ENERGY FLUX INVARIANTS EXPRESSED IN TERMS OF COMPLEX AMPLITUDES APPEARING IN GENERAL SOLUTIONS OF nTH ORDER DIFFERENTIAL EQUATIONS

Publisher Website
Google Scholar
Abstract
Second-order linear differential equations whose coefficients are functions of the independent variable z do not possess energy flux invariants when an energy loss mechanism is present in the medium. When no such mechanism is present, the energy invariant may be expressed in terms of the squared moduli of reflection and transmission coefficients. These concepts are generalized in the present paper to give a clearer idea of the analytical principles involved in the existence and expression of such invariants. When an nth-order linear differential equation is given, we seek conditions that an energy invariant exists, more general equations being derived from a hierarchy of equations of various orders each independent of the others. We then investigate this invariant in terms of solution-amplitudes, showing that a general expression for the invariant is possible for general solutions throughout the whole range of z, and that the concept is not restricted to ranges of z that yield simple exponential functions (‘free space’ wave forms) as solutions of the differential equation.

This publication has 0 references indexed in Scilit: