Announcements 10/7/11 Prayer We’re likely not going to finish dispersion
today, so you might want to wait until after Monday before doing Lab 3 (Dispersion).
I just got the exams from the Testing Center, TA & I will work on grading them today & this weekend.
Non Sequitur
Reading Quiz Which of the following was not a major
topic of the reading assignment?a. Dispersionb. Fourier transformsc. Reflectiond. Transmission
Reflection/transmission at boundaries: The setup
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
Region 1: light string Region 2: heavier string
in-going wave transmitted wave
reflected wave
1 1( )i k x tIA e
1 1( )i k x tRA e
2 2( )i k x tTA e
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e 2 2( )
2i k x t
Tf A e
Goal: How much of wave is transmitted and reflected? (assume k’s and ’s are known)
x = 0
1 1 1 1 1cos( ) cos( )I I R Rf A k x t A k x t 2 2 2cos( )T Tf A k x t
Boundaries: The math
1 1 1 1 2 2( 0 ) ( 0 ) ( 0 )i k t i k t i k tI R TA e A e A e
2 2( )2
i k x tTf A e
x = 0
1 20 0B.C.1:
x xf f
1 1 2i t i t i tI R TA e A e A e
I R TA A A and 1 2
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
Boundaries: The math
1 1 2( ) ( ) ( )1 1 2
0 0
i k x t i k x t i k x tI R T
x xik A e ik A e ik A e
2( )2
i k x tTf A e
x = 0
1 2
0 0
B.C.2:x x
df df
dx dx
1 1 2i t i t i t
I R Tik A e ik A e ik A e
1 1 2I R Tk A k A k A
1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
Boundaries: The math
Like: and
How do you solve?
x = 0
1 1 2I R Tk A k A k A I R TA A A
Goal: How much of wave is transmitted and reflected?
x y z 3 3 5x y z
2 equations, 3 unknowns??
Can’t get x, y, or z, but can get ratios!y = -0.25 x z = 0.75 x
Boundaries: The results
Recall v = /k, and is the same for region 1 and region 2. So k ~ 1/v
Can write results like this:
x = 0
1 2
1 2
R
I
A k kr
k kA
Goal: How much of wave is transmitted and reflected?
1
1 2
2T
I
A kt
k kA
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
“reflection coefficient” “transmission coefficient”
The results….
Special Cases
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?
x = 0
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
The results….
Power
Recall: (A = amplitude)
Region 1: and v are same… so P ~ A2
Region 2: and v are different… more complicated…but energy is conserved, so easy way is:
x = 0
2 21
2P A v
2R
I
PR r
P
21T
I
PT r
P
r,t = ratio of amplitudesR,T = ratio of power/energy
Quick Writing We saw that
A1cos(kx + 1) + A2cos(kx + 2) gives you a cosine wave with the same k, and hence wavelength.
If you add a third, fourth, fifth, etc., such cosine wave, you still get a simple cosine wave. See
How can you then add together cosine waves to get more complicated shapes? Or can you?
Wave packets, cont.
Results:a. To localize a wave in space, you need lots of spatial
frequencies (k values)b. To remove neighboring localized waves (i.e. to
make it non-periodic), you need those frequencies to spaced close to each other. (infinitely close, really)
Dispersion A dispersive medium: velocity is different for
different frequenciesa. Any real-world examples?
Why do we care? a. Real waves are often not shaped like sine
waves.– Non sine-wave shapes are made up of
combinations of sine waves at different frequencies.
b. Real waves are not infinite in space or in time.– Finite waves are also made up of combinations
of sine waves at different frequencies.Focus on (b) for now… (a) is the main topic of the “Fourier transform” lectures