Previous article Next article Exponential-Like Solutions of Systems of Linear Ordinary Differential EquationsHerbert B. Keller and Joseph B. KellerHerbert B. Keller and Joseph B. Kellerhttps://doi.org/10.1137/0110019PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Herbert B. Keller and , Joseph B. Keller, On systems of linear ordinary differential equations, Research Rep. No. EM-33, New York University, Washington Square College, Mathematics Research Group, 1951iii+28, Institute of Mathematical Sciences, July MR0043969 Google Scholar[2A] H. Bremmer, The propagation of electromagnetic waves through a stratified medium and its W.K.B. approximation for oblique incidence, Physica, 15 (1949), 593–608 10.1016/0031-8914(49)90116-0 MR0034705 0033.23203 CrossrefISIGoogle Scholar[2B] H. Bremmer, The W.K.B. approximation as the first term of a geometric-optical series, Comm. Pure Appl. Math., 4 (1951), 105–115 MR0044696 0043.20301 CrossrefISIGoogle Scholar[3] L. Brillouin, The B. W. K. approximation and Hill's equation. II, Quart. Appl. Math., 7 (1950), 363–380 MR0035352 0035.07701 CrossrefISIGoogle Scholar[4] Rolf Landauer, Reflections in one-dimensional wave mechanics, Physical Rev. (2), 82 (1951), 80–83 10.1103/PhysRev.82.80 MR0043576 0043.21107 CrossrefISIGoogle Scholar[5] Richard Bellman and , Robert Kalaba, Functional equations, wave propagation and invariant imbedding, J. Math. Mech., 8 (1959), 683–704 MR0107469 0090.45301 ISIGoogle Scholar[6] F. V. Atkinson, Wave propagation and the Bremmer series, J. Math. Anal. Appl., 1 (1960), 255–276 10.1016/0022-247X(60)90001-9 MR0128808 0103.05502 CrossrefGoogle Scholar[7] Herbert B. Keller, Ionospheric propagation of plane waves, Research Rep. No. EM-56, Mathematics Research Group, Washington Square College of Arts and Science, New York University, 1953iii+41, Institute of Mathematical Sciences, Aug. MR0059779 Google Scholar[8] Philip Parzen, Electromagnetic wave propagation in bounded electron beams, Quart. Appl. Math., 12 (1954), 309–312 MR0063931 0057.42803 CrossrefISIGoogle Scholar[9] I. Kay, Reflection from inhomogeneous plane stratified media I, II, Research Reports WP-1 and WP-2, Institute of Mathematical Sciences, New York University, 1958, January and March Google Scholar[10] I. Kay, Some remarks concerning the Bremmer series, J. Math. Anal. Appl., 3 (1961), 40–49 10.1016/0022-247X(61)90006-3 MR0148971 0103.30502 CrossrefGoogle Scholar[11] F. R. Gantmacher, The theory of matrices. Vol. 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959, 125–141 MR0107649 0085.01001 Google Scholar[12] K. O. Friedrichs, Recent developments in the theory of wave propagation, Lecture Notes, Institute of Mathematical Sciences, New York University, 1949–1950, 7–, Sec. 3B Google Scholar[13] I. M. Rapoport, O nekotoryh asimptotičeskih metodah v teorii differencial' nyh uravnenii˘, Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1954, 292–, (On Some Asymptotic Methods in the Theory of Differential Equations) MR0075366 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Posteriori Analysis for Iterative Solvers for Nonautonomous Evolution ProblemsSIAM/ASA Journal on Uncertainty Quantification, Vol. 3, No. 1 | 9 June 2015AbstractPDF (560 KB)Stability and Asymptotic Estimates in Nonautonomous Linear Differential SystemsSIAM Journal on Mathematical Analysis, Vol. 16, No. 1 | 17 July 2006AbstractPDF (2208 KB) Volume 10, Issue 2| 1962Journal of the Society for Industrial and Applied Mathematics229-393 History Submitted:21 August 1961Published online:13 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0110019Article page range:pp. 246-259ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

for sufficiently regular curves (1), has a non-trivial solution if and only if X2 has one of an infinite sequence of positive real values, the eigenvalues. In many problems of mathematical physics (e.g. waveguides and vibrating membranes) the smallest such value, X2, called the principal eigenvalue, is of importance.")" In the present paper a method for approximating the principal eigenvalue is considered. The principal eigenvalue of any domain lies between the principal eigenvalues of the inscribed and circumscribed circles, as is well known. If the principal eigenvalue of the circle of equal area is used instead of that of the circumscribed circle, the same statement applies, as is also well known. The latter statement is sharper in that the equal area circle furnishes a better (i.e., greater) lower bound than the circumscribed circle. In this paper we attempt to improve the lower bound, that is, to get a still greater lower bound. We do this by introducing a family of functional of the domain which depend monotonically upon a parameter a. For a = — co, 2, + °° this functional yields the principal eigenvalues of the inscribed, equal area, and circumscribed circles respectively. We then conjecture that there exist values of a which furnish better lower bounds than does the equal circle. For almost circular boundaries we find that a = —1.779 • • • is the best such value. It seems probable that when a = —2, in which case the functional is quite simple, a good approximation which is not necessarily a lower bound, is obtained for X2. As an example of the utility of the result, the exact eigenvalues of a rectangle with arbitrary ratio of sides is compared with the approximate expression, given by an integral, for various values of a. The agreement is within 3% for all rectangles when a = — 1 and even better for nearly square rectangles when a = — 2. 2. The a-mean radius.** The boundary (1) is given parametrically in terms of rectangular coordinates with origin at the pole and x-axis on the ray 0 = 0 by

By an extension of ordinary geometrical optics (or acoustics) the intensity of the reflected and transmitted fields due to a point source in the presence of an arbitrary interface between two media is found. Particular consequences of the solution are the general lens and mirror law and the equations for the caustic surfaces.

A periodic point source in medium 1 is at the center of a spherical shell of medium 2 which is surrounded by medium 3. The densities, sound speeds, and shell radii are arbitrary. For this case an exact, explicit expression for the sound pressure is obtained. If the shell radii become infinite but the thickness remains constant, and if media 1 and 3 are the same, the solution reduces to that of Rayleigh for plane waves normally incident on a flat plate. If the shell radii become infinite and if media 2 and 3 are the same, the solution yields the acoustic Fresnel formulae for normal incidence of a plane wave on a plane interface between two half-infinite media. For finite radii, with media 1 and 3 identical and the shell thickness small compared to a wave-length, a simpler form for the solution is obtained. This solution is found to agree with the approximate solution of H. Primakoff and J. B. Keller for the sound fields reflected and transmitted by a thin shell of any shape, when their result is specialized to the present case.