a llega to 425736
TRANSCRIPT
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Geometry of Waves
Georg [email protected]
Waveguides Derived
Wave-equation:
General form of solution (dAlembert):
022
22
21 !"
#
#
#
#
t
y
cx
y
)()(),( 21 ctxfctxftxy "$$!
(Smith, Cook, et al, 83+)
Standing Waves viasums of travelingwaves
Standing and Traveling Waves
Standing Waves viasums of travelingwaves
Standing and Traveling Waves
Bessel Function
J0(k|r|) = 0 are frequencies
Where is my analytic formula?
0 2 4 6 8 10 12 14 160.5
0
0.5
1
x
y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
0
0.5
1
x
y
Bessel Function
J0(k|r|) = 0 are frequencies
Where is my analytic formula? !After 120 years of acoustics, the zeros or frequencies
are still not understood. (Zelditch03)
0 2 4 6 8 10 12 14 160.5
0
0.5
1
x
y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
0
0.5
1
x
y
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Isolated Membrane Oscillations
(0,1) (0,2) (0,3)
(1,1) (1,2) (2,1)
How Big A Mistake?
Discovered that theerror even for lowfrequencies is small
(mth Bessel, nth zero)0.1%4
0.1%3
0.3%2
1.0%11
0.1%4
0.2%3
0.4%2
2.0%10
errornm
(Keller & Rubinov60)
Schlieren Visualization ofLiquid Tank
(Chinnery, Humphrey and Beckett97)
Topology of a Square Drum
Glue all reflectingedges together
Discontinuous lineswith difficultconnectivity book-keeping
Straight lines and
immediateconnectivity
Homeomorphism: Cartesian toCylindrical Coordinates
Paths in a Circular Room
Resonance is a path in a room that returnsto its starting point
(Keller & Rubinov60)
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Resonant Paths
2 :1= 1 3: 1= 1. 566 2 4 :2= 1. 87 35 4: 1= 2.2 508 5 :2= 2. 389 1
6 :3 =2 .7 47 1 5 :1 =2 .9 74 1 6 :2 =3 .0 05 9 7 :3 =3 .2 40 3 8 :4 =3 .6 20 6
7 :2 =3 .6 75 7 6 :1 =3 .7 13 2 8 :3 =3 .8 16 7 9 :4 =4 .1 01 4 8 :2 =4 .3 75 2
9 :3 =4 .4 45 7 7 :1 =4 .4 60 1 1 0: 5= 4. 49 41 1 0: 4= 4. 65 17 9 :2 =5 .0 92 7
1 0: 3= 5. 10 92 8 :1 =5 .2 11 1 1 0: 2= 5. 82 17 9 :1 =5 .9 64 5 1 0: 1= 6. 71 94
Unreachable Spots in CircularRooms
2 :1= 1 3: 1= 1. 56 62 4 :2= 1. 873 5 4 :1 =2 .25 08 5: 2= 2.3 891
6 :3 =2 .7 47 1 5 :1 =2 .9 74 1 6 :2 =3 .0 05 9 7 :3 =3 .2 40 3 8 :4 =3 .6 20 6
7 :2 =3 .6 75 7 6 :1 =3 .7 13 2 8 :3 =3 .8 16 7 9 :4 =4 .1 01 4 8 :2 =4 .3 75 2
9 :3 =4 .4 45 7 7 :1 =4 .4 60 1 1 0: 5= 4. 49 41 1 0: 4= 4. 65 17 9 :2 =5 .0 92 7
1 0: 3= 5. 10 92 8 :1 =5 .2 11 1 1 0: 2= 5. 82 17 9 :1 =5 .9 64 5 1 0: 1= 6. 71 94
Gouy Phase
Phase shift throughfocal point
But not for wave-lengths longer thanfocal length!
Along the axis of symmetry,at points where the intensityis zero, the phase isdiscontinuous.
Short wavelengths, itspi/2. Very largewavelengths it vanishes.Nobody understands thein-between.
(Bergervoet)
(Gbur et al. 02)
What a catastrophe: Caustic
Morning CoffeeCaustic
Cardioid-likeshape
Focal regions
of reflectedwave rays
(Picture courtesy of Erin Panttaja)
What a catastrophe: Caustic
Morning CoffeeCaustic
Cardioid-likeshape
Focal regionsof reflectedwave rays
(Picture courtesy of Erin Panttaja)
Construction
Domain
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Construction
Domain
Interaction Point
Construction
Domain
Interaction Point
Pencil of Rays
Construction
Domain
Interaction Point
Pencil of Rays
Ray Angles
Construction
Domain
Interaction Point
Pencil of Rays
Ray Angles
Segment
Distance
Holditchs Caustic (First Order) Cusp (Semi-cubical)
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Lifted Cusp Huygens & Malus
Final frontas sum of circularexpansions
Front normal to ray
Maintain normal through
reflections
Construction
Domain
Interaction Point
Pencil of Rays
Ray Angles
Segment
Distance
Front Set
Construction
Domain
Interaction Point
Pencil of Rays
Ray Angles
Segment
Distance
Front Set
Waves and Rays Pearcey Diffraction
(Berry)
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Lineland
The Topology
Traveling on the Line is a Loop
Lift double lineto loop
Lift reflection statesto double-loop
Double Loops indicate DynamicalProperties
Dirichlet Neumann Mixed
Integration properties Frequency properties
Interactions in Lineland
Interaction Modeling
A Notion of Coincidence Excitation Points in 1-D
Center Excitation Far Off-Center Excitation
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1-D Topology Examples
Strings & Tubes
Marimba Bars
" Same topology
" Different propagation Wine Glasses
Tibetan Singing Bowls
" Similar topology
" Same dynamics
Circular cylinder = string
Flatland
Some Membranes andPlates are
Nice Roomsin 2-D
(but less nice than in1-D)
(Abbott1884)
2-D Outline
Background" Numerical Efficiency
Why we want to walk on lines
The topology" How we walk on lines in 2-D
" On Drums, round and square Why 3-D can be nicer than 2-D
Why 2-D is tricky" How we interact in 2-D
Flatland
Background
Rooms in Flatland can beexpensive
Mesh Methods: O(n2)
Can we have a Flat room thatscheap?
The cost of walking on a line isindependent of its size!
Are there ways to walk a flat room vialines?
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Rays and Wavefronts
Short Wavelengthslocalize well
Ray Assumption Short-waveAsymptotics
Hamiltonian Room
Well run with itand see
RayRay
Waterfronts
Flatland
The Topology
Rectangular Flatland
On Square Drums
Covering Space of a Square Drum
Single Ray Ray Family
Path-connected orbit on a Torus Path-connected orbit on a Torus
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Modes and Orbits on Tori
Winding around torus for closed orbits: length of orbit
Index counting the number of special points(boundary reflections): phase changes there.
-> Resonance (Chazarain74, Duistermaat-Guillemin75)
Circular Flatland
On Circular Drums
Drum!
Circular Membrane
Uniform (or non-uniform) thickness
Interacting in Flatland
Interaction Modeling
Similar paths on torus
Torus does not itself provide reference
Mirror symmetric
Rotationally symmetric
Center Excitation
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Center Excitation Off-Center Excitation (1)
Off-Center Excitation (1) Off-Center Excitation (2)
Off-Center Excitation (2)Stability of Coincidence Knotsunder Perturbation
Stable! Unstable!
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Demo
Excitation on different points of theTorus
Open Problems
Relationship of topological spectrumand conventional spectrum
What is the proper treatment ofexcitations on caustic? (unknown)
Can asymptotic assumption beremoved? (hard)
Domain shape (guitar, violin top plates,) (Kac 66: Can we hear the shape ofa drum? hard!)
The end (for now)
Questions?
Nice Room in Flatland
At last, to complete a series of minor indiscretions, at ameeting of our Local Speculative Society held at thepalace of the Prefect himself, - some extremely sillyperson having read an elaborate paper exhibiting theprecise reasons why Providence has limited the numberof Dimensions to Two.
- Edwin A. Abbott,
Flatland: A Romance of Many Dimensions
Burning Regions in a CircularRoom
Caustics is formed by ray family reflecting atone angle
If you know the Resonance,correct your stride
You may have to skip a step to get the
Resonance
(Essl&Cook02-04)
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Why bother?
Actual science to behad
Faster and more
furious
Talks with prettypictures
(And cool new math)
(Robinson)
Vibration of a Rectangular Drum
Frequenciesfunction ofthe square-root ofdimensions
Shapesinusoidal
(2,1)
(1,1)
(1,2)
(2,2)
Chladni Figures - Square Drum
Sand on membraneand proper fingerplacement
-> Chladni figures
(Chladni, Heller)
What to do?
We lie, cheat, steal.
What to do?
We lie, cheat, steal.
Triple-A
"Adjust
"Approximate
"Asymptotics
Expansions of Bessel function known
Expansions of Short-Wave Asymptotics known
Same!
High Mode Troubles
High modes dense
High modes distribution look random
Berry-Tabor conjecture: Use Poissontail! (Like Schroeder does for room-acoustics)
Alternative: Low-resolution mesh(Banded Waveguide Mesh, Serafin &Smith 01)
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Interaction Points (CoveringSpace)
Need reference toboundary (how closeto reflection)
Load at four mirrorsymmetric points(string we had two)
Toroidal Topologies
Introduction:
Rectangular Membrane" Ray methods & Topology
Challenge:
Circular Membranes" Bessel, Asymptotics and other catastrophes
Application:
Interaction Modeling
Nice Rooms in Flatland
I call our world Flatland,
not because we call it so,
but to make its nature clearer to you,
my happy readers,
who are privileged to live in Space.
- Edwin A. Abbott,Flatland: A Romance of Many Dimensions, 1884
A Rough Route Through Flatland
Whats so cool about studying flatrooms?
Nice Rooms in 1-D" The lines and circles of strings, tubes, bars,
shells and bells
Nice Rooms in 2-D" Square, round and on doughnuts (2-D)
What makes a nice room?
Symmetric
Simple
Nice sounding
Efficiently computable
Whats so cool about studying flatrooms?
Interactive Applications:" Music Performance
" Games
" Motion Pictures (Especially CG)
Requirements:" Interactive Rates (44100 iterations/second)
" Commodity Hardware
" Proper Numerical Behavior
" Ease of Use for Nonexperts
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Publications
Banded Waveguides
" Solid Bars: Essl & Cook99, Essl & Cool00
"Glasses & Bowls: Essl & Cool02, Serafin etal02
" Membranes: Essl02, Kapur et al02, Essl etal04
" Interactions: Essl04 submitted
Circular Rooms in Flatland
ButIts harder to live in a round room in
flatland
Huygens in Flatland
Initial disturbance
Final frontas sum of circular
expansions
Whats so hard about it?
Huygens Principle
" Proven to not hold in 1-D, 2-D (us!) and2*n-D (all even dimensional rooms)
Only odd dimensional rooms of 3 or
more are generically nice!
(Veselov02)
Whats so hard about it?
Restful places in a round room are the
zeros of the
Bessel function
"Special means rich and difficult in certainsituations, unfortunately this is one !
(Watson22)
Many paths sound the same
2 :1= 1 3: 1= 1. 56 62 4 :2= 1. 873 5 4 :1 =2 .25 08 5: 2= 2.3 891
6 :3 =2 .7 47 1 5 :1 =2 .9 74 1 6 :2 =3 .0 05 9 7 :3 =3 .2 40 3 8 :4 =3 .6 20 6
7 :2 =3 .6 75 7 6 :1 =3 .7 13 2 8 :3 =3 .8 16 7 9 :4 =4 .1 01 4 8 :2 =4 .3 75 2
9 :3 =4 .4 45 7 7 :1 =4 .4 60 1 1 0: 5= 4. 49 41 1 0: 4= 4. 65 17 9 :2 =5 .0 92 7
1 0: 3= 5. 10 92 8 :1 =5 .2 11 1 1 0: 2= 5. 82 17 9 :1 =5 .9 64 5 1 0: 1= 6. 71 94
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Covering of Ray Families
2 :1= 1 3: 1= 1. 566 2 4 :2= 1. 87 35 4: 1= 2.2 508 5 :2= 2. 389 1
6 :3 =2 .7 47 1 5 :1 =2 .9 74 1 6 :2 =3 .0 05 9 7 :3 =3 .2 40 3 8 :4 =3 .6 20 6
7 :2 =3 .6 75 7 6 :1 =3 .7 13 2 8 :3 =3 .8 16 7 9 :4 =4 .1 01 4 8 :2 =4 .3 75 2
9 :3 =4 .4 45 7 7 :1 =4 .4 60 1 1 0: 5= 4. 49 41 1 0: 4= 4. 65 17 9 :2 =5 .0 92 7
1 0: 3= 5. 10 92 8 :1 =5 .2 11 1 1 0: 2= 5. 82 17 9 :1 =5 .9 64 5 1 0: 1= 6. 71 94
The end (for now)