Your Comments
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 topic
Picked 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!
(80.25%)
(73.45%)
(74.22%)
Mechanics Lecture 24, Slide 2
Clicker Question
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.
B: I have checked my schedule already. I will take the conflict 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
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 17, Slide 3
Show wave movie from last class
http://www.youtube.com/watch?v=5_sfnQDr1-o
Mechanics Lecture 24, Slide 4
Physics 211
Lecture 24
Today’s Concepts:
A) Superposition
B) Standing Waves
Mechanics Lecture 24, Slide 5
The Wave Equation
Mechanics Lecture 24, Slide 6
CheckPoint
y
v y ( x, t ) = P( x - vt)
Case A
x
y
Case B
y ( x, t ) = ?
v
x
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?
A) y(x,t) = P(x + vt)
B) y(x,t) = -P(x + vt)
C) y(x,t) = -P(x - vt)
Mechanics Lecture 24, Slide 7
y
v y( x, t ) = P( x - vt)
Case A
A) y(x,t) = P(x + vt)
B) y(x,t) = -P(x + vt)
C) y(x,t) = -P(x - vt)
x
y
Case B
y ( x, t ) = ?
v
x
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 negative
C) The initial amplitude is negatice, so it is -P,
and the velocity is negative, so it is -vt.
Mechanics Lecture 24, Slide 8
Superposition
Q: What happens when two waves “collide?”
A: They ADD together!
Movie (super_pulse)
Movie (super_pulse2)
“what happens when two waves that aren't the same amplitude collide?”
Mechanics Lecture 24, Slide 9
Superposition
The wave equation we derived last time is
linear. (It has no terms where the variables
x,t are squared.)
2
2
d x 1 d x
= 2 2
2
dy
v dt
For linear equations, if we have two separate solutions, A and B, then A + B
is also a solution!
A(1t)
B(2t)
C(t) = A(t) + B(t)
DESTRUCTIVE
INTERFERENCE
CONSTRUCTIVE
INTERFERENCE
Mechanics Lecture 24, Slide 10
Beats
Can we predict this pattern mathematically? Of course!
Just add two cosines and remember the identity:
A cos(1t ) + A cos(2t ) = 2 A cos(Lt )cos(H t )
where  L =
1
(1 - 2 )
2
and  H =
1
(1 + 2 )
2
Demo
cos(Lt)
cos(Ht)
how do we determine that the frequency is in rad/s or just /s or hertz
Mechanics Lecture 24, Slide 11
Standing Waves
What happens when two waves having the same frequency but
moving in the opposite direction meet?
Movie (super)
Wave 1
Wave 2
zero (Nodes)
Superposition
Wave 2
Wave 2
Wave 1
Wave 1
Mechanics Lecture 24, Slide 12
How it Works
A cos( kx - t ) + A cos( kx + t ) = 2 A cos( kx) cos(t )
Stationary
wave
Changing
amplitude
How to make it:
Demo
Mechanics Lecture 24, Slide 13
Standing Waves
(N = 1)
Mechanics Lecture 24, Slide 14
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 15
Standing Waves
How do wind instruments work? Do they use air as a
medium for standing waves?
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html
Mechanics Lecture 24, Slide 16
CheckPoint
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?
24
20
frets
16
12
8
A) Yes
B) No
4
Mechanics Lecture 24, Slide 17
Clicker Question
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”
24
D) 12”
20
frets
16
12
8
4
Mechanics Lecture 24, Slide 18
Clicker Question
Is there a fret that you can push on this guitar to
make the string 6” long?
A) Yes
24
B) No
20
frets
16
12
8
6
4
Mechanics Lecture 24, Slide 19
Clicker Question
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
v=f
vlow
vhigh
Mechanics Lecture 24, Slide 20
Clicker Question
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:
A) mlow = mhigh*2
B) mlow = mhigh
C) mlow = mhigh
*4
Recall
v =
2
*16
T
m
T
m= 2
v
vlow
vhigh
Mechanics Lecture 24, Slide 21
Clicker Question
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?
L
A) dlow = dhigh*2
B) dlow = dhigh
*4
M = mV = m L r 2
C) dlow = dhigh*16
dbig
dsmall
Mechanics Lecture 24, Slide 22
CheckPoint
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?
A) dlow = dhigh*2
B) dlow = dhigh*4
C) dlow = dhigh*16
dlow
dhigh
Mechanics
Lecture24,
24,Slide
Slide 23
Mechanics
Lecture
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?
A) dlow = dhigh*2
B) dlow = dhigh*4
C) dlow = dhigh*16
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.
dlow
dhigh
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:
.050”
.012”
dhigh
dlow
Mechanics Lecture 24, Slide 25
Different Example – 12 String Guitar
The lighter string in these pairs is one
octave higher (2x frequency).
Their diameters are about half of the
adjacent strings.
.028 .052
.022 .042
.014 .030
Mechanics Lecture 24, Slide 26
Homework Problem
v = L/t
T
v =
m
2
T =v m
2
m = m/L
Can we go over how changing the length
affects things. Specifically the slinky
problem on the homework last night.
Mechanics Lecture 24, Slide 27
Homework Problem
Average speed = distance / time
v= f
= 4A/P = 4Af
v
=
f
Mechanics Lecture 24, Slide 28
Homework Problem
Tnew = 2Told
m new
Lnew
t=
vnew
1
= mold
2
Lnew = 2 Lold
vnew =
Tnew
mnew
2Told
Told
=
=2
1
mold
mold
2
= 2vold
Mechanics Lecture 24, Slide 29