[ { "id": "authors:j4gae-5yv10", "collection": "authors", "collection_id": "j4gae-5yv10", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CLApof83", "type": "article", "title": "Characteristic modes and fundamental singularities of partial differential equations", "author": [ { "family_name": "Clauser", "given_name": "Francis H.", "clpid": "Clauser-F-H" } ], "abstract": "Systems of linear partial differential equations with constant coefficients, like their ordinary differential equation counterparts, can be characterized by the properties of the matrices that form the coefficients of the differential operators. The question arises: Do the matrix operators that result from partial differential equations possess eigenvalues and eigensolutions in the same way that ordinary differential matrix operators do? The answer to this question is explored in some detail using as an example the linearized flow of a viscous fluid. It is shown that eigenfactors do exist for these equations, and that, of necessity, these involve hypercomplex algebra. This fact introduces significant new features to the problem. It is shown that eigenmodes exist and that each of these has its distinctive fundamental singularity. The fluid mechanical significance of these is examined in some detail. In addition, a representative group of other partial differential equations is examined and their eigenmodes and fundamental singularities are determined. It is shown that a number of basic differences exist between the eigenfunction theory for ordinary and for partial differential equations.", "doi": "10.1063/1.864045", "issn": "0031-9171", "publisher": "Physics of Fluids", "publication": "Physics of Fluids", "publication_date": "1983-10-01", "series_number": "10", "volume": "26", "issue": "10", "pages": "2787-2800" }, { "id": "authors:qj72h-5ve47", "collection": "authors", "collection_id": "qj72h-5ve47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CLApof63", "type": "article", "title": "Concept of field modes and the behavior of the magnetohydrodynamic field", "author": [ { "family_name": "Clauser", "given_name": "Francis H.", "clpid": "Clauser-F-H" } ], "abstract": "A method for studying the behavior of fields by splitting their behavior into independent field modes is presented. The method is used to explore the characteristics of steady, two-dimensional, linearized magnetohydrodynamic fields with finite viscosity and resistivity and arbitrary orientation of the magnetic vector relative to the velocity vector.It is shown that in general boundary layers and wakes cease to exist in magnetohydrodynamics. Their place is taken by diffusing waves which, in reality, are the fields of a set of viscous-resistive sources, vortices, poles and currents whose field lines are strongly oriented along the characteristic wave directions. When the viscosity and resistivity are equal, these waves diffuse in a simple and independent way, but when these quantities are not equal, the diffusing waves generate a new kind of wake which is located, veil-like, in the fan-shaped region between the two wave directions. These wakes are fed from the differential diffusion of the primary waves. In the special case for which the resistivity is much greater than the viscosity, a new type of pseudo boundary layer is shown to exist in the velocity field. When the viscosity is much greater than the resistivity, this pseudo boundary layer occurs in the magnetic field.", "doi": "10.1063/1.1706722", "issn": "0031-9171", "publisher": "Physics of Fluids", "publication": "Physics of Fluids", "publication_date": "1963-02", "series_number": "2", "volume": "6", "issue": "2", "pages": "231-253" }, { "id": "authors:pkxcq-g0n10", "collection": "authors", "collection_id": "pkxcq-g0n10", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120925-153051814", "type": "article", "title": "The Transient Behavior of Nonlinear Systems", "author": [ { "family_name": "Clauser", "given_name": "Francis H.", "clpid": "Clauser-F-H" } ], "abstract": "It is shown that the classical perturbation procedure for treating nonlinear systems leads to solutions expressed as Fourier-like series with slowly varying coefficients. These slowly varying coefficients contain the information about the long term behavior of the system. Inconsistently, the classical perturbation procedure expresses these coefficients as power series, a mode of expression which has notoriously poor long term validity. An operational procedure is presented for treating oscillations having slowly variable amplitudes and frequencies. An extension of the usual impedance concepts is presented for expressing the frequency characteristics of both linear and nonlinear elements when oscillations with many frequencies are present simultaneously and when these oscillations vary in both frequency and amplitude. From these methods, a perturbation procedure is devised which permits the behavior of systems to be computed with any order of accuracy, using only the algebraic processes which are characteristic of operational procedures. This procedure avoids expressing its results in terms of the local time. Instead, it expresses them in terms of the fundamental characteristics of the oscillations which axe present. As a consequence, the final solutions have the much desired long term validity and they may be used to obtain asymptotic estimates of the behavior of the system. The method is able to treat systems containing nonlinear perturbing elements and elements which we have described as moderately nonlinear. By means of examples it is shown that it is a straightforward process to treat systems to second order accuracy. This level of accuracy covers a large number of the intercoupling effects that characterize the more sophisticated nonlinear phenomena.", "doi": "10.1109/TCT.1960.1086711", "issn": "0096-2007", "publisher": "Institute of Radio Engineers", "publication": "IRE Transactions on Circuit Theory", "publication_date": "1960-12", "series_number": "4", "volume": "7", "issue": "4", "pages": "446-458" } ]