Department of Physics, Chemistry and Biology

Master’s Thesis

Computer simulations of open acoustic Sinai

billiards

Lina Fälth

LiTH-IFM-EX–05/1518–SE

Department of Physics, Chemistry and BiologyLinköpings universitet, SE-581 83 Linköping, Sweden

Master’s ThesisLiTH-IFM-EX–05/1518–SE

Computer simulations of open acoustic Sinai

billiards

Lina Fälth

Adviser: Irina YakimenkoTheoretical Physics

Karl-Fredrik BerggrenTheoretical Physics

Examiner: Irina YakimenkoTheoretical Physics

Linköping, 26 October, 2005

Avdelning, Institution

Division, Department

Theoretical PhysicsDepartment of Physics, Chemistry and BiologyLinköpings universitet, SE-581 83 Linköping, Sweden

Datum

Date

2005-10-26

Spr̊ak

Language

� Svenska/Swedish

� Engelska/English

�

�

Rapporttyp

Report category

� Licentiatavhandling

� Examensarbete

� C-uppsats

� D-uppsats

� Övrig rapport

�

URL för elektronisk version

ISBN

ISRN

Serietitel och serienummer

Title of series, numberingISSN

Titel

TitleEmulering av öppna akustiska Sinai biljarder

Computer simulations of open acoustic Sinai billiards

Författare

AuthorLina Fälth

Sammanfattning

Abstract

In this work we have studied energy flow in acoustic billiards, focusing on irregularbilliards with and without current effects. The open systems were modeled withan imaginary potential as a source and drain. We have used the finite differencemethod to model the billiards. General features of the systems are reported andeffects of the measuring probe on the wave function are discussed.

Nyckelord

KeywordsAcoustic, Sinai billiard

�

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-4774

—

LiTH-IFM-EX–05/1518–SE

—

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-4774

Abstract

In this work we have studied energy flow in acoustic billiards, focusing on irregularbilliards with and without current effects. The open systems were modeled withan imaginary potential as a source and drain. We have used the finite differencemethod to model the billiards. General features of the systems are reported andeffects of the measuring probe on the wave function are discussed.

v

Acknowledgements

I would like to thank my supervisors Irina Yakimenko and Karl-Fredrik Berggrenfor giving me an interesting diploma work, and for all encouraging help they gaveme during the work. I also would like to thank Dragan Adamovic for support andvaluable discussions. I am also very grateful to Johan Larsson and Jani Hakanenfor all their computational help.

For useful comments on my report I thank my opponent Vivianne Deniz.

vii

Contents

1 Introduction 1

2 Theory 32.1 Billiards in general . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Classical chaos and billiards . . . . . . . . . . . . . . . . . . 32.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Basic acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Imaginary potential and energy flow . . . . . . . . . . . . . 6

2.3 Microwave billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Statistics in acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Distribution of eigenvalues . . . . . . . . . . . . . . . . . . . 72.4.2 Wave functions and amplitudes . . . . . . . . . . . . . . . . 8

3 Modelling 113.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Finite difference method . . . . . . . . . . . . . . . . . . . . 113.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Neumann . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Solving the matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4.1 Wave intensity . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.2 Energy flow arrows . . . . . . . . . . . . . . . . . . . . . . . 143.4.3 Energy flow densities . . . . . . . . . . . . . . . . . . . . . . 14

4 Results 154.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Closed billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Rectangular billiard . . . . . . . . . . . . . . . . . . . . . . 174.2.2 Sinai billiards . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Open billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Source and drain . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Antenna added . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Statistics extracted from our models . . . . . . . . . . . . . . . . . 254.4.1 Spacings between energy levels . . . . . . . . . . . . . . . . 25

ix

x Contents

4.4.2 Wave function statistics . . . . . . . . . . . . . . . . . . . . 26

5 Conclusions and future work 315.1 Concluding comments . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A Derivation of the energy flow 33

Bibliography 35

Chapter 1

Introduction

In the past few years, interest has grown regarding a certain type of systems, whichare called billiards. A vast amount of numerical studies of quantum mechanicaland microwave billiards [1]-[3] and also simulations for electrical resonance circuits[4] have been performed with similar results.

The history of billiard experiments goes as far back as the end of the eighteenthcentury when Ernst Florens Chladni (1756-1827) made an experiment which canbe considered as precursor to todays research. He had dust randomly distributedon glass or metal plates, which he set in vibration by means of a violin bow. Hethen studied how the dust arranged itself in characteristic figures on the plates.What he found there was the nodal lines of the standing wave which was createdin the plates, i.e. where the amplitude of the wave function equals zero. He had“made the sound visible”.

Acoustic research plays a big part in the engineering area. However, within thisarea of acoustics, experiments and simulations [5, 6] are more rare but experimentsregarding acoustic chaos were quite recently studied by Schaadt [7, 8].

The aim of this report is to examine the energy flow in acoustic, irregularsystems which includes imaginary potentials mimiking source and drain and anantenna. The antenna represents a measuring probe, and it is interesting to seehow the acoustic wave is affected when a measurement is made on it. We will alsostudy the behavior of the statistics for eigenvalues and -functions.

The disposition of this thesis begins with a chapter dealing with the theory con-cerning the acoustics and statistics used in the report. The next part of the reportis a chapter regarding the model used in the calculations, boundary conditions andvisualisation. At last but not at least we present our results.

1

2 Introduction

Chapter 2

Theory

2.1 Billiards in general

In this chapter we will present the foundations of billiards and chaos.

2.1.1 Classical chaos and billiards

This work will focus on three types of two-dimensional billiards. Theoretically abilliard is a perfect billiard table on which a ball bounces from wall to wall withoutexperiencing any friction or other energy loss. In this work we study acoustic wavesinstead of particles, but the equations are still the same. Analytical expressions forthe wave functions and eigenvalues may be derived if the billiard is, for example,rectangular. Then the solutions can be described by two independent equations,one describing the horizontal- and one describing the vertical movement.

a) c)b)

Figure 2.1. Three different types of billiards; (a) rectangular billiard (b) Sinai billiardwith broken corner (c) Sinai billiard with internal hole.

Figure 2.1 shows the different billiards we will use in this work. The rectangularbilliard is regular since we can separate the different degrees of freedom.1 This

1The movements in x- and y-direction.

3

4 Theory

billiard is mostly interesting because of the possibility to compare the numericalresults with analytical calculations. The rectangle with a piece of one corner goneand the rectangle with a circular hole inside are called Sinai billiards. The Sinaibilliards are asymmetric and hence the variables can not be seperable in theirdegrees of freedom, they are chaotic.

There is an extreme oversensitivity for initial conditions (i.c.) in chaotic sys-tems. Since, in reality, it is impossible to start experiments in entirely the sameway, different solutions will be obtained for each run. In an entirely symmectricgeometry a tiny change in the i.c. would not affect the result very much, butfor a chaotic system even small changes would create a completely different path.There are two types of chaos, i.e. if we have a system with no energy loss, we speakof deterministic chaos, for example, mecanichal systems which obey the laws ofclassical physics, such as moons and asterioids in our solar system, and in the caseof energy loss we have dissipative chaos [9], for example turbulence in gases andfluids.

Time reversal symmetry (TRS)

Time-reversed acoustic process is possible with the aid of an array of microphonesand loudspeakers. This array can re-create sound and send it back to its source asif time had been reversed in a practical sense. Any sound that goes into the trans-ducers comes back, but played in reverse. The difference between time-reversedacoustics and the simple procedure as playing a tape backward is that in time-reversed acoustics the sound is projected back exactly toward its source.

In an experiment where the time-reversal process was studied through a mediumanalogous to a chaotic pinball machine, interesting results have been obtained. Thetime-reversal process is surprisingly stable when it comes to acoustics. A parti-cle follows a well-defined trajectory and a small change in the initial velocity orposition might make the particle to miss an obstacle and then totally change itstrajectory. On the other hand a wave travels along all possible trajectories, andthat makes it much more steady because the wave amplitude results from the in-terference of all those paths. Thus, in chaotic environments, wave physics is morerobust than particle physics [10].

2.2 Acoustics

The basic concepts of acoustics are presented here. For convenience we only sum-marize the aspects needed for this work.

2.2.1 Basic acoustics

The time independent Helmholz equation describes the properties of a wave func-tion in a classical mechanical system. It may, for example, describe elastic waves insolids including vibrating strings, bars and membranes. It also appears in acoustics[11], where the time independent Helmholz equation is of the form

(∇2 + k2mn)ψmn = 0 (2.1)

2.2 Acoustics 5

where

∇2 = ∂2

∂x2+

∂2

∂y2(2.2)

is the two-dimensional Nabla operator, ψmn(x, y) is the amplitude function of themnth resonance and kmn is the associated wave number

k2mn =ω2mnv2

(2.3)

for the mnth eigenmode, v is the phase velocity of waves and ωmn the angular fre-quency for this eigenmode. In this thesis the time independent Helmholz equationis used since we are only interested in studying standing waves.

Analytical expressions

In this thesis we will compare our results with analytical results for different eigen-modes. Here we will give the analytical expressions for the wave function andfrequency of the wave in the rectangular billiard. We will not go through thederivations of these expressions, which can be found in [12]. For a gas in a 2D boxwith a width Lx and lenght Ly, the expression for the wave function ψmn(x, y) is

ψmn(x, y) = A · cos(

mπ

Lxx

)

cos

(

nπ

Lyy

)

, (2.4)

where A is a normalization constant. The expression for the nodal frequency ωmnis

ω2mnv2

=

(

mπ

Lx

)2

+

(

nπ

Ly

)2

(2.5)

where m and n are resonance modes of the system (m,n=1,2,3,...). We seek thefrequencies ωmn as

ωmn =

√

(

mπ

Lx

)

+

(

nπ

Ly

)

v. (2.6)

From equation 2.4 we may easily plot the analytical solutions. The nodal structureof three low modes are presented in figure 2.2.

Figure 2.2. Three low modes.

6 Theory

2.2.2 Imaginary potential and energy flow

In this work we want to look at the energy flow if we disturb the wave somehow,we will use an imaginary potential to simulate transmitters and an antenna. Aninteresting discussion about the conservation of the position probability densitywhen a well is present were made by Schiff [13]. The imaginary potential helpsus to model the flow, since, as in other areas of physics,2 the imaginary part ofthe potential denotes an emission or absorption of particles. This was studied byFeshbach et al. in connection with inelastic nuclear scattering as far back as thefifties [14].

Energy flow

The probability energy flow of the system is given by the Poynting vector [2]

S = Im[

ψ∗(r)∇ψ(r)]

. (2.7)

For a derivation of equation 2.7, see Appendix A.

2.3 Microwave billiards

Figure 2.3. A two-lead cavity.

Microwave cavities are the most common system describedas billiards that billiard experiments have been performedon. Experiments on microwave cavities have been performedby Barth [3], and Kim [15]. A billiard shaped as in figure 2.3was also studied recently using the finite difference methodwhich we also will use [1].3 The equation used in that workwas the Schrödinger equation which is the quantum mechan-ical counterpart to the Helmholz equation. Please, note thesimiliarities between the Helmholz and Schrödinger equa-tions. The Helmholz equation for the perpendicular electricfield E⊥, in a planar microwave resonator is [16]

[

∇2 + k2 + (nπd

)2]

E⊥ = 0. (2.8)

For frequencies υ < c2d only modes with n=0 are possible.4

So, for small d equation 2.8 reduces to

(∇2 + k2)E⊥ = 0. (2.9)

Equation 2.9 has a similar form as the Schrödinger equation for a particle in abox, with infinite walls

(

∇2 + 2mE~2

)

ψ = 0, (2.10)

2For example in quantum mechanics.3This method will be further explored in next chapter.4Corresponding to wave numbers k < π

d.

2.4 Statistics in acoustics 7

if k in equation 2.9 corresponds to√

2mE~2

in equation 2.10 and E⊥ corresponds

to ψ. Thus, in this particular case it is clear that the equations 2.9 and 2.10 areclosely related.

2.4 Statistics in acoustics

Statistics are used to study a systems properties. Objects interesting to studymight be if a system or a particular state is chaotic or if TRS is present, in thiscontext it means that the wave function is complex or real [16, 17].

2.4.1 Distribution of eigenvalues

In this chapter we will study the statistics of spacings between eigenenergies. Thisis done after solving equation 2.1 in the case of a closed billiard, i.e. the casewith no source and drain. What we do is line up all the eigenenergies, Ei, inorder of increasing energy. First we want to normalize the mean distance betweenthe energies to one, which can be done by calculating the positive and normal-ized spacings, si = (Ei+1 − Ei)/∆E, between neighbouring energies. In order tovisualize this in a histogram an array of slots is created, each slot representingintervals of spacings. We place each of the spacings in their correct slot and countthe amount of spacings in each slot. When this is done we have a histogram, P(s),which we scale to have a total probability of one.

By studying these histograms we can extract important information about oursystem. For an integrable, i.e. non-chaotic, system with TRS the eigenfrequenciesare distributed according to the Poisson distribution

P (s) = e−s (2.11)

and when the system undergo chaos effects, but still with TRS present,5 we findthe Wigner-Dyson distribution [9]

P (s) =sπ

2e−

πs2

4 . (2.12)

The Poisson distribution has its maximun at zero. This means that there existmany nearby states, i.e. states with nearly the same or the same frequency. TheWigner-Dyson distribution on the other hand shows no degeneracies at all. Thiscan be understood quite easily, by considering a wave in a square box, wherethe time-independent Helmholz equation can be separated in both directions, onerealizes that many states are similar. For a chaotic system, on the other hand,the states come more randomly, the more chaotic the system is, the less amountof degeneracies present. This is known as level repulsion, i.e. the frequency levelsrepel each other as the degree of chaos increases [9, 16].

5TRS is present because of the fact that we look at the squared absolute value of the eigen-functions.

8 Theory

2.4.2 Wave functions and amplitudes

We have two different possibilities for how the statistics for the spacings si canfollow. It all depends on whether TRS is present or not. If the system has TRS,Random Matrix Theory governs that the statistics follow the Gaussian OrthogonalEnsemble (GOE), and when not, the Gaussian Unitary Ensemble (GUE). Theform of the wave and the type of statistics linked to this are connected, and thetwo cases can be distinguished just by computing statistics for the amplitude ofthe wave in each point.6

We want to study the amplitude for the wave ψ, and in order to do so we haveto normalize ψ according to

∫

S

|ψ|2dS = 1 (2.13)

where S is the area or volume where the wave propagates. We can from thisinformation obtain a plot P (ρ), where ρ = S|ψ|2. This gives the probability tofind a certain amplitude. The plot is made by measuring the amplitudes, and thencounting how many amplitudes one has in certain intervals. According to RMTP (ρ) follows the Porter-Thomas distribution given by

P (ρ) =1√2πρ

e−ρ

2 (2.14)

for GOE statistics, and

P (ρ) = e−ρ (2.15)

which is the Rayleigh distribution for GUE statistics [9]. If we divide the complexwave function in two parts,

ψ = u+ iv (2.16)

where u is the real and v the imaginary part, treated as Gaussian fields, we canalso extract a Gaussian distribution given by

P (s) =1

σ√

2πe−

(s−µ)

2σ2 (2.17)

for u and v, σ is the standard deviation and µ the mean value of the desired curve[18].

If the Porter-Thomas distribution is found in a system,7 the real part can bemade dominant over the imaginary by multiplying the wave function by eiα. Welet 〈...〉 indicate the mean value, α can be written as

α =1

2arctan

2〈uv〉〈u2〉 − 〈v2〉 (2.18)

6Since both GOE and GUE are related to chaotic modes, we can not expect fully reliablestatistics studying them.

7Thus, for GOE statistics in the case of TRS.

2.4 Statistics in acoustics 9

giving a rotation of all points by the angle α in the complex plane [17, 19]. Thisrotation is in fact derived from the expression 〈uv〉 = 0, thus also ensuring thatthe fields are independent. Note that it is also possible to make the imaginarypart dominant by adding one more rotation with an odd multiple of π/2.

If, on the other hand, TRS is not present and we have an intermediate dis-tribution or the Rayleigh distribution, the real and imaginary parts of the wavefunction are of equal magnitude. Hence, neither of the two fields can be madedominant over the other, and the rotations only make the fields independent [19].Thus, if we have GOE statistics two phases are dominant, and if we have GUEthere exists no such dominance.

10 Theory

Chapter 3

Modelling

In this chapter we present the billiards that we modeled and we also show howthey may be implemented. MatLab was used both in the calculations and for thevisualisation.

Since analytical expressions only exist for the rectangular billiard, all the cal-culations will be performed numerically.

It takes four steps to do this. First we must transform our continuous system toa discrete one, then apply the boundary conditions to this discrete system. Whenthis is done we have achieved a matrix from which we will extract our frequencies(eigenvalues) and the corresponding wave-functions (eigenvectors). Finally theresults are visualised.

3.1 Discretization

The continuous system has to be discretized in order to make our numerical calcu-lations. The two most common methods to do this are the finite difference method(FDM) and the finite element method (FEM). The FDM is very easy to under-stand and to implement, while FEM is harder. Why one would like to use FEM isbecause it gives faster calculations, although for this work we have chosen FDMbecause of its simplicity and numerical robustness.

3.1.1 Finite difference method

As the name of the method implies, the FDM uses finite differences to produce agrid of equidistant points.1

The derivatives in FDM can be calculated in several ways but the most commonones are the five points- and the nine points approximation. In this work the fivepoint approximation is used which means that the derivative depends on the valueat a given point (fi,j) and its four nearest neighbours.

1Methods that uses non-equidistant points do exist but that will not be further discussed inthis thesis.

11

12 Modelling

a

a

13 3

3

3

22

2

2

Figure 3.1. Grid showing the discretization of the system with FDM. The point 1 isdependent on its nearest neighbours; 2. 3 are points on the boundary, thus with Neumannconditions, and a is the distance between the grid points.

The first two derivatives in the five point approximation are given by, withrespect to x

∂f(x, y)

∂x→ fi+1,j − fi−1,j

2a(3.1)

∂2f(x, y)

∂x2→ fi+1,j − 2fi,j + fi−1,j

a2(3.2)

and with respect to y∂f(x, y)

∂y→ fi,j+1 − fi,j−1

2a(3.3)

∂2f(x, y)

∂y2→ fi,j+1 − 2fi,j + fi,j−1

a2(3.4)

Thus for ∇2 we have

∇2f(x, y) → fi+1,j + fi,j+1 − 4fi,j + fi−1,j + fi,j−1a2

(3.5)

where a is the distance between the grid points.

3.2 Boundary conditions

Boundary conditions (b.c.) are necessary in order to model our systems. Assumea surface S with a boundary ∂S, and a value ψ(x,y) at the point (x,y) on S.

There exist two kinds of b.c. we may use, one is the Dirichlet b.c. which tellsthat the function should have a certain value at the boundary, most often zero. Inthis work we will use another type of b.c., the Neumann b.c., which instead tellsthat the normal derivative of the function should be zero at the boundary.

3.3 Solving the matrix 13

3.2.1 Neumann

The Neumann b.c. means that the normal derivative of the function should bezero at the boundary [11], i.e.

ψ′(xb, yb) = 0; (xb, yb) ∈ ∂S (3.6)Assume a point on the boundary at i=M, then the second derivative becomes

∇2f(x, y) → fM,j+1 − 4fM,j + fM,j−1 + 2fM−1,ja2

(3.7)

The Neumann b.c. is neutral to the wave function, i.e., not repulsive nor attractive.This property gives that the Neumann b.c. is good to use when the boundary isopen to the outside, for example for systems with source and drain. For furtherdiscussion about the Neumann b.c., see Fletcher et al. [12].

3.3 Solving the matrix

The FDM together with the Neumann b.c. gives a penta-diagonal matrix, seefigure 3.2. The matrix represents the operator ∇2, and can be of three differenttypes depending on which of of the Sinai billiards that is managed. Solving theeigenvalue problem ∇2ψmn = −k2mnψmn gives the solution where we can extractthe frequency for the system for the eigenstate mn from k2mn, and ψmn is the wavefunction for that particular eigentstate. Because of its convenience in handlingmatrices, MatLab was a natural choice to solve this eigenvalue problem.

When the system is solved, the eigenfunctions have to be ordered accordingto some criterion. Since we are interested in the frequency, we choose to sort thestates according to the increasing frequency. In some of our solutions we get acomplex part of the frequency, however, we choose to sort only according to thereal part.

5 10 15 20 25 30 35

5

10

15

20

25

30

35

−5

0

5

10

15

x 105

Figure 3.2. 6x6 point-system matrix with Neumann boundary conditions representingthe rectangular billiard.

3.4 Visualisation

There are several interesting methods of visualisation we want to examine. Thissection presents the different kinds we are interested in.

14 Modelling

3.4.1 Wave intensity

One thing we might be interested in is looking at the wave distribution, i.e. |ψ|2 ofthe system. The wave density plots show this and gives the probability of findinga certain amplitude. The color in the plots represents the wave density, with whitecolor representing the highest density.

3.4.2 Energy flow arrows

The plots showing the flow arrows help us a great deal in understanding howthe energy flows in the system, although they only show relative values, not theabsolute magnitude of the flow. The plots were done with the MatLab functionquiver, which given four matrices (x, y, dx, dy) plots the gradient of the matrices.The arrows show the direction and the relative magnitude of the flow.

Figure 3.3 presents to us where the potentials representing the transducers andthe antenna are situated.

x

y

Ly

Lx

-V’

-V’’

+V

Figure 3.3. The potentials used in this work. The positive V represents a transmitterand the negative V ′ a reciever, i.e. the energy flow in the x-direction. The negative,small V ′′ represents the effects of an antenna. The sum of the potentials equals zero, i.e.V − V ′ − V ′′ = 0.

3.4.3 Energy flow densities

In order to easily see where the highest energy flow densities are we use plots of the

absolute value of the flow in each point. To do this we calculate |S| =√

S2x + S2y .

This is then plotted the same way as the wave intensity, i.e. the lighter the color,the higher currrent density.

Chapter 4

Results

In this chapter we present the results of the calculations for the three billiards:a) a rectangular billiard; b) a Sinai billiard with broken corner; c) a Sinai billiardwith an internal hole. The first section contains general results about the modelitself.

4.1 General results

In this section we present some general results for the model itself, i.e. tests withthe rectangular billiard, for which there are analytical expressions with which wecan compare our results.

4.1.1 Resolution

In order to decide which resolution to use, we performed some calculations on thesystem for which there exists analytical expressions for the wave functions and theenergies, i.e. the rectangular billiard. In figure 4.1 we compare the frequencies ofour calculations with the analytical expressions for the frequencies. It is clear thata grid with 17x25 points is not able to give the right frequencies except for thefirst few frequencies. A grid consistent of 37x55 points gives fairly good resultsbut we have used the 59x87 point grid because it is even closer to the analyticalvalues and the percentual error is smaller.

15

16 Results

0 20 40 60 80 100 120 140 160 180 2000

1000

2000

3000

4000

5000

6000

7000

Eigenvalue number

Ang

ular

freq

uenc

y [r

ad/s

]

Analytical17x25 point grid37x55 point grid59x87 point grid

(a) Frequency.

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

14

16

Eigenvalue number

Rel

ativ

e er

ror

in fr

eque

ncy

[%]

17x25 point grid37x55 point grid59x87 point grid

(b) Error in frequency.

Figure 4.1. Plots of frequencies for a standing wave, as a function of the eigenvaluenumber for different resolutions of the grid.

4.2 Closed billiards 17

4.2 Closed billiards

First of all we want to examine our system in its normal state without any distur-bance to be able to compare the changes when we add our potentials.

4.2.1 Rectangular billiard

In order to verify the numerical model used herein, we compare three examplemodes, figure 4.2, with corresponding analytical solutions, figure 2.2, and concludethat they are in excellent agreement

(a) (b) (c)

Figure 4.2. Three low modes of the rectangular billiard, plotted as |ψ|2.

4.2.2 Sinai billiards

In all the calculations throughout this work the 96th eigenmode is used and wewill not state this in the figures. In this section we present the standing waves infigure 4.3 (a), (c) and (e) and the intensity in (b), (d) and(f).

An interesting aspect to look at is the shape of the wave function. We wantto study the nodal lines, where the amplitude is equal to zero, ψmn(x, y) = 0. Forthe rectangular shaped billiard, these lines form a perfect symmetric net. If weinstead consider a skew billiard, as the Sinai billiards, the nodal lines turn into anasymmetric complex pattern. For a complete asymmetric system, no nodal linecross another. Note that in the intensity plots one can clearly see the nodal lines,i.e. where the wave function equals zero.

18 Results

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

(a) w = 4.5413 · 103rad/s

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(b)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

(c) w = 4.7549 · 103rad/s

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(d)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

(e) w = 4.7146 · 103rad/s

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

(f)

Figure 4.3. (a),(c) and (e) shows the real part of our wave functions at the 96theigenmode, (b),(e) and (f) maps |ψ|2.

4.3 Open billiards 19

4.3 Open billiards

In order to create an energy flow through the system we open up our billiards andstudy two cases, when there is one in- and one outflow and when there also existsa well, representing an antenna. Experiments have shown that the presence ofantennas broaden and shifts the resonances somewhat, but this does not changethe distributions in chaotic systems [3].

In both cases the total in- and outflow are equal in size. For a better veiw howthe energy flow through the system see figure 3.3 where the potentials simulatingthe transducers are shown. Please note that in the first case we have not includedan antenna and thus V ′′ = 0.

4.3.1 Source and drain

In this part of our work we have added two potentials representing transducers,one positive, V , and one negative, V ′, to model a transmitter and reciever. Thisgives rise to an energy flow in our system. In this case, when there only exist oneinflow and one outflow, we have that V = −V ′.

The in- and outflow potentials are located a bit below the center of the sideof the billiard in order to reduce the possibility to be placed on a node. In figure4.4 (a) and 4.5 (a) it is nice to see how the energy flow through the billiard. Onecan clearly see the higher intensity between the potentials and how it weakensthe further along the y-axis and from the source and drain we go. Figures 4.4(b)-(c) and 4.5 (b)-(c) does not show the same symmetry although it is possible todistinguish a relation between how the energy flows and the intensity of the wavefrom figure 4.3.

20 Results

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

(a) w = 4.5413 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]y

[m]

(b) w = 4.7549 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

(c) w = 4.7146 · 103rad/s.

Figure 4.4. Current created by the in- and outflow, represented by the complex poten-tials.

4.3 Open billiards 21

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

1

2

3

4

5

6

x 10−14

(a)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.5

1

1.5

2

2.5

3x 10

−14

(b)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

x 10−14

(c)

Figure 4.5. Energy density flow of the corresponding currents in figure 4.4.

22 Results

4.3.2 Antenna added

It is interesting to see how much we disturb the system when we perform measure-ments with a probe (antenna). Experiments performed on systems with antennasexperience problems with determination of flow distributions. The probe antennagives rise to a leakage flow spoiling the statistical properties of the energy flowdistribution.

We simulate the effect of an antenna by adding the imaginary potential V ′′,and V = −(V ′+V ′′). By comparison between figure 4.6 and figure 4.7 it is clearlyvisible that the energy flow changes when the antenna is added. The effect ofthe antenna is most noticable in our rectangular system, where it seems like thecurrent is pushed away from the antenna. In the two chaotic Sinai billiards thevisible effect is smaller although it is still present.

The influence from the antenna can be made smaller by using a considerablelarger flow through the system [3]. This was also confirmed during the calculations.The potential representing the antenna is fairly large compared with the potentialsrepresenting the transducers, around 18% of the size of the positive transducer andaround 22% of the size of the negative transducer. This is because we actuallywant it to affect the energy flow.

Different locations for the antenna were tested, and depending on where theantenna was located the effects of it varied. If the antenna was placed on a nodalline no effect on the current was noted. The antenna in the current plots in thisthesis is situated where it will give an effect on the energy flow.

4.3 Open billiards 23

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

(a) w = 4.5413 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]y

[m]

(b) w = 4.7549 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

(c) w = 4.7146 · 103rad/s.

Figure 4.6. Changes in the current with an antenna added to the system.

24 Results

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

1

2

3

4

5

6

7

8

x 10−14

(a)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.5

1

1.5

2

2.5

x 10−14

(b)

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−14

(c)

Figure 4.7. Energy density flow of the corresponding currents in figure 4.6.

4.4 Statistics extracted from our models 25

4.4 Statistics extracted from our models

Acoustic resonators were studied by Schaadt [7, 8] where the same statisticalproperties as in this work were investigated. The statistics are only valid formodes in a certain class of symmetry. The procedure to isolate such a class isexplored in [4]. Since this work mostly focuses on chaos effects we have chosennot to look at the symmetric billiard, but to only examine the two Sinai billiards.Thus this chapter will only present results for our Sinai billiards.

4.4.1 Spacings between energy levels

It is probable that the correspondence betwen angular frequencies ω and the en-ergy E in the quantum mechanical case is given by equations 2.9 and 2.10. Formicrowave resonators, it is stated that k2 = ω2/c2 corresponds to the energy Ein quantum mechanics. If this is the case, we should find the eigenmodes corre-sponding to the angular frequencies, square the angular frequencies and computethe spacings, and proceed as outlined above to produce a histogram.

The Wigner-Dyson distribution was discovered when analyzing the energy lev-els for atomic cores. Later on experiments and analytical simulations confirmedthat the energy levels in all examined quantum mechanical system which are clas-sical chaotic follow the Wigner-Dyson distribution law. Thus, what we have hereis a classical system which has been quantisized and as a consequence of this theenergy levels of that system follow the Wigner-Dyson distribution law.

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

P(s

)

s

(a)

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

P(s

)

s

(b)

Figure 4.8. Statistics for the angular frequency spacings, corresponding to the spacingsbetween eigenenergies in quantum mechanics. (a) represents energies extracted fromthe first Sinai billiard and (b) is achieved from the second Sinai billiard and thus havea Wigner-Dyson distribution (equation 2.12). The histograms are generated of 4789respectively 4755 spacings.

The dotted line in figure 4.8 represents a Poisson curve, showing how the

26 Results

spacings would be distributed if the system would be symmetrical. Consideringthe shapes of the first and second Sinai billiards one can presume that the secondis of a more chaotic nature than the first. If this is so, the spacings from thefirst Sinai billiard does not necessarily have to follow the Wigner-Dyson curve asstrictly as the spacings from the second. One can actually see in figure 4.8 (a) thatthe histogram is pushed to the left, that is, the histogram is under the influence ofboth the Poisson and Wigner-Dyson distributions. By comparison with figure 4.8(b) one can see that the histogram consisting of data from the second and morechaotic Sinai billiard is mostly controlled by the Wigner-Dyson distribution.

4.4.2 Wave function statistics

The procedure to create a histogram that show the probability of finding a certainamplitude is as follows. At first a mode is specified, giving the eigenfunction forthat particular mode. The function values in each grid point are lined up, andnormalized according to equation 2.13. When this is done, one just places thenormalized function values in discrete trays. The result is yet another histogramwhich indicates how common the amplitudes of that single wave function are.

A nice way of showing the difference between states following either GOEstatistics och GUE statistics is to study the phase plots. When GOE is present,there are only two phases dominant, that means that a phase plot are dominatedby two colors only, and the difference in color corresponds to π. In our systemswe find only distributions which follow GOE statistics, see figure 4.9 and 4.10.

4.4 Statistics extracted from our models 27

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(a) w = 4.7549 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

−3

−2

−1

0

1

2

3

(b) The phase in each point.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P(ρ

)

ρ

(c) Wave function statistics. The amplitudes of the wavefunction are plotted along the x-axis and the probabilityof finding a certain amplitude is plotted along the y-axis.

Figure 4.9. Statistics regarding the first Sinai billiard. Figure (a) maps the intensity ofthe wave function. (b) Shows the phase of the wave function in each grid point. (c) Thecurve is the Porter-Thomas distribution given by equation 2.14.

28 Results

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

(a) w = 4.7146 · 103rad/s.

−1 −0.5 0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

x [m]

y [m

]

−3

−2

−1

0

1

2

3

(b) The phase in each point.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P(ρ

)

ρ

(c) Wave function statistics. The amplitudes of the wavefunction are plotted along the x-axis and the probabilityof finding a certain amplitude is plotted along the y-axis.

Figure 4.10. Statistics regarding the second Sinai billiard. Figure (a) maps the intensityof the wave function. (b) Shows the phase of the wave function in each grid point. (c)The curve is the Porter-Thomas distribution given by equation 2.14.

The statistics for the real and imaginary parts of the amplitudes follow a Gaus-sian distribution, this has been validated for different modes. Figures 4.11 and 4.12show distributions for the real and imaginary parts of the amplitudes for the twoSinai billiards, for a high mode. The plots show the distribution both before and

4.4 Statistics extracted from our models 29

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(R

e(ψ

))

Re(ψ)

(a) Real part, before the rotation. σ =0.9, and µ = −0.9.

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(I

m(ψ

))

Im(ψ)

(b) Imaginary part, before the rotation.σ = 0.9, and µ = −0.9.

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(R

e(ψ

))

Re(ψ)

(c) Real part, after the rotation. σ = 0.9,and µ = −0.9.

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P(I

m(ψ

))

Im(ψ)

(d) Imaginary part, after the rotation.σ = 0.045, and µ = 0.00001.

Figure 4.11. This figure presents statistics for the first Sinai billiard, with w = 4.7549 ·103rad/s. Figures (a)-(d) show the distributions of the real and imaginary part of ourGaussian fields, before and after the rotation by the angle α. The curves are Gaussian,given by equation 2.17. The real part of the function spans over an energy spectra 1010

times larger than the imaginary part does.

after the rotation given by equation 2.18. The histograms representing the imagi-nary part of the wave function do look equal in size to the histograms representingthe real part. This is because the plots are not scaled with respect to each other.The imaginary part is so much smaller than the real one so it would not be possibleto see any noticable change after the rotation. As the plots show, the imaginarypart becomes very small after the rotation. Although, since the real party is somuch larger than the imaginary one, the rotation is pretty much unnecessary inpractice.

30 Results

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(R

e(ψ

))

Re(ψ)

(a) Real part, before the rotation. σ =0.9, and µ = −0.9.

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(I

m(ψ

))

Im(ψ)

(b) Imaginary part, before the rotation.σ = 0.79, and µ = −0.796.

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

P(R

e(ψ

))

Re(ψ)

(c) Real part, after the rotation. σ =0.86, and µ = −0.9.

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P(I

m(ψ

))

Im(ψ)

(d) Imaginary part, after the rotation.σ = 0.045, and µ = 0.00001.

Figure 4.12. This figure presents statistics for the second Sinai billiard, with w =4.7146 · 103rad/s. Figures (a)-(d) show the distributions of the real and imaginary partof our Gaussian fields, before and after the rotation by the angle α. The curves areGaussian, given by equation 2.17. The real part of the function spans over an energyspectra 1010 times larger than the imaginary part does.

Chapter 5

Conclusions and future work

5.1 Concluding comments

The objective of this work was to examine systems representing different kindsof billiards which includes imaginary potentials mimiking source, drain and anantenna. What we noted here was that the size of the antenna had to be fairlylarge, around 20% of the source and drain potentials, in order to give a visibledisturbance on the energy flow. The tool used in these studies was based on finitedifference approximations.

The electron cavity (see figure 2.3) studied by Hakanen [1] was modelled ina similar fashion to our systems. It also had a source and drain, and the equa-tion used was the the quantum mechanical correspondence to Helmholz equation(equation 2.1), the Schrödinger equation (equation 2.10).

Statistics show that the first Sinai billiard is of less chaotic nature than thesecond one. Histograms showing the spacings between eigenenergies imply thatthe distribution for the first Sinai billiard is affected by both chaotic and non-chaotic distribution while the second Sinai billiard is mostly governed by a chaoticdistribution.

One conclusion of this chapter is that our Sinai billiards have GOE statistics.Experiments have already shown that the time-reversal process is stable and thusour result was expected.

5.2 Future work

There are still interesting work to be done using the present model. One shouldimprove the speed of the calculations so larger systems can be used. This wouldin turn allow higher frequencies to be included. This could be done by using abetter eigenvalue solver for large systems.

In order to make comparison between our acoustic results and quantum theorymore straight forward one could do the same analysis but in a system equivalentto figure 2.3. This system has already been studied using both electrical circuits

31

32 Conclusions and future work

[4] and using quantum theory [1]. It would be interesting do study this both withand without an antenna.

One last thing that would be interesting is to apply the Schrödinger equationto the billiards in this work, yet another way to be able to perform analog studiesbetween classical and quantum mechanical systems.

Appendix A

Derivation of the energy flow

The time-independent Helmholz equation can be considered as a complex ampli-tude ψ in a scalar single-frequencied wave field. Presume that the medium inwhich it propagates is isotropic. The field is expressed by the complex analyticsignal1

φ(r, t) = ψ(r)e−iωt, (A.1)

where ω is the angular frequency and r = (x, y, z). In a source-free region, thecomplex amplitude satisfies

∂2ψ

∂x2+∂2ψ

∂y2+∂2ψ

∂z2+ω2

v2ψ = 0, (A.2)

which we recognize as the three dimensional version of equation 2.1. The analyticsignal φ may represent the complex time-varying sound pressure amplitude P ,with which we can extract energetic quantities of the field. The sound potentialenergy density time averaged over one period is given by [20]

wpot =1

2κ|P |2, (A.3)

where κ denotes the compressibility. The kinetic time-averaged energy density ofthe wave is given by

wkin =1

2ρV · V ∗, (A.4)

where V is the sound velocity vector and ρ is the density of the medium. Newton’srelation

iωρV = ∇P (A.5)where ∇ = x̂ ∂

∂x+ ŷ ∂

∂y+ ẑ ∂

∂z, together with A.4 give

wkin =1

2ρω2· |∇P |2. (A.6)

1This derivation has earlier been done by Ebeling [5].

33

34 Derivation of the energy flow

Now, we seek the time-averaged sound energy flux density, S. From Morse andIngard [20] and a simple dimension analysiswe know that we can express S as PV .But P , V ∈ C so

S =1

2P ∗V +

1

2PV ∗. (A.7)

We know from equation A.5 that

V =1

iωρ∇P (A.8)

and

V∗ =

1

−iωρ∇P∗. (A.9)

So we have from equations A.7, A.8 and A.9 that

S =i

2ωρ(P∇P ∗ − P ∗∇P ). (A.10)

We can express the energy and energy flux densities in terms of the analytic signalφ or the complex amplitude ψ and their gradients. For convenience, we suppressunimportant proportionality constants and define our energy densities with theaid of the complex amplitude. The potential energy density is given by

wpot = |ψ|2, (A.11)and the kinetic energy density by

wkin = |∇ψ|2. (A.12)and thus the energy flux from A.3, A.6, A.11, and A.12 by

S = i(ψ∇ψ∗ − ψ∗∇ψ)/2. (A.13)We can simplify this equation by using ψ = a + ib and ψ∗ = a − ib, which givesthat

S = i[

(a+ ib)∇(a− ib) − (a− ib)∇(a+ ib)]

/2. (A.14)

We know that z = Re(z)+iIm(z) where z in our case equals the expression withinthe brackets in equation A.14 and the real and imaginary parts are:

Re(z) = (a∇a+ b∇b) − (a∇a+ b∇b) = 0 (A.15)

Im(z) = (−a∇b+ b∇a) + (−a∇b+ b∇a) = 2(b∇a− a∇b) (A.16)which gives that equation A.14 can be written as

S = (a∇b− b∇a) (A.17)which is the same as

S = Im[

(a− ib)∇(a+ ib)]

(A.18)

i.e. equation A.13 can be written as

S = Im[

ψ∗(r)∇ψ(r)]

. (A.19)

Bibliography

[1] J. Hakanen. Modeling of nanostructures with complex source and drain.Master’s thesis, Linköping University, 2003.

[2] A.F. Sadreev and K.-F. Berggren. Current Statistics for Wave Transmissionthrough an Open Sinai Billiard - Effects of Net Currents. Physical Review E,70(026201) ,2004.

[3] M. Barth and H.-J. Stöckmann. Current and vortex statistics in microwavebilliards. Physical Review E, 65(066208), 2002.

[4] O. Bengtsson, J. Larsson, and K.-F. Berggren. Emulation of Quantum Me-chanical Billiards from Electrical Resonance Circuits. Physical Review E.71(056206), 2005.

[5] K.J. Ebeling. Statistical Properties of Random Wave Fields. Academic press,New York, 1984. ISBN 0 12 477917 4.

[6] F.J. Fahy. Sound Intensity 2nd ed.. St Edmundsbury Press, Bury St Ed-munds, Suffolk, 1995. ISBN 0 419 19810 5.

[7] K. Schaadt The Qauntum Chaology of Acoustic Resonators. M.Sci. Thesis,University of Copenhagen, 1997.

[8] K. Schaadt Experiments on Acoustic Chaology and Statistical Elastodynam-ics. Ph.D. Thesis, University of Copenhagen, 2001.

[9] K.-F. Berggren. Kvantkaos. In Kosmos, 2001.

[10] M. Fink. Time-Reversed Acoustics. Scientific American, November 1999.

[11] G.B. Arfken and H.J. Weber. Mathematical Methods for Physicists 4th ed..Academic Press, Inc., San Diego, 1995. ISBN 0-12-059815-9.

[12] N.H. Fletcher and T.D. Rossing. The Physics of Musical Instruments 2nded.. Springer-Verlag New York, Inc., New York, 1998. ISBN 0-387-98374-0.

[13] L.I. Schiff Quantum Mechanics 3rd ed.. McGraw-Hill Kogakusha, ltd., Tokyo,1968. ISBN 68-25665.

35

36 Bibliography

[14] H. Feshbach, C.E. Porter, and V.F. Weisskopf Model for Nuclear Reactionswith Neutrons. Physical Review 65, 1954

[15] Y-H. Kim Open quantum dots modeled with microwave cavities. Ph.D.Thesis, Philipps-University Marburg, 2004.

[16] H.J. Stöckmann. Quantum Chaos: an introduction. Cambridge UniversityPress, Cambridge, 1999. ISBN 0521 59284 4.

[17] A.I. Saichev, K.-F. Berggren, and A.F. Sadreev. Distribution of nearest dis-tances between nodal points for the Berry function in two dimensions. Phys-ical Review E, 64(036222), 2001.

[18] H. Ishio, A.I. Saichev, A.F. Sadreev, and K.-F. Berggren. Wave functionstatistics for ballistic quantum transport through chaotic open billiards: Sta-tistical crossover and coexistence of regular and chaotic waves. Physical Re-view E, 64(056208), 2001.

[19] A.I. Saichev, H. Ishio, A.F. Sadreev, and K.-F. Berggren. Statistics of interiorcurrent distribution in two-dimensional open chaotic billiards. Journal ofPhysics A. 35(L87), 2002.

[20] P.M. Morse and K.U. Ingard. Theoretical Acoustics. McGraw-Hill, New York,1968.

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IntroductionTheoryBilliards in generalClassical chaos and billiards

AcousticsBasic acousticsImaginary potential and energy flow

Microwave billiardsStatistics in acousticsDistribution of eigenvaluesWave functions and amplitudes

ModellingDiscretizationFinite difference method

Boundary conditionsNeumann

Solving the matrixVisualisationWave intensityEnergy flow arrowsEnergy flow densities

ResultsGeneral resultsResolution

Closed billiardsRectangular billiardSinai billiards

Open billiardsSource and drainAntenna added

Statistics extracted from our modelsSpacings between energy levelsWave function statistics

Conclusions and future workConcluding commentsFuture work

Derivation of the energy flowBibliography