Synopses & Reviews
Synopsis
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.
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