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Excerpt from Wave Propagation in Variable Media
We shall see that there are two kinds of methods used to solve problems of wave propagation in random media. These may be called 'honest' and 'dis honest' methods respectively. In an honest method the solution is first determined for each value of w. This solution may sometimes be found exactly and explicitly, but more often it is expressed in the form of a series in some parameter, or as a sequence of iterates, or by some other approximation procedure. In the process of solving for randomness plays no role and therefore it provides no advantage. The second step in an honest method is to compute the mean value of as well as its variance and other statistics, from.the explicit expression. In this step randomness may have the helpful effect of yielding simpler expressions for the statistics of u than those for u itself.
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Excerpt from Wave Propagation in Variable Media
We shall see that there are two kinds of methods used to solve problems of wave propagation in random media. These may be called 'honest' and 'dis honest' methods respectively. In an honest method the solution is first determined for each value of w. This solution may sometimes be found exactly and explicitly, but more often it is expressed in the form of a series in some parameter, or as a sequence of iterates, or by some other approximation procedure. In the process of solving for randomness plays no role and therefore it provides no advantage. The second step in an honest method is to compute the mean value of as well as its variance and other statistics, from.the explicit expression. In this step randomness may have the helpful effect of yielding simpler expressions for the statistics of u than those for u itself.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Reseña del editor
Excerpt from Wave Propagation in Variable Media
Wave propagation is one of the means by which energy travels. 'Propagation' is the process whereby the energy moves from one region of space to an adjacent region, and 'wave' is a general term for a moving spatial distribution of energy. The matter in the region of space through which the propagation occurs is called the transmission medium. A random or stochastic medium is a family of media together with a probability distribution which specifies the probabilities of the various members of the family. Thus wave propagation in a random medium refers to propagation in each member of the family of media, together with the probability of each member. This probability, when associated with the wave motion in each medium, characterizes a random wave motion.
Mathematically a wave motion is described by a vector-valued function u(x,t) of the position vector x and the time t. As a consequence of the physical laws governing the wave motion, the function u(x,t) satisfies certain equations. Usually they are partial differential equations of hyperbolic type and often of symmetric hyperbolic type. The transmission medium is characterized by a vector-valued function n(x,t) which enter the coefficients of the equations. A wave propagation problem is that of determining a solution u(x,t) of the equations which satisfies certain auxiliary conditions. These conditions are usually initial and boundary conditions. The problem is said to be well set, well posed or properly posed if it possesses a unique solution which depends continuously, in an appropriate norm, upon the data given in the auxiliary conditions and upon the coefficients in the equation.
A random medium is characterized by a family of functions n(x,t,w) depending upon a parameter w.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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