Announcements 10/1/12 Prayer Exam going on, ends Thurs night No office hours for me today, sorry. (Clement still has his.) The four Taylors series that all physics majors need to know Complex numbers next time. The Colton “Complex numbers

summary” handout on class website should be helpful.

The Far Side

From warmup Extra time on?

a. partial derivatives:– What do the funny little backward sixes mean?– I don't understand partial derivations at all so

that all just confused me.– what do we do for the partial derivatives in

section 16.6?!?! – What is that symbol that looks like an upside

down e in section 16.6?b. Can we have a demo on the wave transmittance

between a heavier string to a lighter one? Other comments?

a. rate my professor you put a generous curve on the tests. Will you?

Review: k and

Reminder: what’s the difference between these:

General form of cosine wave:

…sometimes written as:

k = “wavenumber”; = “angular frequency”

cos( 5 )s x t

cos(2( 5 ))s x t

cos( ( ) )s A k x vt

cos( )s A kx t

v = /k

k = 2 = 2

v = f

Clicker question: A wave pulse traveling on a string hits the

end of the string, which is tied to a post. What happens?

a. The pulse reflects, flipped overb. The pulse reflects, not flipped over

Demos Rubber tubing

From warmup: String attached to a wall. Which of the following will decrease the time required for the the pulse to reach the wall? Mark all that apply. Moving your hand up and down more quickly but by the same amount Moving your hand up and down more slowly but by the same amount Moving your hand the same speed but farther up and down Moving your hand the same speed but a shorter distance up and down Using a heavier string of the same length under the same tension Using a lighter string of the same length under the same tension Using the same string of the same length but under more tension Using the same string of the same length but under less tension

Shive wave machine Web demo:

http://www.colorado.edu/physics/phet/simulations/stringwave/stringWave.swf

The (Linear 1D) Wave Equation

Why is it called the wave equation?a. Because traveling waves are solutions of the

equation!

2 2

2 2

s sC

t x

2 2

2 2

2

sin( ) sin( )

sin( )

cos( ) cos( )

cos( )

s sA x vt A x vt v

x tAv x vt

s sA x vt Av x vt v

x t

Av x vt

Any function that has “x-vt” will work! …or “x+vt”

cos( )s A x vt

What’s that funny symbol?

C = v2

Analysis: A section of rope

T2

T1

x x+x

1 1 2 20 cos cosxF T T 1 1

= small; cos 1

2 2 1 1sin siny y yF ma T T ma T2

2

T1

1

= small; sin tan

1 2T T

22 1

2

tan tanT y

x t

= mass/length“linear mass density”

2 1sin sin yT x a

22 1

2

sin sinT y

x t

A section of rope, cont.

What is tan2 in picture? tan2 = opp/adj = rise/run

= slope! (at x +

x)

Tv

T2

T1

x x+xT2

2

T1

1

22 1

2

tan tanT y

x t

2

2x x x

y y

x xT y

x t

2 2

2 2

T y y

x t

Demo: wave speed vs tension

Can we predict how fast wave will travel on spring (or slinky)?

We need to measure some things:a. mass of springb. length of springc. tension of spring

Do the experiment! Time the wave!

Clicker question: A wave pulse traveling on a string meets an

interface, where the medium abruptly switches to a thicker string. What happens?

a. The pulse continues on, but flipped overb. The pulse continues on, not flipped overc. The pulse reflects, flipped overd. The pulse reflects, not flipped overe. 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 transfer2 21

2P A v

What does everything stand for? Proved in book; most important thing for us now is P ~ A2

From warmup: Explain how each of the following (keeping everything else the same) would impact the energy transfer rate of a wave on a string:

a. reducing the mass density of the string– reduces P linearly, assuming that the velocity is

unchangedb. doubling the wavelength

– irrelevant (if no dispersion)c.doubling the tension

– increases v and therefore P (by sqrt(2))d.doubling the amplitude

– increases by 4

The Wave Equation: Linear

Why is it called the linear wave equation?a. Because we don’t have nonlinear terms like f2,

x2, xf, ex, etc., in the equation itself. Properties of linear differential equations:

a. If f1 is a solution, then so is C f1

b. If f1 and f2 are solutions, then so is (f1 + f2)Consider a medium with v = 3 m/s:

2 22

2 2

f fv

t x

Any function that has “xvt” will work!

cos(2( 3 ) 5.5) cos(9( 3 ) 0.3)f x t x t

It’s a perfectly acceptable wave!

Clicker question: What happens when two wave pulses on a linear

medium run into each other head on?a. They reflect off of each other and go back the way

they came.b. Part of each wave is reflected and part transmitted.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. Graphically, complex conjugate = ? Polar vs. rectangular coordinates

a. Angle notation, “A” Euler’s equation…proof that ei = cos +

isina. must be in radiansb. Where is 10ei(/6) located on complex

plane?

What is the square root of 1… 1 or -1?