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We examine the scattering of Ostrovsky wave packets, generated from an incident solitary wave, in a two layered waveguide with a delamination in the centre and soft (imperfect) bonding either side of the centre. The layers of the waveguide are assumed to consist of different materials, and the strains are described by a system of coupled Boussinesq equations. A semi-analytical approach consisting of matched asymptotic multiple-scale expansions is applied, leading to Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the delaminated region. This semi-analytical method has good agreement with direct numerical simulations, validating the approach. In the delaminated regions, Ostrovsky wave packets evolve into a train of solitary waves, which subsequently evolve into Ostrovsky wave packets in the second bonded region. Analysis of the phase shift in the wave packet, introduced from the delaminated region, allows us to predict both the position and the length of the delamination; the first time this has been achieved using nonlinear waves. These results motivate experiments to validate the theoretical results, with the aim of creating a tool to monitor the integrity of layered structures., Comment: 9 pages, 7 figures

In this paper we examine the effect of delamination on wave scattering, with the aim of creating a control measure for layered waveguides of various bonding types. Previous works have considered specific widths of solitary waves for the simulations, without analysing the effect of changing the soliton parameters. We consider two multi-layered structures: one containing delamination "sandwiched" by perfect bonding and one containing delamination but "sandwiched" by soft bonding. These structures are modelled by coupled Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead to coupled Ostrovsky equations in soft bonded regions and Korteweg-De Vries equations in the perfectly bonded and delaminated region. We use the Inverse Scattering Transform to predict the behaviour in the delaminated regions. In both cases, numerical analysis shows that we can predict the delamination length by changes in the wave structure, and that these changes depend upon the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case of perfect bonding, we derive a theoretical prediction for the change and confirm this numerically. For the soft bonding case, we numerically identify a similar relationship using the change in amplitude. Therefore we only need to compute one curve to determine the behaviour for any incident solitary wave, creating a framework for designing measurement campaigns for rigorously testing the integrity of layered structures., Comment: 12 pages, 7 figures

The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron-Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples., Comment: 19 pages, 4 figures

Standard methods for describing the intensity distribution of mechanical and acoustic wave fields in the high frequency asymptotic limit are often based on flow transport equations. Common techniques are statistical energy analysis, employed mostly in the context of vibro-acoustics, and ray tracing, a popular tool in architectural acoustics. Dynamical energy analysis makes it possible to interpolate between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. In this work a version of dynamical energy analysis based on a Chebyshev basis expansion of the Perron-Frobenius operator governing the ray dynamics is introduced. It is shown that the technique can efficiently deal with multi-component systems overcoming typical geometrical limitations present in statistical energy analysis. Results are compared with state-of-the-art hp-adaptive discontinuous Galerkin finite element simulations.

We report on a sub-stellar companion search utilizing interferometric fringe-tracking astrometry acquired with Fine Guidance Sensor 3 (FGS 3) on the Hubble Space Telescope. Our targets were Proxima Centauri and Barnard's Star. We obtain absolute parallax values for Proxima Cen pi_{abs} = 0.7687 arcsecond and for Barnard's Star pi_{abs} = 0.5454 arcsecond. Once low-amplitude instrumental systematic errors are identified and removed, our companion detection sensitivity is less than or equal to one Jupiter mass for periods longer than 60 days for Proxima Cen. Between the astrometry and the radial velocity results we exclude all companions with M > 0.8M_{Jup} for the range of periods 1 < P < 1000 days. For Barnard's Star our companion detection sensitivity is less than or equal to one Jupiter mass for periods long er than 150 days. Our null results for Barnard's Star are consistent with those of Gatewood (1995)., Comment: 35 pages, 13 figures, to appear in August 1999 AJ

We investigate the form of the one-point probability distribution function (pdf) for the density field of the interstellar medium using numerical simulations that successively reduce the number of physical processes included. Two-dimensional simulations of self-gravitating supersonic MHD and hydrodynamic turbulence, and of decaying Burgers turbulence, produce in all cases filamentary density structures and a power-law density pdf with logarithmic slope around -1.7. This suggests that the functional form of the pdf and the general filamentary morphology are the signature of the nonlinear advection operator. These results do not support previous claims that the pdf is lognormal. A series of 1D simulations of forced supersonic polytropic turbulence is used to resolve the discrepancy. They suggest that the pdf is lognormal only for effective polytropic indices $\gamma=1$ (or nearly lognormal for $\gamma\not=1$ if the Mach number is sufficiently small), while power laws develop at high densities if $\gamma<1$. We evaluate the polytropic index for conditions relevant to the cool interstellar medium using published cooling functions and different heating sources, finding that a lognormal pdf may occur at densities between 10$^3$ and at least 10$^4$ cm$^{-3}$. Several applications are examined. First, we question a recent derivation of the IMF from the density pdf by Padoan, Nordlund & Jones because a) the pdf does not contain spatial information, and b) their derivation produces the most massive stars in the voids of the density distribution. Second, we illustrate how a distribution of ambient densities can alter the predicted form of the size distribution of expanding shells. Finally, a brief comparison is made with the density pdfs found in cosmological simulations., Comment: Submitted to ApJ. 23 pages, 15 postscript figures