waves 9.1.1 The (classical or Mechanical) waves equation Given the initial shape of the string, what is the subsequent form, The displacement at point z, at the later time t,

More information

kg 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.

kg 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

Section 3: Complex numbers

Section 3: Complex numbers Essentially: Section 3: Complex numbers C (set of complex numbers) up to different notation: the same as R 2 (euclidean plane), (i) Write the real 1 instead of the first elementary unit vector e 1 = (1,

More information

MITOCW ocw feb k

MITOCW ocw feb k MITOCW ocw-18-086-13feb2006-220k INTRODUCTION: The following content is provided by MIT OpenCourseWare under a Creative Commons license. Additional information about our license and MIT OpenCourseWare

More information

Physics 505 Homework No. 12 Solutions S12-1

Physics 505 Homework No. 12 Solutions S12-1 Physics 55 Homework No. 1 s S1-1 1. 1D ionization. This problem is from the January, 7, prelims. Consider a nonrelativistic mass m particle with coordinate x in one dimension that is subject to an attractive

More information

32. Interference and Coherence

32. Interference and Coherence 32. Interference and Coherence Interference Only parallel polarizations interfere Interference of a wave with itself The Michelson Interferometer Fringes in delay Measure of temporal coherence Interference

More information

Superposition of electromagnetic waves

Superposition 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 information

Innovation and Development of Study Field. nano.tul.cz

Innovation and Development of Study Field. nano.tul.cz Innovation and Development of Study Field Nanomaterials at the Technical University of Liberec nano.tul.cz These materials have been developed within the ESF project: Innovation and development of study

More information

Differential Equations

Differential Equations Electricity and Magnetism I (P331) M. R. Shepherd October 14, 2008 Differential Equations The purpose of this note is to provide some supplementary background on differential equations. The problems discussed

More information

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:

More information

Chapter 15 Mechanical Waves

Chapter 15 Mechanical Waves Chapter 15 Mechanical Waves 1 Types of Mechanical Waves This chapter and the next are about mechanical waves waves that travel within some material called a medium. Waves play an important role in how

More information

Lecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6

Lecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6 Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength

More information

Numerical Methods I Orthogonal Polynomials

Numerical Methods I Orthogonal Polynomials Numerical Methods I Orthogonal Polynomials Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 Nov. 4th and 11th, 2010 A. Donev (Courant Institute)

More information

Second-Order Linear ODEs

Second-Order Linear ODEs Chap. 2 Second-Order Linear ODEs Sec. 2.1 Homogeneous Linear ODEs of Second Order On pp. 45-46 we extend concepts defined in Chap. 1, notably solution and homogeneous and nonhomogeneous, to second-order

More information

Sinusoids. Amplitude and Magnitude. Phase and Period. CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

Sinusoids. Amplitude and Magnitude. Phase and Period. CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Sinusoids CMPT 889: Lecture Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 6, 005 Sinusoids are

More information

Course Secretary: Christine Berber O3.095, phone x-6351,

Course Secretary: Christine Berber O3.095, phone x-6351, IMPRS: Ultrafast Source Technologies Franz X. Kärtner (Umit Demirbas) & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 Email & phone: franz.kaertner@cfel.de, 040 8998 6350 thorsten.uphues@cfel.de, 040 8998

More information

The Schroedinger equation

The Schroedinger equation The Schroedinger equation After Planck, Einstein, Bohr, de Broglie, and many others (but before Born), the time was ripe for a complete theory that could be applied to any problem involving nano-scale

More information

Instructor (Brad Osgood)

Instructor (Brad Osgood) TheFourierTransformAndItsApplications-Lecture26 Instructor (Brad Osgood): Relax, but no, no, no, the TV is on. It's time to hit the road. Time to rock and roll. We're going to now turn to our last topic

More information

Topic 4: Waves 4.3 Wave characteristics

Topic 4: Waves 4.3 Wave characteristics Guidance: Students will be expected to calculate the resultant of two waves or pulses both graphically and algebraically Methods of polarization will be restricted to the use of polarizing filters and

More information

Assignment #1 Chemistry 314 Summer 2008

Assignment #1 Chemistry 314 Summer 2008 Assignment #1 Due Thursday, July 17. Hand in for grading, including especially the graphs and tables of values for question 2. 1. This problem develops the classical treatment of the harmonic oscillator.

More information

Uniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation

Uniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume

More information

CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 26, 2005 1 Sinusoids Sinusoids

More information

1 Maxwell s Equations

1 Maxwell s Equations PHYS 280 Lecture problems outline Spring 2015 Electricity and Magnetism We previously hinted a links between electricity and magnetism, finding that one can induce electric fields by changing the flux

More information