Chaotic systems and their quantization
Researchers: Per Dahlqvist
Sponsors: NFR
Chaotic systems and their quantum counterparts are studied. Classical and semiclassical properties are studied in a unified formalism involving concepts such as dynamical zeta functions and evolution operators. The emphasis is on bound systems exhibiting intermittency. Classical properties like Lyapunov exponents, decay of correlations, diffusion rates are computed. Asymptotics measures on the set of periodic orbits are developed.
Publications: 14,15,16,17,102
Propagation of nonlinear acoustic waves and beams
Researchers: Bengt Enflo, Graduate student: Claes Hedberg.
Sponsor: TFR
The project is to study theoretically some basic problems of nonlinear acoustic wave propagation and generalize the results to more complicated problems such as propagation of nonlinear sound beams and nonlinear wave propagation dispersive or layered media. Basic wave propagation problems being studied are , using equations of Burgers' type , asymptotic waveforms originating from cylindrical sine waves and propagation of nonlinear limited sound beams (B. Enflo) and propagation of plane biharmonic waves and the influence of phase between original frequencies on the wave profiles during nonlinear propagation (C. Hedberg). For describing nonlinear wave propagation in the sea or in the atmosphere the theory developed on basic problems will be modified to be applied to dispersive or layered media.
Publications: 21,70,71,72,76,77,79,109
Theoretical acoustical investigations with applications in musical acoustics
Researcher: Christer Nyberg
Sponsor: LUFT
The purpose of this project is to investigate nonlinear generation of combination frequencies in cavities. The tone generation in musical instruments is often described in terms of a clearly defined nonlinear element which can excite the rest of the instrument, treated as a linear, passive, multimode cavity. However, linear theory, which requires small amplitudes, seems to be inadequate for describing the sound field in a cavity close to a resonance, as finite amplitudes are predicted even with dissipative effects included. If the sound field is excited by two frequencies close to resonance, nonlinear interaction is therefore expected to become important. Starting with a nonlinear generalization of d'Alembert's wave equation together with appropriate boundary conditions, the acoustic wave field in the cavity is calculated and can then, in the case of periodicity, be decomposed into its Fourier-components.
Publications: 38
Dynamical systems and multi-body modelling
Researchers: Martin Lesser, Hanno Essén, Arne Nordmark, Graduate Students: Mats Fredriksson, Anders Lennartsson
Sponsor: TFR, LUFT
The purpose of this study is to integrate modern methods of modelling multi-body systems with recent results in the theory of dynamical systems. Thus far most dynamical systems treated under the new heading of chaos theory has involved concentration on very low degree of freedom models derived in a somewhat ad-hoc fashion as representations of more complex mechanisms. Our aim is to combine our new techniques for dealing with complex mechanisms by computer algebra and Kanes equations with the methods of dynamical systems theory to achieve useful and readily interpretable representations, e.g. by means of center manifold ideas. One particular area of interest is problems of impact in subassemblies of complex mechanisms.
Publications: 19,31,32,33,46,59,86,87,90
Design of complex mechanical systems
Researchers: Martin Lesser, Sören Andersson (Machine Elements), Lennart Karlsson (Computer Aided Design, Luleå), Tore Risch, (Computer Science Linköping), Volvo Corporation. Graduate student at KTH: Claes Tisell
Sponsor: NUTEK
Complex mechanical systems are treated by a combination of modern methods for simulation, computer aided design tools and object oriented data base technology. The aim of the project is to assemble all of these techniques into a usable design tool. Several particular problems are being used as test cases. These include blade mountings in jet engines and exhaust manifold in automobiles.
Publications: 96
Generation and focusing of converging shock waves of a desired shape
Researchers: Martin Lesser, Nicholas Apazidis, Graduate student: Josefina Dellby
Sponsors: TFR, LUFT.
This project deals with propagation of shock waves in liquids and liquid impact problems. Generation, reflection and convergence of shock waves in confined chambers of various forms is investigated on the basis of Whitham's non-linear theory of geometrical shock dynamics. It is shown that by an appropriate choice of the form of the reflector boundary one may obtain reflected shock waves having desirable shapes, for example a near-square shape. Also reflectors with parabolic geometry are considered. A cylindrical wave is generated at the focus of the parabolic cross-section. It is shown that contrary to the linear case the reflected wave is no longer planar. The case of an impact of a liquid drop on a plane surface is a related phenomenon. The shape and strength of the shock wave created during the impact and traveling at a supersonic speed into the liquid is investigated. It is shown that the Mach number distribution and thus the pressure decreases at the central part of the shock. Technological and medical applications of the project may be found within the fields of shock wave propagation, shock induced collapse of cavities, erosion, disintegration of kidney and bladder stones by means of a shock wave attenuation in lithotriptor devices.
Publications: 3,100,103
Non-Newtonian phenomena and kinetic theory of gases
Researcher: Lars Söderholm.
Sponsor: LUFT.
There are two basic limitations of the Navier-Stokes equations. One is that they are parabolic in time: some wave modes are lacking. The other one is that they are valid only if the mean free path is a factor 100 to 1000 smaller than the macroscopic scale. This is important for high altitude flight. It is also important for flow around very small objects, an area which is becoming increasingly important at present.
General flow equations for gases can be obtained from the Boltzmann equation via the Chapman-Enskog method and the Grad method. For slow, already relaxed, flows, the Chapman-Enskog method gives more accurate results than the corresponding Grad approximation. But for the (fast) relaxation, the Grad method is more reliable than the Chapman-Enskog method.
An intermediate method is developed, in order to combine the advantages of both. A new 13 moments system of equations, which has been obtained, is now studied. The system is hyberbolic like the Grad system (so that a full set of wave modes is included), but gives the correct values for viscocity and heat conduction, in contrast with the Grad system. A new type of regularization of the Burnett equations is also studied.
Non-linear waves and integrability
Researcher: Lars Söderholm.
Sponsor: LUFT.
An important class of non-linear wave equations turns out to be completely integrable. Most of these equations are weakly non-linear approximations to non-linear equations such as the Korteweg-de Vries equation and the non-linear Schrödinger equation (valid for long and short water waves, respectively). The fully non-linear Euler equations are, however, exactly integrable (via the hodograph transformation) and so are the field equations of general relativity when sufficient symmetry reduces to two independent variables.
Integrable equations can be considered intrinsically linear as they via (non-local) transformations can be reduced to linear equations. This calls for an intrinsic, geometric description. In general relativity such methods have been employed for a long time. Some important insight into the class of integrable equations should be possible from a study of the equations of general relativity. A result is that the relativistic Euler equations are reduced to a linear equation (in a hodograph plane). Especially its ultra-relativistic limit is studied. - The field equations of general relativity are studied for gravitational plane waves, special interest is focussed on the continuum contribution in the inverse scattering scheme. - Some of these questions are pursued in cooperation with Michael Bradley, University of Umeå.
Asymptotic methods in dynamical stability theory and quantum mechanics
Researchers: Karl-Erik Thylwe, Ian Cohen, Enrico Gravador (Illigan Inst. of Technology, Philippines), Örjan Dammert (theor. phys., UU) and Anders Hökback (theor. phys., UU) .
Sponsor: 'Rörlig resurs, KTH'.
Higher-order asymptotic methods relevant for parametrically excited, linear and non-linear, coupled oscillator models are developed. Certain uniform matching techniques are needed to describe the behaviour near time-localised resonances of the coupled oscillators. In this process, further developments of an amplitude-phase theory for coupled linear oscillators with arbitrarily varying coefficients is in progress. This generalises the Floquet theory, where Floquet theory is restricted to oscillators with time-periodic coefficients rather than arbitrarily varying ones.
Publications: 20,52,53,106
Dynamics of time-dependent Hamiltonian systems
Researchers: Karl-Erik Thylwe, Per Dahlqvist and F. Bensch (University of Campinas, Brazil)
Sponsor: 'Rörlig resurs', KTH and NFR.
Generic Hamiltonian model systems and their semiclassical quantizations are studied from complementary views. The irregular part of phase space is adequately quantized by the Gutzvillers periodic orbit theory, and the regular part may be treated by approximate vortex tube quantizations and normal form expansions. In the irregular case we focus on eliminating the problems caused by intermittency and quasi regular parts of phase space. In the regular case we focus on a generalisation of the periodic orbit theory, which may extend the success of the theory to include stable periodic orbits. A complementary study of the regular islands based on new results on Lewis invariants and normal-form expansions may help to support progress of the periodic-orbit approach. Publications: 50,51
Black-hole scattering of scalar waves
Researchers: K.-E. Thylwe, in collaboration with N. Andersson (Cardiff University, Wales)
Sponsor: 'Rörlig resurs', KTH.
A description of scalar waves scattered off a Schwarzschild black hole in terms of complex angular momenta is discussed. In the new picture the scattering amplitude is split into a supposedly smooth background integral and a sum over the so-called Regge poles. It seems
reasonable to hope that we can learn more about the intricate details of general relativity by studying how incoming waves are affected by the presence of a black hole. This is especially important considering the fact that we only have indirect evidence for the actual existence of astrophysical black holes. By studying how a black hole interacts with its environment one may understand whether there are ways of making direct observations of such, in principle invisible, objects. Most studies of black-hole scattering have used the standard partial-wave paradigm. In that description it is, however, difficult to interpret the results.
An alternative description of scattering, that is celebrated for its interpretative power, is based on the use of complex angular momenta (CAM). Considering the success of the CAM approach it is surprising that no attempts have been made to use it in the black-hole case. Chandrasekhar and Ferrari have discussed CAM in the context of stellar pulsations, but they do not make use of the full potential of the technique.
Publications: 2