Physics 213 2007 Lecture summaries and lecture notes.

These are the lecture notes for Physics 213 - Waves and Optics taught in Fall 2007. This is a sophomore class taught from French and Walter Smiths book in preparation, and is a preparation for quantum mechanics in the spring semester. As this was my second time teaching this course, and the first time was my first ever course at Haverford, in this version I slowed down to a more reasonable pace. Also, geometric optics is not covered in these lectures as I shifted it to our second semester intro course.

How to read these lecture notes: Each page has 4 panels, representing a chunk of blackboard about 3 feet wide. The sequence of panels on a page is TL, TR, BL, BR. There are a variable number of pages in these lectures - sometimes I overran by up to 15 minutes which I no longer do. A 55 minute lecture is usually ten panels, 2.5 pages, less if its a lot of algebra, but you shouldn't really just talk algebra for an hour unless you like the students to get their beauty sleep in your class.

Note on errata: There are errata in these notes! I was lucky to have several students who were diligent correctors of factors of two, minus signs etc. etc. I do not stop in lecture an correct my notes, so errors I picked up in lecture or the students picked up in lecture are still there in the notes. I'd be grateful, if you find a mistake, if you let me know so I can note it here.

Acknowledgements: Thanks to Kay Warner for scanning these notes.

- Lecture 1 The simple harmonic oscillator. Hookes law and restoring forces. Amplitude, frequency and phase. Solving the equation of motion. Initial conditions. Reading: French pp 3-7
- Lecture 2 Linearity. ArcTan function. Different forms for solution of SHO eqn of motion. Addition of two sine waves of the same frequency. The pendulum. Nonlinearity and small angle approximation. Reading: French pp 19-22
- Lecture 3 Taylor expansions. Ubiquity of SHM for small displacements around a minimum. Circular motion. Picture of phase. Idea that both initial posn. and velocity required to determine point on circle. Reading: French pp 7-9
- Lecture 4 Review of complex numbers. Reading: French pp 10-16 (Note that French uses $j=\sqrt{-1}$), French pp 43-45
- Lecture 5 Use of complex notation for oscillators. Finding the argument of a complex number. Reading: French pp 41-43, French pp 45-60
- Lecture 6 Bead on a string example. Unify description by using natural frequency. Energy. Simple and non-simple pendulums.
- Lecture 7 Energy conservation in oscillators. LC Circuits. Kirchoff's laws review. Analogous and homologous DEQ's. RLC circuits and damping. Definition of $\gamma$. Equation of DHO. Timescale $1/\gamma$ as well as timescale $2\pi/\omega_0$. Reading: Wolfson and Pasachoff pp 860-865 (posted on blackboard), French pp 62-64
- Lecture 8 Guess solutions for DHO. Verify using complex notation to simplify. Amplitude decay and lower frequency. Decay of amplitude in DHO. Reading: French pp 64-66
- Lecture 9 Quality factor. Effects of large damping. Energy of decaying oscillations. Transient solutions. Reading: French pp 66-68
- Lecture 10 Driving of an LC circuit. Amplitude as a function of drive frequency. Resonance. Homogeneous and inhomogeneous DEQs. Order of DEQ. Second order DEQ need 2 adjustable constants. General solutions and particular solutions, initial conditions. Equation of Damped Driven Oscillator (DDO). Reading: French pp 78-82, {\em Forced undamped oscillations}, French pp 82-83 {\em Complex exponential method}, French pp 83-89 {\em Solving the damped driven oscillator}.
- Lecture 11 Solving the DDO. Dependence of resonance on $Q$. Sharpness of resonance peak. Lorentzian and FWHM. Estimating oscillator parameters from resonance peak. Reading French pp 89-92 {\em Effect of varying the resistive term}, French pp 92-96 {\em Transient phenomena}.
- Lecture 12 Average power dissipated per cycle. Velocity resonance. Dissipated energy per cycle and stored energy. Reading: French pp 96-101{\em Power absorbed by a damped driven oscillator}.
- Lecture 13 Power equation. Mean power dissipated per cycle. Lorentzian. Dependence of width of resonance on $Q$. Reading: French pp 89-92, pp 96-101
- Lecture 14 Driving with superposed forces of different frequencies. Beats. Coupling. CLASS DEMO. DEQ's for two coupled oscillators. Solving the DEQ's after demonstration of modes. Reading: French pp 119-124
- Lecture 15 General motion of two coupled oscillators is superposition of normal modes. Beat phenomena. Vectorial description of normal modes. Configuration space and phase space. Reading: French pp 124-129
- Lecture 16 Solving for beats solution with zero initial velocities. General normal modes procedure. Inner products and Hilbert space. Reading: French pp 129-132
- Lecture 17 Hilbert space - formal definition. General solution of two symmetric coupled oscillators. General solution of asymmetric coupled oscillators. Eigenvalue problem.
- Lecture 18 Non-zero initial velocities. Finding real and imaginary parts of complex amplitudes given initial data. Meaning of eigenvectors and the eigenvalue problem.
- Lecture 19 The normal modes procedure for asymmetric coupled oscillators - an example. Finding the frequencies and amplitudes (eigenvalues and eigenvectors) for a specific system
- Lecture 20 Initial value problems for normal modes. Zero initial velocities. Non-zero initial velocities. Finding real and imaginary parts of complex amplitudes given initial data.
- Lecture 21 Initial value problems. Beats. Driven coupled oscillators - lab demo.
- Lecture 22 Three coupled oscillators. Symmetry of matrix of couplings. Determinants: elimination, computing $3\times3$ determinants.
- Lecture 23 Examples of eigenvector/eigenvalue calculations. Applications of normal modes.
- Lecture 24 Exam Review.
- Lecture 25 Orthonormality of eigenvectors. Symmetric force matrices and Newtons third law. Unequal masses - rescaling eigenvectors by the mass. Reading: Boas, {\em Mathematical Methods in The Physical Sciences} pp 81-144, (posted on Blackboard) this is a useful reference for Matrix related math required in Physics.
- Lecture 26 From masses coupled by springs to the beaded string. Beaded string eigenvectors. Spatial dependence of displacement in beaded string eigenvectors and existence of cutoff frequency. Reading: French pp $136-141$.
- Lecture 27 Beaded string dispersion relation. Continuum limit of dispersion relation and eigenvectors. Reading: French pp$141-147$.
- Lecture 28 From beaded string to continuous string Orthonormality for eigenfunctions of continuous string.
- Lecture 29 Initial value problem for continuous string. Class demonstration (movie) of solution of recitation problem. Bases of functions and $k$ space. Gibb's phenomena.
- Lecture 30 A complete basis for periodic functions - Fourier series.
- Lecture 31 More applications of Fourier series.
- Lecture 32 Thanksgiving break
- Lecture 33 Thanksgiving break
- Lecture 34 From Fourier series to Fourier transforms. Fourier transform of a Gaussian. Introduction of the wave equation by limit of beaded string.
- Lecture 35 Solutions of the wave equation. Fourier transforms as the normal modes of the wave equation. Boundary conditions. Advanced, retarded and standing waves. Reading: French Chapter 7: ``What is a wave?'' and ``Normal modes and travelling waves'' pp 201-206 ``Progressive waves in one direction'' pp 207-209
- Lecture 36 New idea of wave speed as a characteristic speed. Idea of what is propagating in a wave - points of constant phase. Phase velocity, group velocity and signalling velocity. Idea that modulation is required for signalling. Reading: French Chapter 7 ``Wave speeds in specific media'', pp 209-212 ``Superposition'' pp 213-216, ``Wave Pulses'' pp 216-223
- Lecture 37 Beats and group velocity. Dispersion, linear and nonlinear, normal and anomalous. Phase velocity is not nesc. signalling velocity.
- Lecture 38 E/M waves. Waves at 1-D boundaries for stretched string. Boundary conditions for E/M waves
- Lecture 39 Derivation of Snells law from E/M wave boundary conditions. Total internal reflection and evanescent waves.
- Lecture 40 Overview lecture plus mysteries - instability of ``Newtonian'' atoms. Double slit experiment. Wave-particle duality. Fourier transform of posn. space wavefunction is momentum space wavefunction. Prokudin-Gorsky photographs.

Physics 213 2006 Lecture summaries and lecture notes.

These are the lecture notes for my very first class at Haverford - Fall 2006. This is a sophomore class taught from French and Walter Smiths book in preparation, and is a preparation for quantum mechanics in the spring semester.

How to read these lecture notes: Each page has 4 panels, representing a chunk of blackboard about 3 feet wide. The sequence of panels on a page is TL, TR, BL, BR. There are a variable number of pages in these lectures - sometimes I overran by up to 15 minutes which I no longer do. A 55 minute lecture is usually ten panels, 2.5 pages, less if its a lot of algebra, but you shouldn't really just talk algebra for an hour unless you like the students to get their beauty sleep in your class.

Note on errata: There are errata in these notes! I was lucky to have several students who were diligent correctors of factors of two, minus signs etc. etc. I do not stop in lecture an correct my notes, so errors I picked up in lecture or the students picked up in lecture are still there in the notes. I'd be grateful, if you find a mistake, if you let me know so I can note it here.

- Lecture 1 Mass on a spring - SHM. Position, velocity, acceleration. Newton's law for SHM. Sinusoidal solutions, angular frequency and amplitude. Addenda about solving diff eqs.
- Lecture 2 Linearity. Addition of two sinusoids of same frequency. Phase and amplitude. Pendulum. Small angle approximation. Ubiquity of SHM for small displacements around a minimum. Modified version of notes
- Lecture 3 Circular motion. Picture of phase. Idea that both initial posn. and velocity required to determine point on circle. Complex number review. ulers equation and ``real part of'' notation.
- Lecture 4 Procedure for using Newton's laws. Bead on a string example. Unify description by using natural frequency. Energy. Damping (false analogy with friction). Equation of DHO. Amplitude decay and lower frequency.
- Lecture 5 Difference between damping and friction. Equation of DHO. Amplitude decay and lower frequency. Guess solutions for DHO. Verify using complex notation to simplify.
- Lecture 6 Decay of amplitude in DHO: timescale $1/\gamma$. As well as timescale $2\pi/\omega_0$. Quality factor.
- Lecture 7 RLC Circuits. Kirchoff's laws review. Analogous and homologous DEQ's. Driving. Low frequency limit, superposition.
- Lecture 8 Steady state and transient motion. Steady state motion independent of initial conditions. Obtain amplitude and phase of steady state motion. Amplitude and phase as a function of $\omega_d$. Resonance. CLASS DEMO.
- Lecture
9 Recap resonance. Energy in DDO.

- Lecture
10
Average power dissipated per cycle. Velocity resonance. Sharpness of
resonance peak. Lorentzian and FWHM. Dissipated energy per cycle and
stored energy.

- Lecture 11 Mathematical details. Linear and nonlinear, Homogeneous and inhomogeneous DEQs. Order of DEQ. Second order DEQ need 2 adjustable constants. General solutions and particular solutions, initial conditions. Phase space.
- Lecture 12 Application of ideas of previous lecture to DDO. General soln of inhomo. DEQ. Driving with superposed forces of different frequencies. Beats.
- Lecture 13 Phase space of two uncoupled oscillators. Coupling. CLASS DEMO. Solving the DEQ's after demonstration of modes.
- Lecture 14 General motion of two coupled oscillators is superposition of normal modes. Beat phenomena. Vectorial description of normal modes. Solving for beats solution with zero initial velocities. (Confusion caused here between zero velocity slice of phase space and the configuration space, next time introduce difference between config. space and phase space explicitly.)
- Lecture 15 General normal modes procedure. Inner products and Hilbert space.
- Lecture 16 Hilbert space - formal definition. General solution of two symmetric coupled oscillators.
- Lecture 17 General solution of asymmetric coupled oscillators. Eigenvalue problem.
- Lecture 18 Non-zero initial velocities. Finding real and imaginary parts of complex amplitudes given initial data. Meaning of eigenvectors and the eigenvalue problem. Determinants. Example problem (with integers).
- Lecture 19 Politzer song. Three coupled oscillators. Class exercises. Determinants and elimination. Symmetry of matrix of couplings. Computing $3\times3$ determinants. Class visit Chris Hooley
- Lecture 20 A three by three determinant example. Other tricks for three by three determinants. Mini ppt lecture on applications of normal modes.
- Lecture 21 Initial value problems. Orthonormality of eigenvectors. Kronecker delta.
- Lecture 22 From masses coupled by springs to the beaded string
- Lecture 23 Beaded string eigenvectors and beaded string dispersion relation with Mathematica demo
- Lecture 24 Class taken by Jerry Gollub
- Lecture 25 Recap eigenvectors and dispersion relation for beaded string, spatial dependence and existence of cutoff frequency. Continuum limit of dispersion relation and eigenvectors.
- Lecture 26 Orthonormality for eigenfunctions of continuous string. Class demonstration (movie) of solution of recitation problem.
- Lecture 27 Initial value problem for continuous string and Gibbs phenomena. Movie of fundamental mode Movie of second normal mode Movie of third normal mode Movie of sum of first three Mathematica demo
- Lecture 28 Energy of continuous systems as a sum of normal mode energies.
- Lecture 29 Fourier series introduction.
- Lecture 30 Fourier series formal details.
- Lecture 31 Applications of Fourier series
- Lecture 32 From Fourier series to fourier transforms
- Lecture 33 Introduction of the wave equation by limit of beaded string. New idea of wave speed as a characteristic speed. Idea of what is propagating in a wave - points of constant phase.
- Lecture 34 Solutions of the wave equation. Advanced, retarded and standing waves. Phase velocity and signalling velocity. Idea that modulation is required for signalling.
- Lecture 35 Beats and group velocity. Dispersion, linear and nonlinear, normal and anomalous. Phase velocity is not nesc. signalling velocity.
- Lecture 36 E/M waves: Integral Maxwells eqns review. Meaning of Maxwells equations. Accelerating charges emit e/m waves. Derivation of wave equation from Maxwells equations.
- Lecture 37 Waves at 1-D boundaries for stretched string. Boundary conditions for E/M waves.
- Lecture 38 Derivation of Snells law from E/M wave boundary conditions. Total internal reflection and evanescent waves.
- Lecture 39 Ray optics from Snells law. Refractive index. Spherical reflecting surfaces. Paraxial approximation.
- Lecture 40 Spherical refracting surfaces. Virtual images. Thin lens and lensmakers formula - no time for thin lens derivation. Variation of refractive index with wavelength, prisms and chromatic aberration. Prokudin-Gorsky photographs.
- Lecture
41
Overview lecture plus mysteries - instability of ``Newtonian'' atoms.
Double slit experiment. Wave-particle duality. Fourier transform of
posn. space wavefunction is momentum space wavefunction.