Special effects accompanying the wave propagation through material geometries in space-time are analyzed by dynamic (spatio-temporal) laminates for screening the extended domains. This renewed look has revealed a conceptually new mechanism of relaxation of material optimization problems in dynamics, which has led to additional resources for optimization previously concealed in the property layouts.ĭynamic materials are studied in this book from the following perspectives: ability to appear in dissimilar implementations, universality as formations that are thermodynamically open, and unusual effects supported by dynamic materials in mechanical and electromagnetic implementations. Since the release of the first edition, a number of new results have created a more complete picture of unusual effects hidden in spatio-temporal material geometry. Systems with one spatial coordinate and time are used to illustrate essentials of temporal property change in this setting and prompt forthcoming extensions and technical improvements. This new edition emphasizes the differences between material optimization techniques in statics and dynamics. Mathematical treatment to properties of dynamic materials, material substances whose properties are variable in space and time are examined in this book. It will also be useful for researchers in the field of smart metamaterials and their applications to optimal material design in dynamics. This book is intended for applied mathematicians interested in optimal problems of material design for systems governed by hyperbolic differential equations. Some unusual applications are listed along with the discussion of some typical optimization problems in space-time via dynamic materials. The book discusses some general features of dynamic materials as thermodynamically open systems it gives their adequate tensor description in the context of Maxwell’s theory of moving dielectrics and makes a special emphasis on the theoretical analysis of spatio-temporal material composites (such as laminates and checkerboard structures).
Of special significance is the participation of dynamic materials in almost every optimal material design in dynamics. These materials have diverse applications: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth technology, to name a few. Unlike conventional composites that are often found in nature, dynamic materials are mostly the products of modern technology developed to maintain the most effective control over dynamic processes. This book gives a mathematical treatment of a novel concept in material science that characterizes the properties of dynamic materials-that is, material substances whose properties are variable in space and time. We conclude with a class project for students to derive a generalization of Snell's law of refraction for a particular type of a moving interface.
We demonstrate that incident concentric spherical waves reflect as non-concentric spherical waves, and thereby derive the Doppler effect formula. In this paper, we develop a new geometric approach which shows that the relativistic reflection from a plane mirror is equivalent to the ordinary reflection from a certain hyperbolic mirror, and that the incident and reflected rays are the focal rays of the hyperbola.
While modern derivations rely on Lorentz transformations or Maxwell's equations, the original derivation was an elegant application of geometric optics.
For a moving mirror, the corresponding law is known as the relativistic reflection law however, it was first inferred prior to relativity. Euclid's law of reflection, r = i, rests on three assumptions: Fermat's principle of least time, constancy of speed of light, and that the mirror is stationary.