Announcements 10/5/12




Prayer
Handout – Adding together two cosine waves
Colloquium: Did you notice “Fourier transforms”?
I just got the exams from the Testing Center, TA
& I will work on grading them today & this
weekend.
Non
Sequitur
From warmup

Extra time on?
a. how exactly can an amplitude absorb a complex
number when it itself is not complex? Is it related
to the way you lump a constant into +C after
taking an integral?

Other comments?
a. (none in particular)
Adding together two cosine waves


In short: “The amplitude and phase of the answer
were completely determined in the step where we
added the amplitudes & phases of the original two
cosine waves, as vectors.”
Don’t worry about writing each step completely.
a. Don’t write “Real( )”
b. Don’t write “e i (3x)”
HW 16.5: Solving Newton’s 2nd Law

Simple Harmonic Oscillator
(ex.: Newton 2nd Law for mass on spring)
2
d x
k
 x
2
m
dt

Guess a solution like
x(t )  Ae
it  
what it means, really: x(t )  A cos(t   )
 there’s an understood “Real{ … }”
Complex numbers & traveling waves




Traveling wave: A cos(kx – t + )
Write as:
Often:
…or
f (t )  Ae
i kx t  
i i kx t 
f (t )  Ae e
f (t )  Ae
i kx t 
– where A = “A-tilde” = a complex number
 the amplitude of which represents the amplitude of the
wave
 the phase of which represents the phase of the wave
– often the tilde is even left off
Clicker question:

Which of these are the same?
(1) A cos(kx – t)
(2) A cos(kx + t)
(3) A cos(–kx – t)
a.
b.
c.
d.

(1) and (2)
(1) and (3)
(2) and (3)
(1), (2), and (3)
Which should we use for a left-moving wave: (2) or (3)?
a. Convention: Use #3, Aei(-kx-t)
b. Reasons:
– (1) All terms will then have same e-it factor.
– (2) Whether you have kx then indicates the direction the wave
is traveling.
c. “Wavevector” k  k iˆ
From warmup
What was wrong with the first solution that was
tried in the reading today (PpP section 3.2)? What
assumption did it start with and how could Dr.
Durfee tell that that assumption was wrong?
a. it started by assuming that the wave passed
straight from one rope to the next and was
wrong because that would lead to the wave
having the same velocity on both ropes.
 How did the next guess (section 3.3) build on the
first?
a. He then guessed that a wave was partially
reflected, instead of solely transmitted

Reflection/transmission at boundaries: The setup
x=0
Region 1: light string
in-going wave
reflected wave
i ( k1x1t )
transmitted wave
AI e
i (  k1x1t )
ARe
i ( k1x1t )
f1  AI e
Region 2: heavier string
Goal: How much of wave is
transmitted and reflected?
(assume k’s and ’s are known)
i ( k1x1t )
 ARe
f1  AI cos(k1x  1t  I )  AR cos(k1x  1t  R )


AT ei ( k2 x2t )
i ( k2 x2t )
f2  AT e
f 2  AT cos(k2 x  2t  T )
Why are k and  the same for I and R? (both labeled k1 and 1)
“The Rules” (aka “boundary conditions”)
a. At boundary: f1 = f2
b. At boundary: df1/dx = df2/dx
Boundaries: The math
x=0
i ( k1x1t )
f1  AI e
B.C.1:
i ( k1x1t )
i ( k2 x2t )
 ARe
f1 x0  f 2
f2  AT e
Goal: How much of wave is
transmitted and reflected?
x0
AI ei (k1 01t )  ARei (k1 01t )  AT ei (k2 02t )
AI e
i1t
 ARe
AI  AR  AT
i1t
 AT e
and
i2t
1  2
Boundaries: The math
x=0
i ( k1xt )
f1  AI e
B.C.2:
df1
dx
x 0
i ( k1xt )
 ARe
df 2

dx
ik1 AI e
f2  AT e
Goal: How much of wave is
transmitted and reflected?
x 0
ik1 AI ei ( k1xt )  ik1 AR ei (  k1xt )
it
i ( k2 xt )
 ik1 ARe
x 0
it
k1 AI  k1 AR  k2 AT
 ik2 AT ei ( k2 xt )
 ik2 AT e
it
x 0
Boundaries: The math
x=0
Goal: How much of wave is
transmitted and reflected?
AI  AR  AT
k1 AI  k1 AR  k2 AT
2 equations, 3 unknowns??
x y  z

Like:

How do you solve?
and
3x  3 y  5z
Can’t get x, y, or z, but can get ratios!
y = -0.25 x
z = 0.75 x
Boundaries: The results
x=0
The results….
Goal: How much of wave is
transmitted and reflected?
AR k1  k2
r

AI k1  k2
“reflection coefficient”


AT
2k1
t

AI k1  k2
“transmission coefficient”
Recall v = /k, and  is the same for region 1 and region 2.
So k ~ 1/v
Can write results like this:
AR v2  v1
r

AI v1  v2
AT
2v2
t

AI v1  v2
Special Cases
x=0
The results….
AR v2  v1
r

AI v1  v2



AT
2v2
t

AI v1  v2
Do we ever have a phase shift in reflected or
transmitted waves?
a. If so, when? And what is it?
What if v2 = 0?
a. When would that occur?
What if v2 = v1?
a. When would that occur?
Reflected & Transmitted Power
x=0


1
Recall: P  m 2 A2v
2
Region 1: m and v are same
… so P ~ A2
PR
R
 r2
PI

(A = amplitude)
r,t = ratio of amplitudes
R,T = ratio of power/energy
Region 2: m and v are different… more complicated
…but energy is conserved, so easy way is:
PT
T
 1 r2
PI
Clicker question:

A wave at frequency ω traveling from a string to a rope.
At the junction, 80% of the power is reflected. How
much power would be reflected if the wave was going
from the rope to the string instead?
a. Much less than 80%
b. A little less than 80%
c. About 80%
d. More than 80%
e. It depends on the color of the rope.
AR v2  v1
r

AI v1  v2
AT
2v2
t

AI v1  v2
R  r2
T  1 R
Demo

Reflection at a boundary. Measure v1 and v2.
v2  v1
r
v1  v2
2v2
t
v1  v2
Now, on to sound!
Clicker question:

Sound waves are typically fastest in:
a. solids
b. liquids
c. gases
Sound Waves





What type of wave? What is waving?
Demo: Sound in a vacuum
Demo: tuning fork
Demo: Singing rod
Sinusoidal?
a. Demo: musical disk
vsound  343 ms



T
293K
Speed of sound
Speed of sound…
a. in gases: ~300-1200 m/s
b. in liquids: ~1000-1900 m/s
c. in solids: ~2000-6000 m/s
v = sqrt(B/r)
compare to v = sqrt(T/m)
Speed of sound in air
a. 343 m/s for air at 20C
b. Dependence on temperature (eqn in book
and also given on exam)
vsound  343 ms
T
293K