Reaction-diffusion equations occur in the description of chemical reactions of reactants which are distributed in space. The motion of a surface along its normal, with a velocity proportional to its mean curvature, is called "flow by curvature" by geometers. Their results on such flows can be applied to the motion of fronts. Those results are primarily for motions in unbounded domains, and the strongest results are for curves embedded in the plane. One part of that lecture concerned the stability of crystals that grow or evaporate by step propagation. The step mechanism of crystal growth was proposed by W. K. Burton, N. Cabrera and F. C. Frank in 1951. It leads to a Stefan-like problem for the motion of a step on a crystal surface. This problem has solutions in which a single step moves at constant velocity, and other solutions in which an infinite number of equally spaced parallel steps move at constant velocity.

A common problem in engineering and science is to derive simple equations governing complicated phenomena. Often complicated governing equations are known, but they are too difficult to analyze. An example of a simplified equation is Darcy’s law, which describes flow of a viscous fluid through a porous medium. The more complicated equation for the same phenomenon is the Navier-Stokes equation. As an example of a general method for simplifying equations, This chapter shows how to derive Darcy’s law from the Navier-Stokes equation. Simplified equations are often called “homogenized equations,” and the procedure of replacing the original equations by them is often called “homogenization.” The chapter discusses the two-space method for deriving simplified equations by using an example of the flow of a compressible viscous fluid through a rigid porous medium.

PreviousNext No AccessClassical and Modern Diffraction Theory4. Foundations of Modern Diffraction TheoryAuthors: John CoulsonG. G. BecknellN. G. Van KampenR. N. BuchalJ. B. KellerRobert G. KouyoumjianPrabhakar H. PathakK. D. Klem-MusatovA. M. AizenbergM. BerteroP. BoccacciM. PianaJohn Coulson, G. G. Becknell, N. G. Van Kampen, R. N. Buchal, J. B. Keller, Robert G. Kouyoumjian, Prabhakar H. Pathak, K. D. Klem-Musatov, A. M. Aizenberg, M. Bertero, P. Boccacci, and M. Pianahttps://doi.org/10.1190/1.9781560803232.ch4 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract Chapter 4 addresses the cornerstones of modern theory. The use of high-frequency asymptotic approximations of the wavefield is a common theme in this chapter. The centerpiece of Chapter 4 is the geometrical theory of diffraction. This chapter includes defining works by Keller and others, along with the experimental results by Becknell and Coulson, which inspired Keller's theory. Permalink: https://doi.org/10.1190/1.9781560803232.ch4FiguresReferencesRelatedDetails Classical and Modern Diffraction TheoryISBN (print):978-1-56080-322-5ISBN (online):978-1-56080-323-2Copyright: 2016 Pages: 352 publication data© 2016 All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher.Publisher:Society of Exploration Geophysicists HistoryPublished in print: 01 Jan 2016 CITATION INFORMATION John Coulson, G. G. Becknell, N. G. Van Kampen, R. N. Buchal, J. B. Keller, Robert G. Kouyoumjian, Prabhakar H. Pathak, K. D. Klem-Musatov, A. M. Aizenberg, M. Bertero, P. Boccacci, and M. Piana, (2016), "4. Foundations of Modern Diffraction Theory," Geophysics Reprints Series : 213-301. https://doi.org/10.1190/1.9781560803232.ch4 Plain-Language Summary PDF DownloadLoading ...

More than 70 ways to show resilienceRuben Dahm and his colleagues call for greater "flood resilience" in delta cities (

Previously we studied the two-dimensional problem of planing of a flat plate on the free surface of an incompressible inviscid fluid of infinite depth. We assumed that the Froude number U /(gL)(1/2) was large, with U the horizontal velocity of the plate, L the length of the plate, and g the acceleration of gravity. We solved this problem by the method of matched asymptotic expansions, with gLU(-2) as the small parameter. The solution contained a jet flowing up along the forward-facing surface of the plate. When the jet reached the upper edge of the plate, it became a free jet that followed a parabolic path and fell back onto the free surface. Now we consider the case where the jet falls off the plate before it reaches the upper edge, and we determine where it leaves the plate.

The water waves produced by a partially submerged oscillating body are calculated when their wavelength is small compared with the dimensions of the body. The calculation involves ray methods, the 'double body' flow produced by the body motion and the 2D solution for waves on a sloping beach. It is based upon the ray method, which was used by Hermans (1973, A perturbation method for the radiation of short surface waves in three dimensions. J. Engrg Math., 7, 75-84.) for bodies vertical at the waterline, and upon Holford's (1965, On the generation of short two-dimensional surface waves by oscillating surface cylinders of arbitrary cross-section. Technical Report, No. 4. Dept. Math., Stanford University.) method for cylinders oblique at the waterline.

A jogger's ponytail sways from side to side as the jogger runs, although her head does not move from side to side. The jogger's head just moves up and down, forcing the ponytail to do so also. We show in two ways that this vertical motion is unstable to lateral perturbations. First we treat the ponytail as a rigid pendulum, and then we treat it as a flexible string; in each case, it is hanging from a support which is moving up and down periodically, and we solve the linear equation for small lateral oscillation. The angular displacement of the pendulum and the amplitude of each mode of the string satisfy Hill's equation. This equation has solutions which grow exponentially in time when the natural frequency of the pendulum, or that of a mode of the string, is close to an integer multiple of half the frequency of oscillation of the support. Then the vertical motion is unstable, and the ponytail sways.

A method is presented for evaluating authors on the basis of citations. It assigns to each author a citation score which depends upon the number of times he is cited, and upon the scores of the citers. The scores are found to be the components of an eigenvector of a normalized citation matrix. The same method can be applied to citation of journals by other journals, to evaluating teams in a league [1], etc.

The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each corresponding codimension is the number of conditions which a matrix of the given type must satisfy in order to have the specified multiplicities.

The probability density of the resistance of a two dimensional rectangular network between two conducting plates is calculated. The nodes form an $M$ by $N$ lattice, and each edge has a random resistance. The Monte Carlo method is used.

Stirling's formula, the asymptotic expansion of n! for n large, or of Γ(z) for z→∞, is derived directly from the recursion equation Γ(z+1) =z Γ(s) and the normalization condition Γ (1/2) =√(π).

We report the experimental observation of a well-defined rippling of the air cavity entrained by a rapidly moving solid object entering the free surface of a liquid (water or ethanol). The ripples are fixed in the lab frame, and begin just after the pinch-off (deep seal) of the cavity, simultaneous with the acoustic emission. This acoustic resonance corresponds approximately to the Minnaert frequency for volume oscillations of the bubble. We present an irrotational model which explains the ripples as a spatial rectification of these volume oscillations by the surface of the moving object.