Announcements 10/3/11 Prayer The four Taylors series that all physics majors need to know Complex numbers next time. The Colton “Complex numbers summary” handout should be helpful.

k and  Reminder: what’s the difference between these: General form of cosine wave: …sometimes written as: k = “wavevector”;  = “angular frequency” v =  /k k = 2  = 2  v = f

Reading Quiz A wave pulse traveling on a string hits the end of the string, which is tied to a post. What happens? a. a.The pulse reflects, flipped over b. b.The pulse reflects, not flipped over

Demos Shive wave machine Rubber tubing Web demo:

Reminder: the Wave Equation Why is it called the wave equation? a. a.Because traveling waves are solutions of the equation! Any function that has “x-vt” will work! …or “x+vt”

Analysis: A section of rope T2T2 T1T1 x x+  x 11 11  = small; cos   1 T2T2 22 T1T1 11  = small; sin   tan   = mass/length “linear mass density”

A section of rope, cont. What is tan  2 in picture?  tan  2 = opp/adj = rise/run = slope! (at x +  x) T2T2 T1T1 x x+  x T2T2 22 T1T1 11

Demo: wave speed vs tension Can we predict how fast wave will travel on spring (or slinky)? We need to measure some things: a. a.mass of spring b. b.length of spring c. c.tension of spring Do the experiment! Time the wave!

Reading Quiz A wave pulse traveling on a string meets an interface, where the medium abruptly switches to a thicker string. What happens? a. a.The pulse continues on, but flipped over b. b.The pulse continues on, not flipped over c. c.The pulse reflects, flipped over d. d.The pulse reflects, not flipped over e. e.The pulse partially reflects and partially transmits Advertisement: We’ll figure out the equations for reflection and transmission in the class after next

Power: energy transfer What does everything stand for? Proved in book; most important thing is P ~ A 2

The Wave Equation: Linear Why is it called the linear wave equation? a. a.Because we don’t have nonlinear terms like f 2, x 2, x  f, e x, etc., in the equation itself. Properties of linear differential equations: a. a.If f 1 is a solution, then so is C  f 1 b. b.If f 1 and f 2 are solutions, then so is (f 1 + f 2 ) Consider: Any function that has “x  vt” will work!

Thought Question What happens when two wave pulses on a linear medium run into each other head on? a. a.They reflect off of each other and go back the way they came. b. b.Part of each wave is reflected and part transmitted. c. c.They pass right through each other. Demo: Shive wave machine interference Web demo again

Complex Numbers – A Summary What is “i”? What is “-i”? The complex plane Complex conjugate a. a.Graphically, complex conjugate = ? Polar vs. rectangular coordinates a. a.Angle notation, “A  ” Euler’s equation…proof that e i  = cos  + isin  a. a.  must be in radians b. b.Where is 10e i(  /6) located on complex plane? What is the square root of 1… 1 or -1?