announcements 2/9/11 prayer complex numbers next time. the colton “complex numbers summary”...

Announcements 2/9/11 Prayer Complex numbers next time. The Colton

“Complex numbers summary” handout should be helpful.

k and

What’s the difference between these:

General form of cosine wave:

…sometimes written as:

k = “wavevector”; = “angular frequency”

cos( 5 )s x t

cos(2( 5 ))s x t

cos( ( ) )s A k x vt

cos( )s A kx t

= _______

Reading Quiz 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 Shive wave machine Web demo:

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

Reminder: the Wave Equation

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

equation!

2 2

2 2

f fC

t x

2 2

2 2

2

sin( ) sin( )

sin( )

cos( ) cos( )

cos( )

f fA x vt A x vt v

x tAv x vt

f fA x vt Av x vt v

x t

Av x vt

2C v Any function that has “x-vt” will work!…or “x+vt”

cos( )f A x vt

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

I measured mass of spring to be 0.28 kg.

Can we predict how fast wave will travel?

Reading Quiz 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 on Monday

Power: energy transfer

2 21

2P A v

(Proved in book; most important is P ~ A2)

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:

2 22

2 2

f fv

t x

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

1 1 1 2 2 2cos( ( ) ) cos( ( ) )f A k x vt A k x vt

Thought 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?

Reminder What is ? What is k? How do they relate to the velocity? Relationship between and T Relationship between k and Consistency check: /k = ?