Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods

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Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering ∗

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning point Xt of the WKBJ approximation, while its quantization is provided by the connection of solutions across Xt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.

Global linear stability analysis of weakly non-parallel shear flows

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

Localized Light Waves: Paraxial and Exact Solutions of the Wave Equation (a Review)

We review the progress in the development of frequency converters based on three-wave mixing in whispering gallery resonators (WGRs) The theoretical description, given in a unified notation for all basic processes, reveals that the phase-matching condition known from conventional devices is replaced by several selection rules and, furthermore, the fact that conversion efficiencies of more than 25% can be reached in the overcoupled regime only Experimentally, the conversion efficiencies exceed 50% already at milliwatt input powers This is achieved, however, so far in bulk resonators only since today the on-chip devices have two orders of magnitude lower quality factors Regarding the stability of the conversion process, one has to consider impurities left from the crystal growth and material specific effects like photoconductivity, photorefractivity, and pyroelectricity The impressive experimental progress paves the way that micrometer-sized frequency converters based on WGRs will find the way out of the lab into real-world applications

Three‐wave mixing in whispering gallery resonators

Short-Wavelength Diffraction Theory

http://link.springer.com/content/pdf/10.1007/978-3-642-83459-2_6.pdf

The Parabolic Equation Method

http://link.springer.com/content/pdf/10.1007/978-3-642-83459-2_9.pdf

Eigenfunctions Concentrated in the Vicinity of a Closed Geodesic

http://link.springer.com/content/pdf/10.1007/978-3-642-83459-2_3.pdf

The Field Near a Caustic

http://link.springer.com/content/pdf/10.1007/978-3-642-83459-2_10.pdf

Vasili M. Babič

Papers

Short-Wavelength Diffraction Theory

The Parabolic Equation Method

Eigenfunctions Concentrated in the Vicinity of a Closed Geodesic

The Field Near a Caustic

Multiple-Mirror Resonators