# 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)

Save this PDF as:

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

### 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

### 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

### 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

### 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

### 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]

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### = 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

### 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

### 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

### 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

### Supporting Information

Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin

### Math Methods for Polymer Physics Lecture 1: Series Representations of Functions

Math Methods for Polymer Physics ecture 1: Series Representations of Functions Series analysis is an essential tool in polymer physics and physical sciences, in general. Though other broadly speaking,

### 5.5. The Substitution Rule

INTEGRALS 5 INTEGRALS 5.5 The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function, making integration easier. INTRODUCTION Due

### MITOCW 6. Standing Waves Part I

MITOCW 6. Standing Waves Part I The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.

### Trig Identities. or (x + y)2 = x2 + 2xy + y 2. Dr. Ken W. Smith Other examples of identities are: (x + 3)2 = x2 + 6x + 9 and

Trig Identities An identity is an equation that is true for all values of the variables. Examples of identities might be obvious results like Part 4, Trigonometry Lecture 4.8a, Trig Identities and Equations

### 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,

### Electrodynamics 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

### Explanations 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

### 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

### Wave 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

### POLARIZATION OF LIGHT

POLARIZATION OF LIGHT OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results. It is not your job to understand every aspect of the theory,

### COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS

COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall

### RELATIVE MOTION ANALYSIS (Section 12.10)

RELATIVE MOTION ANALYSIS (Section 1.10) Today s Objectives: Students will be able to: a) Understand translating frames of reference. b) Use translating frames of reference to analyze relative motion. APPLICATIONS

### Formulas to remember

Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal

### A Guided Tour of the Wave Equation

A Guided Tour of the Wave Equation Background: In order to solve this problem we need to review some facts about ordinary differential equations: Some Common ODEs and their solutions: f (x) = 0 f(x) =

### 3. Maxwell's Equations and Light Waves

3. Maxwell's Equations and Light Waves Vector fields, vector derivatives and the 3D Wave equation Derivation of the wave equation from Maxwell's Equations Why light waves are transverse waves Why is the

### 2 Vector Products. 2.0 Complex Numbers. (a~e 1 + b~e 2 )(c~e 1 + d~e 2 )=(ac bd)~e 1 +(ad + bc)~e 2

Vector Products Copyright 07, Gregory G. Smith 8 September 07 Unlike for a pair of scalars, we did not define multiplication between two vectors. For almost all positive n N, the coordinate space R n cannot

### Introduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)

Introduction to the Fourier transform Computer Vision & Digital Image Processing Fourier Transform Let f(x) be a continuous function of a real variable x The Fourier transform of f(x), denoted by I {f(x)}

### Matter 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

### The Plane of Complex Numbers

The Plane of Complex Numbers In this chapter we ll introduce the complex numbers as a plane of numbers. Each complex number will be identified by a number on a real axis and a number on an imaginary axis.

### EM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves

EM Waves This Lecture More on EM waves EM spectrum Polarization From previous Lecture Displacement currents Maxwell s equations EM Waves 1 Reminders on waves Traveling waves on a string along x obey the

### Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis

Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to

### Chapter 4 Wave Equations

Chapter 4 Wave Equations Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti Instructor: Nayer Eradat Spring 2009 3/11/2009 Wave Equations 1 Wave Equation Chapter Goal: developing the

### Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0

Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can

### Diffractive Optics. Professor 송석호, Physics Department (Room #36-401) , ,

Diffractive Optics Professor 송석호, Physics Department (Room #36-401) 2220-0923, 010-4546-1923, shsong@hanyang.ac.kr Office Hours Mondays 10:00-12:00, Wednesdays 10:00-12:00 TA 윤재웅 (Ph.D. student, Room #36-415)

### REVIEW REVIEW. Quantum Field Theory II

Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,

### Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 16 The Quantum Beam Splitter

Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 16 The Quantum Beam Splitter (Refer Slide Time: 00:07) In an earlier lecture, I had

### 3.012 Structure An Introduction to X-ray Diffraction

3.012 Structure An Introduction to X-ray Diffraction This handout summarizes some topics that are important for understanding x-ray diffraction. The following references provide a thorough explanation

### 5. LIGHT MICROSCOPY Abbe s theory of imaging

5. LIGHT MICROSCOPY. We use Fourier optics to describe coherent image formation, imaging obtained by illuminating the specimen with spatially coherent light. We define resolution, contrast, and phase-sensitive

### Lecture notes 5: Diffraction

Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through

### Review of scalar field theory. Srednicki 5, 9, 10

Review of scalar field theory Srednicki 5, 9, 10 2 The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate

### Clicker Question. Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no

Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no Is the following

### Linear Operators and Fourier Transform

Linear Operators and Fourier Transform DD2423 Image Analysis and Computer Vision Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 13, 2013

### BASIC 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,

### SURFACE WAVES & DISPERSION

SEISMOLOGY Master Degree Programme in Physics - UNITS Physics of the Earth and of the Environment SURFACE WAVES & DISPERSION FABIO ROMANELLI Department of Mathematics & Geosciences University of Trieste

### Quantum Theory. Thornton and Rex, Ch. 6

Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)

### Trigonometric Identities and Equations

Chapter 4 Trigonometric Identities and Equations Trigonometric identities describe equalities between related trigonometric expressions while trigonometric equations ask us to determine the specific values

### Optical Lattices. Chapter Polarization

Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark

### Typical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields

Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy

### g(2, 1) = cos(2π) + 1 = = 9

1. Let gx, y 2x 2 cos2πy 2 + y 2. You can use the fact that Dg2, 1 [8 2]. a Find an equation for the tangent plane to the graph z gx, y at the point 2, 1. There are two key parts to this problem. The first,

### Math 425 Fall All About Zero

Math 425 Fall 2005 All About Zero These notes supplement the discussion of zeros of analytic functions presented in 2.4 of our text, pp. 127 128. Throughout: Unless stated otherwise, f is a function analytic

### Mathematics for Chemists 2 Lecture 14: Fourier analysis. Fourier series, Fourier transform, DFT/FFT

Mathematics for Chemists 2 Lecture 14: Fourier analysis Fourier series, Fourier transform, DFT/FFT Johannes Kepler University Summer semester 2012 Lecturer: David Sevilla Fourier analysis 1/25 Remembering

### 24. Advanced Topic: Laser resonators

4. Advanced Topic: Laser resonators Stability of laser resonators Ray matrix approach Gaussian beam approach g parameters Some typical resonators Criteria for steady-state laser operation 1. The amplitude

### Calculus/Physics Schedule, Second Semester. Wednesday Calculus

/ Schedule, Second Semester Note to instructors: There are two key places where the calculus and physics are very intertwined and the scheduling is difficult: GaussÕ Law and the differential equation for

### FLUX OF VECTOR FIELD INTRODUCTION

Chapter 3 GAUSS LAW ntroduction Flux of vector field Solid angle Gauss s Law Symmetry Spherical symmetry Cylindrical symmetry Plane symmetry Superposition of symmetric geometries Motion of point charges

### General 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

### Math 12 Final Exam Review 1

Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles

### 1. Given the apparatus in front of you, What are the forces acting on the paper clip?

Forces and Static Equilibrium - Worksheet 1. Given the apparatus in front of you, What are the forces acting on the paper clip? 2. Draw a free body diagram of the paper clip and plot all the forces acting

### Total Internal Reflection & Metal Mirrors

Phys 531 Lecture 7 15 September 2005 Total Internal Reflection & Metal Mirrors Last time, derived Fresnel relations Give amplitude of reflected, transmitted waves at boundary Focused on simple boundaries:

### The Wave Function. Chapter The Harmonic Wave Function

Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that

### Applied Calculus I Practice Final Exam Solution Notes

AMS 5 (Fall, 2009). Solve for x: 0 3 2x = 3 (.2) x Taking the natural log of both sides, we get Applied Calculus I Practice Final Exam Solution Notes Joe Mitchell ln 0 + 2xln 3 = ln 3 + xln.2 x(2ln 3 ln.2)

### Part 1. The simple harmonic oscillator and the wave equation

Part 1 The simple harmonic oscillator and the wave equation In the first part of the course we revisit the simple harmonic oscillator, previously discussed in di erential equations class. We use the discussion

### PHYS 3900 Homework Set #02

PHYS 3900 Homework Set #02 Part = HWP 2.0, 2.02, 2.03. Due: Mon. Jan. 22, 208, 4:00pm Part 2 = HWP 2.04, 2.05, 2.06. Due: Fri. Jan. 26, 208, 4:00pm All textbook problems assigned, unless otherwise stated,

### Chapter 5: Double-Angle and Half-Angle Identities

Haberman MTH Section II: Trigonometric Identities Chapter 5: Double-Angle and Half-Angle Identities In this chapter we will find identities that will allow us to calculate sin( ) and cos( ) if we know

### f(x 0 + h) f(x 0 ) h slope of secant line = m sec

Derivatives Using limits, we can define the slope of a tangent line to a function. When given a function f(x), and given a point P (x 0, f(x 0 )) on f, if we want to find the slope of the tangent line

### Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law

Maxwell s equations and EM waves This Lecture More on Motional EMF and Faraday s law Displacement currents Maxwell s equations EM Waves From previous Lecture Time dependent fields and Faraday s Law 1 Radar

### Chapter 06: Analytic Trigonometry

Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is oneto-one. Are any of our trigonometric

### GRATINGS and SPECTRAL RESOLUTION

Lecture Notes A La Rosa APPLIED OPTICS GRATINGS and SPECTRAL RESOLUTION 1 Calculation of the maxima of interference by the method of phasors 11 Case: Phasor addition of two waves 12 Case: Phasor addition

### Introduction to Vectors

Introduction to Vectors K. Behrend January 31, 008 Abstract An introduction to vectors in R and R 3. Lines and planes in R 3. Linear dependence. 1 Contents Introduction 3 1 Vectors 4 1.1 Plane vectors...............................

### and the radiation from source 2 has the form. The vector r points from the origin to the point P. What will the net electric field be at point P?

Physics 3 Interference and Interferometry Page 1 of 6 Interference Imagine that we have two or more waves that interact at a single point. At that point, we are concerned with the interaction of those

### Dynamics. What You Will Learn. Introduction. Relating Position, Velocity, Acceleration and Time. Examples

Dynamics What You Will Learn How to relate position, velocity, acceleration and time. How to relate force and motion. Introduction The study of dynamics is the study of motion, including the speed and

### Limits of Exponential, Logarithmic, and Trigonometric Functions

Limits of Exponential, Logarithmic, and Trigonometric Functions by CHED on January 02, 2018 lesson duration of 3 minutes under Basic Calculus generated on January 02, 2018 at 01:54 am Tags: Limits and

### PHY3128 / PHYM203 (Electronics / Instrumentation) Transmission Lines

Transmission Lines Introduction A transmission line guides energy from one place to another. Optical fibres, waveguides, telephone lines and power cables are all electromagnetic transmission lines. are

### Longitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation

Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,

### Today in Physics 218: Fresnel s equations

Today in Physics 8: Fresnel s equations Transmission and reflection with E parallel to the incidence plane The Fresnel equations Total internal reflection Polarization on reflection nterference R 08 06

### Angular Momentum, Electromagnetic Waves

Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.

### EM waves: energy, resonators. Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves

EM waves: energy, resonators Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves Simple scalar wave equation 2 nd order PDE 2 z 2 ψ (z,t)

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation

Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations

### THE WAVE EQUATION (5.1)

THE WAVE EQUATION 5.1. Solution to the wave equation in Cartesian coordinates Recall the Helmholtz equation for a scalar field U in rectangular coordinates U U r, ( r, ) r, 0, (5.1) Where is the wavenumber,

### Graphical Analysis; and Vectors

Graphical Analysis; and Vectors Graphs Drawing good pictures can be the secret to solving physics problems. It's amazing how much information you can get from a diagram. We also usually need equations

### c. Better work with components of slowness vector s (or wave + k z 2 = k 2 = (ωs) 2 = ω 2 /c 2. k=(kx,k z )

.50 Introduction to seismology /3/05 sophie michelet Today s class: ) Eikonal equation (basis of ray theory) ) Boundary conditions (Stein s book.3.0) 3) Snell s law Some remarks on what we discussed last

### B 2 P 2, which implies that g B should be

Enhanced Summary of G.P. Agrawal Nonlinear Fiber Optics (3rd ed) Chapter 9 on SBS Stimulated Brillouin scattering is a nonlinear three-wave interaction between a forward-going laser pump beam P, a forward-going

### Reference Text: The evolution of Applied harmonics analysis by Elena Prestini

Notes for July 14. Filtering in Frequency domain. Reference Text: The evolution of Applied harmonics analysis by Elena Prestini It all started with: Jean Baptist Joseph Fourier (1768-1830) Mathematician,

### AP CALCULUS AB 2007 SCORING GUIDELINES (Form B)

AP CALCULUS AB 27 SCORING GUIDELINES (Form B) Question 2 A particle moves along the x-axis so that its velocity v at time 2 t is given by vt () = sin ( t ). The graph of v is shown above for t 5 π. The

### EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply

### Lecture 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,