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Geometry of Waves

Georg Esslgeorg@mle.media.mit.edu

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)