Edward L. Reiss

Bio: Edward L. Reiss is an academic researcher from Northwestern University. The author has contributed to research in topics: Acoustic wave & Buckling. The author has an hindex of 19, co-authored 68 publications receiving 1368 citations. Previous affiliations of Edward L. Reiss include Rensselaer Polytechnic Institute.

Multiple Eigenvalues Lead to Secondary Bifurcation

Abstract: Bifurcation problems are considered where the primary bifurcation points are functions of a parameter $\tau $. It is shown that a multiple bifurcation point, which occurs for $\tau = \tau _0 $, may “split” into two (or more) simple primary bifurcation points and several secondary bifurcation points as $\tau $ varies from $\tau _0 $. These secondary points move along one or more of the primary branches as $\tau $ varies. This is shown to occur for a simple two-degrees-of-freedom system, which is a model for plate and rod buckling. This analysis shows that the splitting occurs in several different and physically significant ways. A new perturbation method is presented for analyzing problems which cannot be explicitly solved. The method is applied to study secondary bifurcations in the axisymmetric buckling of spherical shells. The paper concludes with a brief discussion of some physical problems where this new phenomenon may occur and where the perturbation method may be applicable.

A numerical method for ocean‐acoustic normal modes

Abstract: The method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans. The propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes. Errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field. Such errors can severely degrade the accuracy of the normal mode representation, particularly at long ranges. In this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes. It consists of a combination of well‐known numerical procedures such as Sturm sequences, the bisection method, Newton’s and Brent’s methods, Richardson extrapolation, and inverse iteration. We also introduce a modified Richardson extrapolation procedure that substantially increases the speed and accuracy of the computation.

Singular Perturbations of Bifurcations

Abstract: An asymptotic theory is presented to analyze perturbations of bifurcations of the solutions of nonlinear problems. The perturbations may result from imperfections, impurities, or other inhomogeneities in the corresponding physical problem. It is shown that for a wide class of problems the perturbations are singular. The method of matched asymptotic expansions is used to obtain asymptotic expansions of the solutions. Global representations of the solutions of the perturbed problem are obtained when the bifurcation solutions are known globally. This procedure also gives a quantitative method for analyzing singularities of nonlinear mappings and their unfoldings. Applications are given to a simple elasticity problem, and to nonlinear boundary value problems.

A numerical method for bottom interacting ocean acoustic normal modes

Abstract: In this paper we present a finite‐difference method to numerically determine the normal modes for the sound propagation in a stratified ocean resting on a stratified elastic bottom. The compound matrix method is used for computing an impedance condition at the ocean–elastic bottom interface. The impedance condition is then incorporated as a boundary condition into the finite difference equations in the ocean, yielding an algebraic eigenvalue problem. For each fixed mesh size this eigenvalue problem is solved by a combination of efficient numerical methods. The Richardson mesh extrapolation procedure is then used to substantially increase the accuracy of the computation. Two applications are given to demonstrate the speed, accuracy, and efficiency of the method.

Block five diagonal matrices and the fast numerical solution of the biharmonic equation

Abstract: A factoring and block elimination method for the fast numerical solution of block five diagonal linear algebraic equations is described. Applications of the method are given for the numerical solution of several boundary-value problems involving the bi- harmonic operator. In particular, 22 eigenvalues and eigenfunctions of the clamped square plate are computed and sketched.

On the Lambert W function

Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

Seismic Ray Theory

Abstract: 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.

Two-dimensional turbulence

Abstract: The theory of two-dimensional turbulence is reviewed and unified, and some hydrodynamic and plasma applications are considered. The topics covered include some equations of incompressible hydrodynamics, absolute statistical equilibrium, spectral transport of energy and enstrophy, turbulence on the surface of a rotating sphere, turbulent diffusion, MHD turbulence, and two-dimensional superflow. Finally, an attempt is made to assess the status and future of the principal research topics which have been discussed.

Sinc approximation of algebraically decaying functions

Abstract: An extension of sinc interpolation on $\mathbb{R}$ to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of error estimates. First covers a wider class of functions with the algebraic order of decay on $\mathbb{R}$. The second type of error estimates governs the case when the order of function's decay can be estimated everywhere in the horizontal strip of complex plane around $\mathbb{R}$. The numerical examples are provided.