Singularity Phenomena in Ray Propagation

in Anisotropic Media

V. L. TurovaN. D. BotkinA. A. Melikyan

Institute for Problemsin Mechanics, Moscow

Center of Advanced European Studies and Research, Bonn

PhysCon2005, St.Petersburg, August 24-26, 2005

www.caesar.de

caesar Research areas• Smart materials combined

with nanotechnologiesMultidisciplinary research center employing about 200 researchers • Ergonomics: cooperation

between humans and machines1999 – foundation of caesar;2003 – moving in the new building • Coupling of biological and

electronic systems

Motivation

Development of an acousticwave sensor for biologicaland medical applications(microbalance)

Operation principlesQuartz crystal Y-cut

1. Excitation of acoustic shear waves due topiezolectric properties of a substrate

2. Selective binding of the biomolecules from a contacting liqiud due to aptamers

3. Mass estimation through the measurementof the phase shift in the electric signal

Biomolecules

Aptamers

Methods of modelling

Direct modelling with finite elements(recources consuming because of short wave lengths)

Dispersion relations (dependence of the wave propagation velocity on the frequency) for multi-layered anisotropic structures

Description of wave fronts usingHamilton-Jacobi equations

Waves description using optimality principles1. Characteristic surfaces

Wave surface

0

Ve

n

n

wave front

Ve energy velocity

n

Ve

V

Velocity surface

0

Ve .n = |V | - phase velocityn wave vector

Slowness surface (inverse through the origin to the velocity surface)

m = n / |V |

m

n

Ve

0

The energy velocity is normal to the slowness surface at all points.

2. Fermat´s principle

Or, using parametrization

Thus, feasible rays are solutions of the Euler equation

Applicable if Ve is a well-defined function.

3. Pontryagin´s maximum principle

p – adjoint vector,

– wave surface

Applicable if the wave surface does not have swallow tails.

Complicated structure of wave surfacesSlowness surface Wave surface Example of a wave

surface for a crystalwith cubic symmetry

0

θ

θ

θ

a

c

b

a´ c´

1

23

1´

2´ 3´

0

Variation of the curvature of the slowness surface can yield multiplevalues for the energy velocity (swallow tails in the wave surface).

Description of wave propagation usingHamilton-Jacobi equations

Elasticity equations for anisotropic inhomogeneous media

WKB-approximation

ε

Eikonal equation

is equivalent to the following three equations

where

solve the eigenvalue problem

,with

which gives three phase velocities (and three polarizations) for the propagation direction n.

−Using the transformation yields H.-J. equation

If the wave surface has swallow tails, the Hamiltonian is

non-convex in the impulse variable which motivates to

use differential games instead of classical optimality

principles

Usage of differential games

The first player minimizes and the second player maximizes the timeof attaining a given terminal set M.

Value function T(x) satisfies the Hamilton-Jacobi equation

in all points x where T(x) is differentiable. In other points, this equation holds in a viscosity sense. If we find a differential game with the dynamics that provides the equality

then level sets of T yield wave frontsin the wave propagation problem.

Differential games with simple dynamics are appropriate

P and Q are symmetric about the origin convex setsHamilton-Jacobi equation

can be rewritten as

or

Comparison with the eikonal equation yields the condition on P and Q :

Application to the propagation of surface acoustic waves

Velocity contour for shearsurface acoustic wavesx3

Gold layerSiO2- guiding layer

Quartz crystal

Protein layerFluid

x1x2

Multi-layered structure

x1

x2

x1

x2

cα

N.D. Botkin et al. Numerical computation of dispersion relations in multi-layered anisotropicstructures. In: Proceedings of Nanotech, Boston, 2004

Slownesssurface

1. Approximation of the phase velocity contour

The resultChoice of QChoice of P

P

Q

2. Numerical solution to the differential game

u minimizes and v maximizes the timeof attaining the terminal set MM

Backward step-by-step computation of the attainability set on interval [0,θ] with a step ∆

V.S.Patsko, V.L.Turova„Numerical solution of two-dimensional differential games“,Preprint, Institute of Math. & Mech.,Ekaterinburg, 1995.

3. Investigation of singular linesA.A. Melikyan. Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games, Boston: Birkhauser, 1998.

Wave frontsand

singular lines

Behaviorof optimaltrajectories

Acoustic waves in anisotropic crystals

obey differential games!

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