## Surveys in Applied Mathematics by Joseph B. Keller, David W. McLaughlin, George C.

By Joseph B. Keller, David W. McLaughlin, George C. Papanicolaou

Partial differential equations play a critical position in lots of branches of technology and engineering. as a result it is very important clear up difficulties regarding them. One element of fixing a partial differential equation challenge is to teach that it's well-posed, i. e. , that it has one and just one answer, and that the answer relies consistently at the information of the matter. one other point is to procure certain quantitative information regarding the answer. the normal approach for doing this used to be to discover a illustration of the answer as a sequence or imperative of recognized certain features, after which to judge the sequence or quintessential by means of numerical or by way of asymptotic tools. the inability of this technique is that there are fairly few difficulties for which such representations are available. for this reason, the conventional procedure has been changed via equipment for direct answer of difficulties both numerically or asymptotically. this text is dedicated to a specific technique, known as the "ray method," for the asymptotic resolution of difficulties for linear partial differential equations governing wave propagation. those equations contain a parameter, corresponding to the wavelength. . \, that's small in comparison to all different lengths within the challenge. The ray strategy is used to build an asymptotic growth of the answer that is legitimate close to . . \ = zero, or equivalently for okay = 21r I A close to infinity.

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11. from the prescribed boundary conditions. Instead they can be obtained either from the solution of canonical problems or by boundary layer methods [3]. The latter methods for > 0. For some purposes, it might suffice to dealso yield the values of the termine D experimentally. In general, the diffraction coefficient depends on the local geometrical properties of M, the local values of the index of refraction, the directions of both incident and diffracted rays, and the wave number k; and it vanishes in the limit k ---+ oo.

It will be required shortly in our discussion of diffraction by smooth bodies. 236). 17. The Surface Eiconal Equation and Surface Rays In preparation for our study of diffraction by smooth bodies we consider now the initial value problem for the eiconal equation on a surface. We are concerned with a function 8 defined only on a surface S, and with initial values prescribed on a curve 44 Joseph B. Keller and Robert M. Lewis which lies on that surface. Let X = X ( r 1 , r2) be a parametric equation for the regular surface S.

Suppose that a plane wave is normally incident upon a plane screen containing an aperture, the edge of which is an arbitrary regular curve. Then the singly-diffracted rays emanating from each point of the edge lie in a plane perpendicular to the edge. The envelope of these planes is a cylinder with generators normal to the plane of the screen. This cylinder is a caustic surface of the singly-diffracted wave. The other caustic is the edge itself. The cross-section of the cylinder, formed by its intersection with the plane of the screen, is a curve.