I thought we didn't have to do the lecture thoughts before. Why do we have to do them now?What exactly is the topic breakdown for the final? How many questions from each topicPicked up my guitar for the first time in a while and messed around with physics.This is very clear and easy! I love waves!I was confused on the diameters of the string question.Best homework ever. As a guitar player, I thoroughly enjoyed it. Can this lecture have guitars? Please, let's have guitars.I really didnt understand the guitar problem, please go over it in class..Can we go over the Guitar problems? WOO-HOO 7 MORE DAYS OF CLASS LEFT!!!Test scores: 78,78,100. Awwwww yeah.my mom used to work at fermilab, if I told you I grew up literally surrounded by physicist and I STILL am not good at physics, would you give me an A for the recognized lost cause I am?Why do guitar players always get so much attention? Why can't oboe players get some love every once in a while?I've been awake for over 20 hours running on 3 hours on sleep. All of this seems difficult to me.For anyone running the marathon on Saturday (and everyone else as well), come out and support Make-A-Wish Foundation on Friday at Chi Omega's Carbo Load Pasta Dinner! $5 presale $7 at the door!

Your Comments

Mechanics Lecture 24, Slide 2

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Mechanics Lecture 17, Slide 3

Combined final: Wednesday May 9 at 1:30 pm.Conflict final: Thursday May 10 at 8 am. 

A: I have checked my schedule already. I will take the combined final.

C: I have checked my schedule already. I can not take either of the above. I have already notified Paco at rdjain1@illinois.edu

B: I have checked my schedule already. I will take the conflict final.

Clicker Question

D: I have checked my schedule already. I can not take either of the above. I will notify Paco later today at rdjain1@illinois.edu

E: I have not checked my schedule but I will do so today. I will notify Paco today if I have a problem.

Mechanics Lecture 24, Slide 4

Show wave movie from last class

http://www.youtube.com/watch?v=5_sfnQDr1-o

Physics 211Lecture 24

Today’s Concepts:A) SuperpositionB) Standing Waves

Mechanics Lecture 24, Slide 5

The Wave Equation

Mechanics Lecture 24, Slide 6

Suppose a pulse in Case A described by the function y(x,t) = P(x-vt). Which of the following functions described the pulse in Case B?

CheckPoint

Mechanics Lecture 24, Slide 7

A) y(x,t) = P(x + vt)B) y(x,t) = -P(x + vt)C) y(x,t) = -P(x - vt)

)(),( vtxPtxy v

x

y

vx

y

?),( txy

Case A

Case B

A) This is the equation when the wave is traveling in the negative x direction.

B) because its going left it must be x +vt and because it is starts out below the x axis it must be negativeC) The initial amplitude is negatice, so it is -P, and the velocity is negative, so it is -vt.

Mechanics Lecture 24, Slide 8

A) y(x,t) = P(x + vt)B) y(x,t) = -P(x + vt)C) y(x,t) = -P(x - vt)

)(),( vtxPtxy vx

y

vx

y

?),( txy

Case A

Case B

Q: What happens when two waves “collide?”

A: They ADD together!

Superposition

Mechanics Lecture 24, Slide 9

Movie (super_pulse)

Movie (super_pulse2)

“what happens when two waves that aren't the same amplitude collide?”

For linear equations, if we have two separate solutions, A and B, then A + B is also a solution!

C(t) = A(t) + B(t)

Superposition

The wave equation we derived last time is linear. (It has no terms where the variables x,t are squared.)

2 2

2 2 2

1d x d x

dy v dt

CONSTRUCTIVEINTERFERENCE

DESTRUCTIVEINTERFERENCE

Mechanics Lecture 24, Slide 10

A(1t)

B(2t)

Can we predict this pattern mathematically? Of course!Just add two cosines and remember the identity:

where and

cos(Lt)

Demo

cos(Ht)

Beats

212

1 L 212

1 H

ttAtAtA HL coscos2)cos()cos( 21

Mechanics Lecture 24, Slide 11

how do we determine that the frequency is in rad/s or just /s or hertz

Standing Waves

What happens when two waves having the same frequency but moving in the opposite direction meet? Movie (super)

Mechanics Lecture 24, Slide 12

Wave 1 Wave 2

Wave 1Wave 2

Wave 1Wave 2

zero (Nodes)

Superposition

)cos()cos(2)cos()cos( tkxAtkxAtkxA

Changingamplitude

Stationary wave

How it Works

Demo

How to make it:

Mechanics Lecture 24, Slide 13

Mechanics Lecture 24, Slide 14

Standing Waves

(N = 1)

Mechanics Lecture 24, Slide 15

Why is it that the guitar string must vibrate with fundamental frequency? What would happen if the frequency were just a little off? Does this have to do with harmonics on the strings?

(N = 1)

Mechanics Lecture 24, Slide 16

Standing WavesHow do wind instruments work? Do they use air as a medium for standing waves?

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html

Suppose the strings on your guitar are 24” long as shown. The frets are the places along the neck where you can put your finger to make the wavelength shorter, and appear as horizontal white lines on the picture.

When no frets are being pushed the frequency of the highest string is 4 times higher than the frequency of the lowest string. Is it possible to play the lowest string with your finger on any of the frets shown and hear the same frequency as the highest string?

A) Yes B) No4

8

12

16

20

24

frets

CheckPoint

Mechanics Lecture 24, Slide 17

If you want to increase the frequency of a 24” string by a factor of 4 while keeping the tension the same, how long should the string be?

A) 4” B) 6” C) 8” D) 12”

Clicker Question

Mechanics Lecture 24, Slide 18

4

8

12

16

20

24

frets

4

8

12

16

20

24

frets

Is there a fret that you can push on this guitar to make the string 6” long?

A) Yes B) No

6

Clicker Question

Mechanics Lecture 24, Slide 19

The highest string vibrates with a frequency that is 4 times that of the lowest string.

Compare the speed of a wave on the high string and the low string:

A) vhigh = vlow*2

B) vhigh = vlow*4

C) vhigh = vlow*16

vlowvhigh

Clicker Question

v fMechanics Lecture 24, Slide 20

The speed of a wave on the high string is 4 times the speed on the lowest string.

If the tensions are the same, how does the mass per unit length of the strings compare:

vlowvhigh

A) mlow = mhigh*2

B) mlow = mhigh*4

C) mlow = mhigh*16

Recall

Clicker Question

2 Tv

2

T

v

Mechanics Lecture 24, Slide 21

Two cylinders have the same length and are made from the same material. If the mass of the bigger one is 16 times the mass of the smaller one, how do their diameters compare?

dbigdsmall

L

Clicker Question

A) dlow = dhigh*2

B) dlow = dhigh*4

C) dlow = dhigh*16

2M V L r

Mechanics Lecture 24, Slide 22

Mechanics Lecture 24, Slide 23

The 6 strings on a guitar all have about the same length and are stretched with about the same tension. The highest string vibrates with a frequency that is 4 times that of the lowest string.

If the strings are made of the same material, how would you expect the diameters of the lowest and highest strings to compare?

dlow

dhigh

CheckPoint

Mechanics Lecture 24, Slide 23

A) dlow = dhigh*2

B) dlow = dhigh*4

C) dlow = dhigh*16

The highest string vibrates with a frequency that is 4 times that of the lowest string. If the strings are made of the same material, how would you expect the diameters of the lowest and highest strings to compare?

dlow

dhigh

B) My guitar uses standard Ernie Ball slinkies. The low E uses a .36 and the high E uses a .10. I love how guitar helps me do physics. Really, now, the frequency is inversely proportional to the square root of the mass, which means the mass density has to be be 16 times greater for the low string, which means the radius has to be 4 times as large.

A) dlow = dhigh*2

B) dlow = dhigh*4

C) dlow = dhigh*16

Mechanics Lecture 24, Slide 24

We just found that the frequency of a string scales with its diameter if the tension is the same.

This is really what you find if you examine a guitar:

dlow

dhigh

.050” .012”

Mechanics Lecture 24, Slide 25

Their diameters are about half of the adjacent strings.

.028 .052 .022 .042 .014 .030

The lighter string in these pairs is one octave higher (2x frequency).

Different Example – 12 String Guitar

Mechanics Lecture 24, Slide 26

m = m/L

v = L/t

2 Tv

2T v

Mechanics Lecture 24, Slide 27

Homework Problem

Can we go over how changing the length affects things. Specifically the slinky problem on the homework last night.

Average speed = distance / time = 4A/P = 4Af

v f v

f

Mechanics Lecture 24, Slide 28

Homework Problem

Mechanics Lecture 24, Slide 29

Homework Problem

newnew

new

Tv

2new oldT T

1

2new old

new

new

Lt

v

2new oldL L

212

old

old

T

2 old

old

T

2 oldv