Closed complex rays in scattering from elastic voids

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1 Cosed compex rays in scattering from eastic voids N. Søndergaard, Predrag Cvitanović and Andreas Wirzba Division of Mathematica Physics, LTH, Lunds Universitet, Sweden Center for Noninear Science, Schoo of Physics, Georgia Institute of Technoogy, Atanta, USA Institut für Kernphysik, Jüich, Germany Abstract. The scattering determinant for the scattering of waves from severa obstaces is considered in the case of eastic soids with voids. The scattering determinant dispays contributions from cosed ray spitting orbits. A discussion of the weights of such orbits is presented. Keywords: semicassics, zeta function, scattering determinant, eastodynamics PACS: Sq, Mt, Cd, d 1. INTRODUCTION Studies of hemhotz scatterers have shown effects of trapped periodic orbits [1, 2]. We sha discuss the infuence of cosed trapped rays on scattering determinants in a medium with severa poarizations each with their own veocity. The exampe treated wi be the case of eastic wave propagation in a soid fied with a finite number of voids. There a ray hitting a boundary can either refect or refract. In that case ray spitting occurs when the poarization changes. This eads to a ray dynamics which no onger is unique, where in genera a singe poarized ray evoves into a tree of rays. A simiar behaviour is observed in microwave resonators with dieectrica. There rays can either refect or transmit at the boundaries of the dieectrica. 2. ELASTODYNAMICS In isotropic easticity the wave equation in the frequency domain is µ (u + (λ + µ ( u + ρω 2 u = 0, (1 where u(x is the dispacement fied in the body, λ, µ are the materia dependent Lamé coefficients and ρ is the density [3, 4]. This wave equation admits two different poarizations: ongitudina and transverse with veocities c L = λ + 2µ ρ and c T = µ ρ. (2

2 n n FIGURE 1. Genera zone of two dimensiona cavity scattering. The ongitudina and transverse waves correspond to pressure respective shear deformations. This eads to the aw of refraction for incoming pane waves c L c T = sinθ L sinθ T, (3 where θ L, θ T denote the ange of incident or refection of the ongitudina and transverse wave, respectivey, measured with respect to the norma to the surface. The stress tensor in easticity has the form σ i j = λ k u k δ i j + µ ( i u j + j u i. (4 The boundary conditions considered here are free. Hence t(u = σ(u n = 0 (5 for the dispacement fied at the boundary where n denotes the norma to the boundary. The operator t refers to the traction. 3. EXACT SCATTERING DETERMINANT Via the nu fied method a cass of muti scattering probems can be soved exacty. In particuar exact scattering matrix eements are known for the case of severa scatterers of anayticay seperabe shapes for various types of media [1, 5, 6]. As mentioned, in this treatment the medium corresponds to an eastic soid which is assumed isotropic for simpicity. The scattering geometry consists of parae cyindrica voids, see fig. 1. Using ine sources parae to the voids respects this symmetry. This eads to two dimensiona easticity referred to as pane strain.

3 The scattering determinant may be factorized into a part containing singe scattering determinants and the whoe set of scatterers [7] { dets(ω = det [ } [ S (1 j] Det M(ω ] (ω j Cavities Det [ M(ω ]. (6 As the scatterers are moved around ony the atter factor, the custer determinant, changes: M = 1 + A (7 The matrix A j j = (1 δ j j a j a j [ t (J j ] ] (+ j j [t(+ j ] 1, [T ] (+ j j [T = δ σσ σσ H(+ (k σ R j j e iα( j j i (α ( j j π. (+ j j [T ]σσ may interpreted as a transation matrix acting on the scattering states [5, 6]. α ( j i is the ange to the center of cavity i in the coordinate system of cavity j. The singe cavity scattering matrices decompose over anguar momentum due to the rotationa symmetry. They have the genera form: [ S (1 j ] = [ t (+ j ] 1 [ ( j] t with the boundary conditions occurring in two-by-two matrices [ t (Z j ] with the anguar momentum, type Z {+,,J} and j the cavity index. The type refers to outgoing, incoming or reguar scattering states and invoves H (1,H (2 or J Besse functions. Thus, for the outgoing case [ (+ j] 2µ t = (2 z 2 T /2H(1 dh (1 (z (z L z L L dz L i(h (1 dh (1 (z (z T z T T dz T a 2. j i(h (1 dh (z L z (1 (z L L dz L ( 2 z 2 T /2H(1 dh (z T + z (1 (z T T dz T (9 The row index i {r,φ} is a geometric index for poar coordinates and the coumn index is a poarization index σ {L, T }. Hence, the singe cavity scattering matrix connects different poarizations. For a fu discussion, see [8, 9]. The connection to the interior probem of a singe disc is described in [10]. The poes of the custer determinant Det M(ω cance by construction the poes of the singe determinants. Likewise for the poes DetM(ω which cance the zeros of the singe determinants. Thus a scattering resonances corresponding to poes of the scattering determinant can be found from the zeros of the custer determinant. As an exampe consider the resonances in fig. 2 of a two cavity system for a materia with c L = 1950 m/s, c T = 540 m/s, cavity radii equa to 1 m and intercavity separation as measured from the centers equa to 6 m. The reguar spaced horizonta set of resonances in fig. 2 is paced beow an irreguar set. This is opposite to the scaar Hemhotz case (8

4 Im k L a Re k L a FIGURE 2. Eastodynamic scattering resonances for two cavities ( A 1 -representation. for the same geometry where the reguar spaced resonances are above the irreguar [11, 12]. The reguar resonances particuar to the fundamenta A 1 -representation are we described by the foowing condition 0 = 1 exp(ik L L/ Λ (10 with the ength L = 4a and instabiity Λ = L corresponds to that of the shortest orbit moving in a symmetry reduced domain bouncing between a cavity and the center of mass of the two cavities. Λ is obtained from the product of ray matrices as the eading eigenvaue of the corresponding (geometric acoustic ray system. This next raises the question about the effect of the remaining set of orbits. 4. ORBITS IN TIME DELAY For rea frequencies the tota scattering phase Θ is seen to be a sum over custer phase Θ c and singe cavity phases Θ j, see (6. Likewise the derivative with respect to frequency dθ/dω, the Wigner-Smith time deay, decomposes. The numerics of the custer time deay dθ c /dω show fuctuations which are reated to trapped orbits in the scattering geometry, see fig. 3, where the resuts for two identica cavities (same as those of fig. 2 are presented 1. Some of these orbits are diffractive incuding segments of surface propagation of Rayeigh type, ie. earthquakes [13]. For these proceedings we discuss the purey non-diffractive contributions, caed geometrica ray spitting orbits. Athough 1 Due to symmetries the custer deay decomposes further into a sum over four ireps, two of which are shown.

5 ^ Time deay τ Time Spectrum 100 P P2 P3 P4 P5 P6 P7 S S2 S3 S t (sec FIGURE 3. Power spectrum of custer fuctuations in time deay for two cavities. Symbos Pi and Si denote orbits bouncing i times of pressure respective shear poarization. The circuar arcs indicate Rayeigh surface waves. these orbits are essentia in most geometries, our work done on scattering has shown the importance of the Rayeigh orbits at intermediate frequencies. This is indicated in the tite which refers to that such orbits are compex. A short discussion of the diffractive orbits is given in [14]. 5. EXPANDING THE CLUSTER DETERMINANT A first step towards orbits is to consider the definition of the infinite custer determinant in terms of traces: ( z F(z = det(1 + za exp n tr( A n (11 n which at z = 1 evauates to the desired. This hods if M is Fredhom and (11 is caed a Fredhom expansion. This transates into we behaved numerica properties. For exampe the determinant converges as the dimension of the truncation of M increases. To obtain M with such properties a reguarization has been performed. This reguarization may be thought of as a Jacobi preconditioner to the origina probem in which a matrix is repaced by the one divided by the diagona. Presenty the Fredhom properties of M has ony been proved in detai in the scaar case [2]. Nevertheess, we proceed as if this is true aso in our more genera case. As numerica evidence we mention, that the n=1

6 resonances of the Fredhom expansion of the determinant to fourth order z 4 agree we with the exact resonances, the atter potted in fig RAY LIMIT AND ORBITS The expansion (11 indicates that the custer determinant can be obtained from the knowedge of an increasing number of traces. In the sadde point approximation cosed orbits bouncing n times are seen to contribute to traces of powers of the kerne tra n [15]. These orbits fufi the aws of refection and refraction and have phases corresponding to their time periods of revoution T p. In the cacuation of a trace, the singe cavity scattering matrices (8 are inserted as operators between the transation operators. To do so, use the identity J (z = 1 ( H (1 2 (z + H (2 (z giving a simiar identity for the singe cavity matrices that encode the boundary condition: [ (J] 1 ( [t (+] [ ( ] t = 2 + t, (12 which is finay is substituted into A in (7. The ray imit of the singe cavity scattering matrix gives unitary refection coefficients simiar to those of scattering from an infinite haf pane [3, 10]. This eads to an overa ampitude α p defined as a product over a refection coefficients aong the orbit. This ampitude describes the eakage from the orbit due to ray spitting. The cacuation of their geometric ampitudes requires more work. For a genera reference in the interior scaar case, see [16]. Asymptotic wave theory indicates [17 20] that for open trajectories in two dimensions the ampitude goes as (kr 1/2 where k is the wave number in question and R is the radius of curvature of the wave front at the observer. This radius is studied in e.g. geometric optics and it is possibe to keep track of its evoution during free propagation between the scatterers and at impact with possibe refractions using suitabe ray matrices. Indeed, it can be shown that for our probem open segments, in which end points are fixed and intermediate variabes are integrated by the sadde point method, has such an ampitude evoution. This comes about by cacuating the accompanying sparse hessian of this restricted integration. For a fu sadde point integration over a variabes, a cosed orbit p, the ampitude turns out to be expressibe as yet a sparse hessian and can be expanded into hessians corresponding to the previous considered open pieces as in [2]. Using the previous information then aows the fu cacuation with the ampitude going as A p = α p Det(1 J p 1/2 zn p (13 with J p and α p the product of ray matrices respective refection coefficients aong the orbit bouncing n p times. This form is precisey part of the conventiona semicassica density of states [21 23]. However, the forma parameter z is aso present and can be seen as an ordering of the various orbits in the expansion over infinitey many orbits.

7 Incorporating the resuts of the geometric ray spitting orbits gives the foowing factor of the custer determinant: ( 1 e ir ωt p F G (z = exp r αr p 1 J r p 1/2 zrn p. (14 p r=1 The sum over r refers to a sum over repeats of shortest orbits, prime cyces p. Note, that if the ogarithmic derivative with respect to ω is taken a resut very simiar to the spectra density for the interior probem is obtained. This yieds an agreement with the genera resut for the density of states for ray spitting systems described in [24]. Simiar resuts for the case of fexura vibrations in the interior case are given in [25]. As the orbits are unstabe and J p is sympectic it is possibe to expand the instabiity denominator in (14 in the inverse of its eading eigenvaue 1/Λ p for each orbit and obtain a socaed Gutzwier Voros resummed zeta function simiar to those of two-dimensiona Hamitonian fows [26]: F G (z = ζk 1, (15 k=0 where 1/ζ k (z = (1 t p (k with t (k e iωt p p = α p z n p. (16 p Λp Λ k p 7. SUMMARY Detaied studies of Hemhotz scattering determinants at sma wave engths have shown the infuence of cosed geometric rays. The case of scattering from voids in two dimensiona eastodynamics was considered here with a discussion of the anaytica contribution of cosed geometric ray spitting orbits to the scattering determinant. ACKNOWLEDGMENTS N.S. acknowedges discussions with J.D. Achenbach and the European Network on Mathematica aspects of Quantum Chaos which supported his stay at the Weizmann Institute, the Crafoord Foundation and from Det Svenska Vetenskapsrådet. REFERENCES 1. P. Gaspard and S. Rice, J. Chem. Phys. 90, 2225; 2242; 2255 ( A. Wirzba, Phys. Rep. 309, 1 ( L.D. Landau and E.M. Lifshitz, Theory of Easticity (Pergamon, Oxford B.A. Aud, Acoustic Fieds and Waves in Soids I,II (John Wiey and Sons, New York B. Peterson and S. Ström, J. Acoust. Soc. Am. 56, 771 ( A. Boström, J. Acoust. Soc. Am. 67, 399 ( N. Sondergaard, Wave Chaos in Eastodynamic Scattering, nsonderg, thesis, Northwestern University (unpubished. 8. J.L. Izbicki, J.M. Conoir and N. Vekser, Wave Motion 28, 277 (1998.

8 9. Y.H. Pao and C.C. Mow, Diffraction of Eastic Waves and Dynamic Stress Concentrations (Rand Corporation, New York N. Søndergaard and G. Tanner, Phys. Rev. E66, ( G. Vattay, A. Wirzba and P.E. Rosenqvist, Proc. Int. Conference on Dynamica Systems And Chaos, edited by Y. Aizawa, S. Saito and K. Shiraiwa (Word Scientific, Singapore, 1995, Vo. 2, pp , chao-dyn/ A. Wirzba and P.E. Rosenqvist, Phys. Rev. A ( I.A. Viktorov, Rayeigh and Lamb waves (Penum Press, New York A. Wirzba, N. Sondergaard, P. Cvitanović, Europhys. Lett. 72, 534 (2005. nin.cd/ Detais to be pubished. 16. T. Harayama and A. Shudo, Phys. Lett. A 165, 417 ( J.B. Keer and F.C. Kara, Jr., Journ. of App. Phys. 31, 1039 ( J.B. Keer and F.C. Kara, Jr., J. Acoust. Soc. Am. 36, 32 ( B. Ruf, Journ. of Acoust. Soc. Am. 45, 493 ( J.D. Achenbach, A.K. Gautesen and H. McMaken, Ray Methods for Waves in Eastic Soids (Pitman, Boston M.C. Gutzwier, Chaos in Cassica and Quantum Mechanics (Springer, New York M. Brack and R.K. Bhaduri, Semicassica Physics (Addison-Wesey, Reading H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, L. Couchman, E. Ott and T.M. Antonsen, Jr., Phys. Rev. A46, 6193 ( E. Bogomony and E. Hugues, Phys. Rev. E ( P. Cvitanović, R. Artuso, R. Mainieri and G. Vattay, Cassica and Quantum Chaos, Nies Bohr Institute, Copenhagen, 2005.