Non-separable Problems Research Articles

Long-time dynamic response of linearly elastic non-self-adjoint and non-separable systems: A green function approach

The solution of non-self-adjoint and non-separable initial-boundary-value problems is treated by using a Green function approach. Specifically, the analysis of the long-time response of a finite, isotropic, homogeneous, linearly elastic cantilever plate, in a state of plane strain to an antisymmetric and dynamically applied distributed surface load, has been shown to constitute such a non-self-adjoint and non-separable problem. The Green function of the problem is determined, based on the method given by J. Miklowitz for solving non-separable waveguide problems, by using a double Laplace transform and an entirety condition on the solution. The Green functions for two near-field and far-field domains are obtained. Hence, the response of the plate to any antisymmetric dynamically applied distributed surface load is determined in the form of an integral equation for the two domains, with the respective Green functions as the kernels of integration. It is concluded that the “elementary” theories and the “engineering methods” in which a “dynamic load factor” is applied to the static solution tend to underestimate the maximum values of the plate deflections in the vicinity of the free end of the cantilever plate.

Comparative analysis of pinch and operating line methods for heat and power integration

Two methods for approaching the problem of the optimal synthesis of heat-and-power cogeneration systems from a thermodynamic point of view are compared. The first one is the pinch method (Townsend and Linnhoff, 1983, A.I.Ch.E. J. 29, 742–771) which is based on the pinch properties. The second one is the operating line method (Irazoqui, 1986, Chem. Engng Sci. 41, 1243–1255) which considers the heat-and-power cogeneration as a unique, nonseparable problem. Different results for the appropriate placement of power-generating cycles can be derived by applying either method, particularly in the case of a “nose”-shaped grand composite curve. The efficiencies obtained with the operating line method are higher than those calculated with the pinch method, when both are applied to a dominant power load case.

A spectral view of wave propagation

Wave propagation in environments characterized by weakly nonseparable conditions can be described by adiabatic modification of spectra pertaining to the unperturbed separable case. Recent developments are reviewed here, with emphasis on approximate adiabatic transform relations and adiabatic invariants, which permit reduction of the direct weakly nonseparable wave problem to spectral form and, conversely, reconstruction of wave spectra from measured data.

Particle in an equilateral triangle: Exact solution of a nonseparable problem

Particle in an equilateral triangle: Exact solution of a nonseparable problem

A class of nonseparable dynamic programming problems

A solution procedure is proposed for a class of deterministic sequential decision problems whose objective functions are of the form Σfn(xn)+?(Σgn(xn)) where ? is differentiable and either concave or convex. The procedure calls for the collaboration between dynamic programming and c-programming, and is demonstrated in our treatment of a minimum variance type problem.

Coupled mode theory of intrinsic modes in a wedge

Recent theoretical studies of acoustic wave propagation in a model waveguide consisting of a homogeneous wedge with one reflecting and one penetrable boundary have established the utility of the concept of an intrinsic mode, which generalizes for nonseparable problems the normal mode of separable configurations. In this paper, it is shown by explicit calculations on the same model problem that direct orthogonal expansion (in local normal modes) of the intrinsic mode produces the same expansion coefficients as those obtained by a perturbation analysis of the coupled mode equations. The perturbation method used is that of renormalization, in which the mode coupling operator is iteratively diagonalized up to a certain order in the nonseparability parameter, which, in this case, is the wedge angle α.

Generalised power series solutions for non-separable wave equations

Three-dimensional non-separable problems of wave mechanics are solved in terms of generalised power series solutions. Non-central interactions are shown to produce, in general, logarithmic terms in regular wavefunctions. A necessary and sufficient criterion for the appearance of logarithmic terms is derived.

Ray invariants, plane wave spectra, and adiabatic modes for tapered dielectric waveguides

In nonseparable problems resulting from the analysis of wave propagation in longitudinally varying waveguides, such as a wedge‐shaped taper, singularities appear in both ray and coupled mode treatments at the local normal mode cutoff transition. A uniformization of the local normal (adiabatic) mode is proposed, using plane wave spectra, which effectively resolves this difficulty.

Long-time response of finite cantilever plates to dynamic surface loadings

This paper is concerned with the long-time analysis of the response of a finite, isotropic, homogeneous, elastic cantilever plate to different dynamic surface loadings while the plate is assumed to be in a state of plane strain. This is an extension to the solution of the antisymmetric loading case given by the authors [1]. The analysis is based on a method given by J. Miklowitz for solving non-separable waveguide problems in which an entirety condition is used on the solution. The effect of the material properties on the stress singularities at the corners of the fixed end is considered for a calculation of the singularity exponent for a realistic engineering material with its corresponding Poisson ratio being taken into account. Moreover, a change of the loading function is considered in the general solution of the problem. It is concluded that, for the finite plate, the results calculated for antisymmetric loading are good approximations to the unsymmetric case. This work may be considered as a step towards the assessment of the importance of stress wave propagations in finite plates, as such waveguides have some practical significance in structural engineering.

Long-time response of a finite cantilever plate to antisymmetric dynamic surface loading

This paper treats the transient response of a finite, isotropic, homogenous, elastic cantilever plate in a state of plane strain to an antisymmetric surface line load, which is assumed to be a step function in time. This analysis is based on the method given by J. Miklowitz for solving non-separable waveguide problems by using a double Laplace transform and an entirety condition on the solution. The corner stress singularities are considered in the evaluation of the stress distribution at the fixed end. Then, the near field solution is found by means of asymptotic expansion. Moreover, the transverse displacement along the plate is obtained in the travelling wave form as well as in the vibrational one, which is evaluated numerically and discussed. It is concluded that the engineering methods in which a “dynamic load factor” in used in conjunction with the static solution tend to underestimate the values of the deflections beyond the point of load up to the free end of the cantilever plate.

Rays and local modes in a wedge-shaped ocean

Conventional normal mode theory cannot be applied to the nonseparable problem of wave propagation in an ocean with sloping bottom. For small bottom slopes, the sound field may be expressed approximately in terms of adiabatic modes, but this description fails when a mode propagating upslope passes through cutoff. An alternative solution by ray acoustics, valid at high frequencies, contains many multiply reflected contributions, and also undergoes difficult trapped to leaky ray transitions upslope. To address these transition problems, the conventional ray solution is used here as a convenient starting point for collective treatment of ray fields and their conversion into local modes. First, the ray fields are generalized by associating with each ray trajectory from source to observer a bundle of local plane waves that is multiply reflected between the boundaries and remains valid also in the transition regions. When the generalized ray series is subjected to Poisson summation and subsequent asymptotics on the transform integrals, it is found to furnish local modes which coincide with the conventional adiabatic modes where these exist. The local mode integrals also yield transition functions which smoothly continue an originally trapped adiabatic mode through cutoff to the leaky regime. The transition behavior agrees with that found by Pierce [J. Acoust. Soc. Am. 72, 523–531 (1982)] by an entirely different approach, and with that predicted by the spectral Green’s function method of Kamel and Felsen [J. Acoust. Soc. Am. 73 xxx–xxx (1983)]. The ultilization of rays and local modes in each step of the analysis here grants physical insight and therefore clarifies the mechanism of propagation in this range-dependent environment.

Wavefront Analysis in the Nonseparable Elastodynamic Quarter-Plane Problems, I—Part 1: The General Method

This work concerns the transient response in elastodynamic quarter-plane problems. In particular, the work focuses on the nonseparable problems, developing and using a new method to solve basic quarter-plane cases involving nonmixed edge conditions. Of initial interest is the classical case in which a uniform step pressure is applied to one edge, along with zero shear stress, while the other edge is traction free. The new method of solution is related to earlier work on nonseparable wavequide (semi-infinite layer) problems in which long time, low-frequency response was the objective. The basic idea in the earlier work was the exploitation of a solution boundedness condition in the form of an infinite set of integral equations. These equations were generated by eliminating the residues at an infinite set of poles stemming from the zeros of the Rayleigh-Lamb frequency equation. These poles were associated with exponentially unbounded contributions. The solution of the integral equations determined the unknown edge conditions in the problem, hence giving the long time solution. In the new method for the quarter-plane problems the analog of the Rayleigh-Lamb frequency equations is the Rayleigh function that yields just three zeros, i.e., say the squares of cR1, cR2, and cR3 speeds for the Rayleigh surface waves. Of these six roots (and corresponding poles) only one is physical, i.e., cR1. The other five are eliminated since they correspond to: (1) exponentially unbounded waves, or (2) waves with speeds cR2 > cd, cR3 > cd, all inadmissible. They analogously (to the layer case) give four integral equations for the edge unknowns, hence the formal solution for the problems. To date the method has been successful in obtaining all the wavefront expansions for the uniform step pressure case mentioned earlier. The dilatational and equivoluminal wavefronts were obtained for the interior and both edges and the Rayleigh wavefronts for both edges. This asymptotic solution has been verified and hence establishes the credibility of the new method.

Resolvent formulation of the Green's function for a wedge-shaped ocean

By the method of resolvents or characteristic Green's functions [L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N J, 1973), Chap. 3], one may construct general contour integral representations of Green's functions for separable range-independent and range-dependent ocean profiles. From these general integral representations, one may derive a variety of alternative sound field representations, including those involving normal (discrete and continuous) modes, leaky modes, conventional and generalized rays, hybrid ray-mode combinations, etc. The method cannot be applied rigorously to nonseparable problems such as a wedge-shaped ocean with penetrable bottom. However, for small bottom slopes, we have managed to construct an approximate characteristic Green's function integral based on the ray-cycle invariant and on the requirement that the resulting normal mode representation arising from residues at the resonance poles yields the properly normalized and symmetrized trapped adiabatic modes downslope from their cutoff points. For upslope propagation, as a mode approaches cutoff, its resonance pole approaches a branch point and therefore couples strongly to its radiation field. In this transition region, the Green's function may be reduced asymptotically to a canonical integral which is discussed in detail. Beyond the transition region, the previously trapped mode pole becomes a leaky mode pole. The resulting formulation is compared with recent results obtained by other techniques (A.D. Pierce, J. Acoust. Soc. Am., to be published; J. M. Arnold and L. B. Felsen, this meeting). It is also shown how ray, ray-mode, and other relevant field representations can be obtained from the original integral. [Work supported by ONR Ocean Acoustics Branch.]

Consistent coupled mode theory of sound propagation for a class of nonseparable problems

This article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of A. D. Pierce [J. Acoust. Soc. Am. 37, 19–27 (1965)] and D. M. Milder [J. Acoust. Soc. Am. 46, 1259–1263 (1969)], are applied to propagation problems involving range dependent boundaries. This boundary condition approximation requires that the depth derivative, rather than the normal derivative of the field, be continuous across a sloping interface. The approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory. This article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy. It is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope. Moreover, the corrections are not prohibitive in terms of added computational complexity. Numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory.

Multichannel quantization for molecular systems. I. Bound states studies

The usefulness of the Fox—Goodwin propagator for the search of the bound levels in a multichannel situation is demonstrated. Applications to perturbations in diatomic molecules and to a two-mode non-separable problem show that the results are comparable in accuracy with those obtained in a one-channel situation.

A variance‐constrained reservoir control problem

A dynamic programing solution procedure is developed for a class of variance‐constrained reservoir control problems. The variance constraint is incorporated as a penalty term in a nonseparable Lagrangian problem which is solved by a two‐stage procedure. First, the difficulty associated with the nonseparability of the problem is resolved by the consideration of a modified simpler separable problem which is solved by a simple additive dynamic programing recursion. Then the optimal solution to the nonseparable problem is found by a simple search algorithm. The entire discussion is devoted to the mathematical aspects of the problem and no consideration is given to the applicability of variance constraints as far as water resources management problems are concerned. The procedure is applied to a simple numerical example.

Classical limit of a nonseparable problem with exchange: Comparison of classical and quantum mechanics

Classical limit of a nonseparable problem with exchange: Comparison of classical and quantum mechanics

Multipole wavefunctions for photoelectrons in crystals. IV. The irregular functions and the matching to an impurity

Extension of the method of partial waves to scattering by an impurity center in a crystal requires the construction of energy eigenfunctions of the unperturbed crystal that are irregular at the center. These irregular solutions of a nonseparable problem are constructed by utilizing the expansion of the Green’s function into the regular solutions symmetry-adapted about the impurity center, which have been obtained earlier. The wavefunction within an impurity can thus be continued throughout the surrounding crystal by matching it at the impurity boundary to a superposition of regular and irregular solutions. This procedure is compared to the current EXAFS treatment of photoabsorption.

Optimal Reliability Allocation by Branch-and-Bound Technique

The paper presents an efficient method for finding the exact optimal solutions of reliability allocation problems that are formulated as an integer nonlinear programming problem generalized to handle nonlinear constraints and nonseparable problems. The method is based on branch-and-bound and developed by considering separation and relaxation techniques.

Integral equation approach to the problem of scattering by a number of fixed scatterers

In the derivation of the Lippmann–Schwinger integral equation for scattering of a wave ψ(r) by the potential ν(r), one constructs the Green's function for the operator [Formula: see text], and treats νψ as the inhomogeneous term. However, in certain cases, it is desirable to formulate the scattering problem in terms of an integral equation by obtaining the Green's function for the operator [Formula: see text], and by considering (−k2ψ) as the inhomogeneous term. An important aspect of this formulation is that the resulting integral equation can be used to generate a low energy expansion of the wave function for some separable and nonseparable systems. For two-dimensional scattering, if the geometry of the scatterers is simple enough, the Laplace equation with the prescribed boundary conditions on the surface of the scatterers is separable in a certain coordinate system, then one can write the solution of the wave equation as an inhomogeneous integral equation. In this way the problems of scattering by two cylinders, an array of cylinders, and a grating can be formulated in terms of integral equations. For three-dimensional scattering, one can consider either the spherically symmetric cases or nonseparable problems. In the former case, for certain types of force laws, a Volterra integral equation in one variable can be found for the wave function. In the latter case, integral equations in two or three variables can be obtained for scattering by two spheres or by a torus.