Wave Phenomena Physics 15c. Masahiro Morii

Transcription

1 Wave Phenomena Physics 15c Masahiro Morii

2 Teaching Staff! Masahiro Morii gives the lectures.! Tuesday and 1:00 :30.! Thomas Hayes supervises the lab.! 3 hours/week. Date & time to be decided.! Antonio Copete leads the sections.! sections. 1 hour/week. Date & time to be decided.! Please fill out the student survey! Carol Davis is the course assistant

3 Prerequisite Courses! Physics 15b (or 153)! Introductory Electromagnetism! If you have not finished 15b/153, you need to get a written permission from the Head Tutor (Margaret Law)! Mathematics 1b! Linear Algebra and Differential Equations! This can be taken concurrently

4 Textbooks! Introduction to Wave Phenomena (Hirose/Lonngren)! Required.! Simple and practical textbook.! Good illustrations.! The Physics of Waves (Georgi)! Suggested reading.! Deeper understanding of physics behind waves.! Slightly more advanced, and elegant, math.

5 Homework! Problem sets are distributed on Thursdays.! You will get the first one next Thursday.! Work together in groups.! Groups will be assigned according to the Survey.! Each of you must turn in your own report, though.! Reports due Thursday the next week.! I ll collect them at the lecture! Graded reports will be returned on the next Tuesday

6 Exams and Paper! Mid-term & final exams! Mid-term (1 hour) on March 7! Final (3 hours) during the exam period! A term paper between the two exams! You choose the subject (some kind of wave phenomenon)! Decide on your subject with me by March 5! Write 5 10 pages and analyze the phenomenon! There will be examples! Due date: April 15

7 Grades! Weighted average of:! Homework 30%! Mid-term exam 15%! Final exam 30%! Term paper 10%! Laboratory 15%

8 Getting Best Out of Us! Come and talk to us!! Ask questions in the lecture, lab and sections! Come to our office hours! Tell us when something in the class is not working for you Morii Hayes Copete Davis Office Lyman 39 Jefferson 5D Jefferson 66 Lyman 37 Phone

9 Waves What We Study in This Course

10 Common Waves! There are waves everywhere.! Sea waves! Sound! Earthquakes! Light! Radio waves! Microwave! Human waves(?)

11 Features of Waves! Oscillation at each space point.! Something ( medium ) is moving back and force.! Air, water, earth, electromagnetic field, people! Propagation of oscillation.! Motion of one point causes the next point to move.! How does oscillation propagate over distance?! What determines the propagation speed?! We study the general properties of waves focusing on the common underlying physics.

12 Waves and Modern Physics! Modern (= 0 th century) Physics has two pillars:! Relativity was inspired by the absoluteness of the speed of light = electromagnetic waves.! Quantum Mechanics was inspired by the wave-like and particle-like behaviors of light.! Everything is described by wave functions.! Relativistic QM is a theory of generalized waves.! Solid understanding of waves is essential for studying the advanced physics.

13 Goal of This Course! Understand basic nature of wave phenomena.! Intuitive picture of how waves work! How things oscillate. How the oscillation propagates! How do waves transmit energy?! Why are waves so ubiquitous?! Foundation for more advanced subjects! Familiarity with wave equations! Cover a few cool stuff related to waves! Esp. electromagnetic waves

14 Math?! We need, for example:! Taylor expansion! Fourier expansion, Fourier transformation! Complex numbers and functions! Differential equations! We use math to simplify the problems.! Make physics easier to see through.! Use experimental physicists approach.! Solve equations by guesswork and physical intuition.

15 Plan of Attack! Simple harmonic oscillator! Prepare the mathematical tools! Connected (coupled) oscillators! Oscillation of continuous media = waves! Analyze waves on springs, strings, and sound waves! Wave propagation, reflection, standing waves! Electromagnetic waves! Interference, phase/group velocities, dispersion! Optics, etc.

16 Harmonic Oscillators Simple harmonic oscillation

17 Simple Harmonic Oscillators C current L Already familiar with them, aren t we?

18 Spring-Mass System!Mass m is placed on a friction-free floor.!spring pulls/pushes the mass m with force (Hooke s law).!newton s F = law: kx m F d x F = ma = m dt F x m d x dt = kx x

19 Equation of Motion! The equation of motion for the spring-mass system: m d x = kx! We must solve this differential equation. dt! We know that the solution will look like a sine wave.! Try x= x cosωt 0 d m ( x 0cos ωt) = kx0cosωt dt k ω = mx ω cosωt = kx cosωt m 0 0

20 A (Not So General) Solution! We found a solution: cos, where k x= x0 ωt ω = m! Let s remind ourselves how this solution looks like:! How the position and the velocity change with time.! What is the frequency/period of the oscillation.! How the energy is (or is not) conserved.! Then come back and think about the general solution

21 Position and Velocity x= x0 cosωt dx v= = x0ω sinωt dt x 0 x = x(t) ωt! Oscillation repeats itself at ωt = π.! Position and velocity are off-phase by 90 degrees. x 0 x 0 ω x 0 ω v = v(t) ωt

22 Frequency and Period! ω in cosωt is the natural angular frequency.! How much the phase of the cosine advances per unit time.! Unit is [radians/sec].! The period T [sec/cycle] is given by π m π = ωt T = = π ω k ω =! The frequency ν (Greek nu) [cycle/sec] is given by k m 1 ω 1 k ν = T = π = π m a.k.a. Hertz

23 Energy! Spring stores energy when stretched/compressed: 1 1 ES = kx = kx0 cos ωt! Moving mass has kinetic energy: 1 1 EK = mv = mx0ω sin ωt 1 = kx0 sin ωt Remember ω = k/m! Therefore 1 ES + EK = kx0 = constant.

24 Energy Tossing 1 ES = kx0 cos ωt 1 EK = kx0 sin ωt 1 kx 0 E S! Energy moves between the spring and the mass, keeping the total constant. 1 kx 1 kx 0 0 E S E K E K ωt

25 General Solution! We know both cosωt and sinωt are solutions! The general solution is xt () = acosωt+ bsinωt for arbitrary values of a and b! How do we know this solution is complete?! Linearity! Mathematical reasoning! Physical reasoning

26 Linearity d x t () dt! The equation m = kx() t is linear! If you multiply a solution by a constant, it still works! If you add two solutions, the sum works, too! Since cosωt and sinωt are solutions, any linear combination acosωt + bsinωt is also a solution! What if we had one more solution?

27 Linearity! Suppose there is another solution x(t) = ξ(t) that is not a part of the general solution! That is, ξ(t) cannot be expressed by acosωt + bsinωt! Because of the linearity, the general solution becomes: xt () = acosωt+ bsin ωt+ cξ() t i.e., it has 3 free parameters instead of! This is not allowed by two reasons

28 Mathematical Reason! The equation of motion m d x t () dt = kx() t is a nd -order differential equation! Its solution must have free parameters This was too easy

29 Physical Reason! The motion of a harmonic oscillator must be uniquely determined by the initial conditions:! Location x = x 0 at t = 0! Velocity v = v 0 at t = 0! x 0 and v 0 determines what happens to the actual oscillator! From the solution x(t) = acosωt + bsinωt, d x0 = x(0) = a, v0 = x(0) = bω dt v0 xt () = x0 cosωt+ sinωt ω This must be the only possible solution

30 Summary! Analyzed a simple harmonic oscillator.! The equation of motion:! The general solution: xt () = acosωt+ bsinωt! Studied the solution.! Frequency and period.! Energy tossing.! Completeness of the solution. d x t! Will continue with more general questions. m! Expect a bit of math next time () dt ω = = kx() t k m