# Edward L. Reiss's research while affiliated with Keele University and other places

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## Publications (80)

The propagation of acoustic waves from a high-frequency point source in a shear layer flowing over an infinite rigid plate is considered. Asymptotic expansions of the solution are obtained as k = omegaD/c0 --> infinity, using a previously developed method. Here, omega is the circular frequency of the point source, D is the thickness of the shear la...

A mathematical model for the non-linear stability of two-dimensional leakage channels is formulated and analyzed. It consists of an infinite channel with flexible elastic walls containing a flowing viscous, incompressible fluid. Two infinite elastic plates are inserted into the channel, parallel to the walls, to form parallel channels. The walls an...

The propagation of acoustic waves from a high-frequency line source in a shear layer flowing over an infinite elastic plate is considered. The fluid is inviscid and compressible. The Lagrange-Kirchhoff linear plate theory, including structural damping, is used to describe the small amplitude motions of the plate. The resulting problem is solved app...

The initial thrust of our research program was to study the propagation of acoustic waves from a time periodic point source in an ocean whose properties vary slowly with range, and which rests on an elastic bottom (whose properties may also vary slowly with range). The motivations for these investigations were to study the propagation of sound in r...

An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and...

In this paper, a direct method and a simple inverse method, which can be used to determine the velocity profile of a shear layer, are presented. Specifically, an infinite acoustic medium, with constant density and sound speed, containing a free shear layer of infinite extent, is considered. The free shear layer is probed with a two-dimensional plan...

The scattered field produced by a source inside a shear layer of arbitrary profile, flowing above an infinite, rigid wall, is examined. The shear layer is topped by a uniform flow. A representation of the solution is obtained in terms of a pair of functions that satisfy a homogeneous second-order ordinary differential equation, with variable coeffi...

The effects of rotation on secondary transitions of thermal convection
in an infinite layer with finite Prandtl number and rigid boundaries are
investigated analytically, expanding on the numerical results of Clever
and Busse (1979). An asymptotic approach is employed to study the
secondary transitions, to determine the secondary states, and to
eva...

A flexible membrane backed by a rigid cavity is set into an infinite plane baffle. The upper halfspace contains an acoustic fluid, and the cavity which lies in the lower half-space, contains another acoustic fluid. A time harmonic wave in the upper half-space is incident on the plane. When the frequency of the incident wave is bounded away from the...

We consider the Boussinesq theory for convection in a rectangular box with imposed constant, negative, vertical heat and salt gradients. We analyze the bifurcation of two-dimensional convection steady states near a double instability point defined by a critical value of the thermal Rayleigh number and geometrical aspect ratio. We find that for ther...

Passive techniques for nonlinear stability control are presented for a model of fluidelastic instability. They employ the phenomena of lambda-bifurcation and a generalization of it. lambda-bifurcation occurs when a branch of flutter solutions bifurcates supercritically from a basic solution and terminates with an infinite period orbit at a branch o...

A perturbation analysis conceived for thermal convection states in a rotating box with a uniform temperature distribution on the walls is extended to the effect of a slightly nonuniform temperature distribution on a wall of the box, qualitatively altering the response of the convection system. This two-parameter study establishes that the vertical...

Perturbation and asymptotic methods are presented for analyzing a class of subcritical bifurcation problems whose solutions possess minimum transition values. These minimum transition values are determined. In addition, the dynamics of the transitions from the basic state to the larger amplitude bifurcation states are obtained. The effects of imper...

A rigid baffle which separates an acoustic fluid in the upper half-space
from a vacuum in the lower half-space is considered, taking into account
a pulse which is incident on the baffled membrane. The pulse satisfies
the acoustic wave equation in the upper half-space. The pulse scattering
problem is formulated as an initial-boundary value problem f...

DOI:https://doi.org/10.1103/PhysRevA.32.1909

The method of matched asymptotic expansions is used to study the scattering of plane monochromatic acoustic waves from baffled flexible surfaces in the limit as L/lambda approaches 0. Here, lambda is the wavelength of the incident acoustic wave and L is a characteristic size of the flexible surface, such as its maximum diameter. The baffled surface...

The normal modes and their propagation numbers for acoustic propagation in wave guides with flow are the eigenvectors and eigenvalues of a boundary value problem for a non-standard Sturm-Liouville problem. It is non-standard because it depends non-linearly on the eigenvalue parameter. (In the classical problem for ducts with no flow, the problem de...

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 boundar...

We analyze convection in a rectangular box where two ``substances,'' such as temperature and a solute, are diffusing. The solutions of the Boussinesq theory depend on the thermal and solute Rayleigh numbers RT and Rs, respectively, in addition to other geometrical and fluid parameters. As RT is increased, the conduction state becomes linearly unsta...

Perturbation and asymptotic methods are used to obtain the secondary bifurcation of quasi-periodic solutions from periodic solutions for a model problem. It is a two-cell model consisting of a coupled system of van der Pol-Duffing oscillators. The qualitative features of the secondary bifurcation states depend on the detuning of the oscillators. Fo...

The classical optical theorem for scattering by compact obstacles is a forward-scattering theorem. That is, the total cross section of the obstacle is proportional to the imaginary part of the far-field directivity factor evaluated in the forward scattering direction. An analogous theorem is derived in this paper for the scattering of acoustic wave...

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 boundar...

This chapter discusses an asymptotic analysis of acoustic scattering by nearly rigid or soft targets. The qualitative features of the field scattered by a penetrable acoustic target, insonified by a time periodic incident wave, depends on a number of factors. Among them are the frequency of the incident wave: the densities and indicies of refractio...

This chapter discusses new mechanism for the formation of turbulent spots. Because the bursting process is violent, it generates a pressure field with a relatively broad frequency spectrum, as experimental results demonstrate. The bursts as localized sources of energy, among other important effects, propagate acoustic waves in the sublayer. The wav...

An acoustic target of constant density ϱt and variable index of refraction is imbedded in a surrounding acoustic fluid of constant density ϱa. A time harmonic wave propagating in the surrounding fluid is incident on the target. We consider two limiting cases of the target where the parameter ε ≡ ϱa/ϱt → 0 (the nearly rigid target) or ε → ∞ (the nea...

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...

A bifurcation problem is analyzed for a Brussellator boundary value problem, which is a typical reaction-diffusion system. The bifurcation parameter lambda is proportional to the length of the system. Previously developed perturbation methods are employed to analyze the secondary bifurcation of steady solutions that arise from the splitting of mult...

Two mechanisms of drag reduction for flow over flat plates were investigated. The first mechanism employs Bushnell's hypothesis that compliant walls produce drag reduction by interfering with the formation of the turbulent spots in a turbulent boundary layer. It is shown that the amplitudes and frequencies of compliant wall motions for drag reducti...

A mathematical model of two-dimensional flow through a flexible channel is analyzed for its stability characteristics. Linear theory shows that fluid viscosity, modelled by a Darcy friction factor, induces flutter instability when the dimensionsless fluid speed, S, attains a critical flutter speed, S0. This is in qualitative agreement with experime...

Bifurcation Theory is a study of the branching of solutions of equations as a parameter λ, called the bifurcation parameter, is varied. The branching, or bifurcation points, are singular points of the solutions. In Bifurcation Theory solutions are analyzed near bifurcation points.

The propagation of acoustic waves from a high frequency line source in a two-dimensional parallel shear flow adjacent to a rigid wall is analyzed by a ray method. The leading term in the resulting expansion is equivalent to the geometrical acoustics theory of classical wave propagation. It is shown that energy from the source is radiated either dir...

The Brussellator is a simple chemical model describing pattern formation by bifurcation of solutions. In this model, and for a one-dimensional system of length L, the steady states are determined by the solutions of a boundary value problem for a system of two, nonlinear, ordinary differential equations. For a one-dimensional system, the bifurcatio...

An asymptotic expansion which is uniformly valid in space is obtained for the low frequency scattering of a plane wave incident on a localized inhomogeneity. The scattering region, which may be simply or multiply (collection of scatterers) connected, has a characteristic length which is small compared with the wave length of the incident wave. The...

A general bifurcation problem is considered that depends on two parameters in addition to the bifurcation parameter lambda. It is assumed that all primary bifurcation states correspond to steady solutions and that they branch supercritically. Then it is shown that for a range of system parameters and near a triple primary bifurcation point the foll...

Sequential bifurcation of solutions of nonlinear equations, as the bifurcation parameter increases, is called cascading bifurcation. It has been proposed as a mechanism to describe the transition from laminar to turbulent fluid flows, and as a mechanism for chemical and biochemical morphogenesis and pattern formation. The creation of cascading bifu...

Isolas are isolated, closed curves of solution branches of nonlinear problems. They have been observed to occur in the buckling of elastic shells, the equilibrium states of chemical reactors and other problems. In this paper we present a theory to describe analytically the structure of a class of isolas. Specifically, we consider isolas that shrink...

An ″honest″ statistical method is presented to analyze the effects of imperfections and other disturbances on the bifurcation of solutions of nonlinear problems. First, uniformly valid asymptotic approximations of the solutions are obtained for any realization of the imperfections. The approximations are valid as the magnitude of the imperfections...

The effects of small periodic disturbances on the response of a two-degree-of-freedom, nonconservative mechanical system are analyzed. The system is a simple model for panel flutter. The disturbance simulates the pressure fluctuations of a turbulent boundary layer on the panel. Asymptotic expansions of the solutions are obtained for small-amplitude...

The deformation of a thin elastic plate which is initially wrinkled when the plate is subjected to a constant compressive end thrust is considered. The singularly perturbed bifurcation theory of Reiss and Matkowsky is used. It is found that the initial deformation (imperfection) of the plate leads to solutions which explain the experimentally obser...

A perturbation method is employed to obtain a new class of periodic motions for the nonlinear vibrations of rectangular, elastic plates. The dynamic von Karman plate theoryr is used in the analysis. Period solutions bifurcate at the natural frequencies of free vibration of the linearized plate theory. The new solutions bifurcate from these periodic...

The effects of temporal variations of the soundvelocity on acoustic propagation from a time periodic point source in a layer of uniform depth h are studied. It is assumed that the sound speed is a slowly varying function of time, and that the upper and lower surfaces of the layer are free and rigid, respectively. An asymptotic expansion of the acou...

The secondary buckling of rectangular elastic plates is studied as a problem of secondary bifurcation. The nonlinear von Karman plate theory is used in the analysis. The secondary bifurcation points, and the secondary states that bifurcate from them are determined by a previously developed perturbation method. Secondary buckling is related to the p...

The two-dimensional thermal convection of a viscous fluid in a rectangular region is analyzed using the Boussinesq theory, and the Rayleigh boundary conditions. The existence of a unique convention state bifurcating from each eigenvalue of the linearized theory is established by using the Morse lemma. This establishes the validity of the formal per...

The first 21 normal modes and corresponding cutoff frequencies are obtained for the E modes of waveguides with regular hexagonal cross-sections. A previously developed numerical method, using inverse iterations, finite differences, and Richardson's mesh extrapolation procedure, is employed. The inverse iteration method is modified by an orthogonali...

It is shown that small imperfections or impurities always present in
experiments may account for the smooth transitions from conduction to
convection observed at a critical Rayleigh number in viscous fluid
heating. The imperfections embrace several types of thermal noise in the
applied temperature, including those due to irregularities in the fluid...

For planar fields, the Grad–Shafranov theory of magnetohydrodynamic equilibria is reduced to solving Δ&psgr;+kG (&psgr;) =0 for some function G (&psgr;), where Δ is the two-dimensional Laplacian and k is an ’’amplitude’’ of G. This equation is solved on a rectangular region of aspect ratio l, with &psgr;=0 on the boundary, and G satisfying the cond...

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...

This chapter presents a case history of numerical bifurcation and secondary bifurcation. Secondary transitions or secondary bifurcations occur frequently in many nonlinear stability problems. In hydrodynamic stability, secondary bifurcations are called secondary transitions and in elastic stability, they are called secondary bucklings. The chapter...

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 equals tau //0, may ″split″ into two (or more) simple primary bifurcation points and several secondary bifurcation points as tau varies from tau //0. This analysis shows that...

It is shown by using a formal perturbation expansion, that unsymmetric equilibrium states branch from axisymmetric buckled states of the uniformly compressed and clamped circular plate. The process is called secondary buckling and the edge thrusts at which the branchings occur are called the secondary buckling loads. In the neighborhoods of the sec...

An iterative method is presented for solving plane strain and plane stress problems for homogeneous and isotropic elastic materials. Displacements or some combination of displacements and stresses are prescribed on the boundary of the elastic solid. The iterates are evaluated numerically by difference methods. The direct block factoring method is u...

The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call th...

This chapter focuses on the collapse of shallow elastic membranes. The nonlinear theory of buckling of elastic shells is presented in the chapter to study the response of the shell as Q0 increases by using the membrane as a model, where Q0 is a parameter proportional to the total pressure force acting on the membrane. It was reasoned that as the st...

The branching of unsymmetric equilibrium states from axisymmetric equilibrium states for clamped circular plates subjected to a uniform edge thrust and a uniform lateral pressure is analyzed. The branching process is called wrinkling and the loads at which branching occurs are called wrinkling loads. The nonlinear von Karman plate theory is employe...

The shell is subjected to a uniform compressive surface load (either a pressure or a centrally directed load). The results of a numerical study are summarized and compared with experiments.RésuméOn soumet la surface d'une coque à une charge de compression uniforme (soit une pression soit un champ de forces centrales). On résume les résultats et on...

Two stability problems for the nonlinear sine-Gordon equation are studied. The stability of a class of time independent (static) solutions is studied using linear dynamic stability theory. An asymptotic approximation of the nonlinear transient response to small disturbances of an unstable static state is obtained by the two time method. Interpretat...

A numerical method is presented for determining the bending stresses in elastic cantilever plates. Southwell's formulation of the classical small deflection plate theory is employed. The method is applied to obtain numerical solutions for rectangular and swept back plates subjected to a variety of surface and edge loads. Boundary layer effects are...

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...

In this paper we consider the damped linear oscillator with small damping $\varepsilon $. We obtain uniform asymptotic expansions of the solution as $\varepsilon \to 0$ that are uniformly valid for all time $t \geqq 0$, by the multitime method. We show how to determine the expansion coefficients without resorting to intuitive arguments. This is don...

: A nonlinear thin shell theory is derived for the axisymmetric buckling of spherical shells subjected to either a pressure or a centrally directed surface load. The theory is reduced to a boundary value problem for a system of four first order ordinary differential equations. Numerical solutions of this boundary value problem are obtained by the s...

Existence and uniqueness theorems are proved for two boundary value problems for the axisymmetric deformation of a circular
membrane subjected to normal pressure. The nonlinear Föppl membrane theory is employed. The shooting method is used to establish these results. It is also shown that if the edge is
free to move in the plane of the membrane the...

The nonlinear deflections of a thin elastic simply-supported rectangular plate are studied. The plate is deformed by a compressive thrust applied along the short edges. For the boundary value problem considered we prove that the plate cannot buckle for thrusts less than or equal to the lowest eigenvalue of the linearized buckling problem. For large...

A previously developed iterative procedure is applied to obtain numerical solutions of the von Kármán equations for rectangular plates subjected to a uniform normal pressure. On the simply supported boundary, it is assumed that the normal membrane stress and the tangential membrane displacement vanish. Solutions are obtained for a wide range of val...

A boundary layer expansion procedure is presented for constructing approximate two dimensional shell theories from the three dimensional theory of linear elasticity for cylindrical bodies for which the ratio of the shell thickness 2 h to the radius R is small. The basis of the approximation is the thin shell theory of Donnell. It appears, naturally...

This paper consider the secondary and cascading bifurcation of two-dimensional steady and period thermal convection states in a rotating box. Previously developed asymptotic and perturbation methods that rely on the coalescence of two, steady convection, primary bifurcation points of the conduction state as the Taylor number approaches a critical v...

Some recent progress that has been made in the theory and numerical computation of secondary bifurcation is described by considering two problems for the nonlinear buckling of elastic plates.

## Citations

... KRAKEN [40,41] is a normal mode model where range-dependent solutions are obtained using optionally adiabatic or coupled-mode theory [23]. The KRAKEN program takes an ENV file, computes the modes, and writes them to disk for use by other modules. ...

... Wrinkling of circular and annular plates and spherical shells appears in various engineering and biomechanical applications and has been studied by Adachi and Benicek (1964); Bushnell (1981); Panov and Feodosiev (1948). Under some loading types and boundary conditions a ring of large circumferential compressive stress develops near the edge of the plate (or shell) and may cause asymmetrical buckling (Cheo and Reiss, 1973;Voronkova, 2018, 2020)). Nonaxisymmetric buckling of circular plates subjected to surface load was first formulated and studied in Panov and Feodosiev (1948), and, later, existence and uniqueness of unsymmetric equilibrium states were proved by Morozov (1961); Piechocki (1969)). ...

... It is in fact very special and nongeneric for power series to directly solve a boundary value problem, even though the class of exceptions includes all Hough functions. (The range of applicability can be increased by domain decomposition, that is, by applying a swarm of power series, each evaluated only on its associated subdomain, and matching the piecewise approximations together as in Keller and Reiss [32] and Weinitschke [50].) The domain of convergence of power series approximations can be greatly extended by applying sum acceleration schemes such as the Euler sum acceleration [14,16,41] and Padé approximants [3,12]. ...

... The proof of convergence of the estimation procedure is given here (Theorem 2.1). It is expected to obtain good estimates only if the required matrix has a sparse structure (Beu 2015). In Sect. 3 the algorithm for estimating the unknown parameters of a matrix product is described which is based on solving a minimization problem by the SPSA (Simultaneous Perturbation Stochastic Approximation) (Spall 2003). ...

... In the context of elastic membranes, response curves exhibiting multiple solutions were studied by Callegari, Reiss 4nd Keller [8] and by Bauer, Callegari and Reiss [9]. ...

... We refer to the boundary-value problem (2.6) as Problem S. If X < 0 then Problem S has a unique solution [8], If X is positive and sufficiently large then Problem S may have non-unique solutions. We seek unsymmetric solutions of Problem B which branch from a solution of Problem S. For the pure buckling problem this corresponds to studying secondary buckling from a buckled axisymmetric state. ...

Reference: Unsymmetric wrinkling of circular plates

... But, it is a well known^fact that finite Taylor expansions usually furnish asymptotic approximations to exact solutions only on finite time interval though we need perturbation procedures which give asymptotic expansions approximating exact solutions uniformly on the whole time interval 0<ί<oo, which is quite difficult. To overcome this difficulty, one of the most powerful perturbation methods was first developed by Cole and Kevorkian [3] and was simplified by Reiss [38] later. The method is known under the various names, the method of multiple scales, or simply the two-timing method. ...

... Fourth-order boundary value problems which describe the deformations of an elastic beam in an equilibrium state whose both ends are simply supported have been extensively studied in the literature. For classical results obtained on elastic beam equations we refer to [6,12,14,15,38,44], in particular [44] is one of the pioneering works on extensible beams), while [12] settles the existence and multiplicity question for (P f ) in the physical situation p = 2, ρ = 0 and M of the form M (s) = as + b. Recently, the existence of solutions to fourth-order boundary value problems have been studied in many papers and we refer the reader to the papers [8,9,20,22,26,28,29,30,31,36] and the references therein. ...

... As e -0, the regular Hopfbifurcation becomes singular, and the mathematical problem is to investigate the perturbation of a double zero eigenvalue. It appears that this problem is quite different from other double zero eigenvalue bifurcation problems, which results from the coalescence of a Hopf bifurcation point and a steady bifurcation or limit point [12], [10], [8], [6]. In [1] we analyzed the bifurcation problem for (1.1) and discovered that it eventually reduces to the study of the following equations for x and y dx dy (1.2) y+ ef(x, y, L, e), ...

... The layer thickness is constrained to lie between 5 and 100 m, the compressional speed between 1540 and 1800 m/s, the gradient between 0 and 4 s À1 , and density between 1.2 and 2.0 g cm À3 . Wavenumbers for each candidate parameter vector x were computed using the Sturm sequence eigenvalue search 23 (the method used in KRAKEN 24 ) The objective function is minimized by performing a sequential 1D optimization over each parameter, holding the others fixed at their latest optimized value. Six iterations were seen to provide good convergence. ...