Distortionless wave propagation in inhomogeneous media and transmission lines

Abstract

Abstract.Of concern are mechanical or electrical waves in a media which may be nonuniform and dissipative. The problem posed is to find conditions for the undistorted propagation of signals. The electrical transmission line is chosen as the general model. Along the length of the transmission line there are four functions which may be prescribed essentially arbitrarily. These are series resistance, series inductance, shunt conductance, and shunt capacitance. A differential equation is derived relating these functions which gives a necessary and sufficient requisite for distortionless transmission of a voltage wave. Various corollaries of this theorem are developed. For instance, it is shown that simultaneous voltage and current waves can be transmitted without distortion if and only if the characteristic impedance of the transmission line is positive at each point.

Transient analysis and simulation of nonuniform lossy transmission lines

“…This requires the fulfillment of the following equations: or equivalently, From the last equations, obtained in a much more complicated manner in ref. 13, it is evident that…”

Radial Wave Propagation and Distortionless Signalling

“…n= andM=O, or B. n is an odd integer, M (n 1)/2 and ak(r) ckr 2+k-n where c are constants, or C. the solution is nonelementary built out of solutions of the form given in (4). (2) In the above for all reasonable f means that land its derivatives f, f f are sufficiently independent for the following proof to work. One possible precise assumption is to assume that (2) is a solution for all C functions f Proof.…”

“…Suppose there is an integer M and sufficiently smooth functions (r), ao(r), ..., at(r), with/(0) 0,/ '(r) > 0 such that for all reasonable functionsf(t) the function M (2) u(r, t)= ak(r)f(t--(r)) k=0 is a solution of (1). Then a person at a distance ro from the origin can reconstruct the function f by solving a linear ordinary differential equation, provided initial conditions on fare specified.…”

“…Explicit solutions of the form (2) in dimensions 5 and 7 are given in [5], where a general technique for generating solutions of the form (2) in all odd dimensions is given; see also [1 ]. The case M 0 for the general transmission line was solved in [2]. Received If we have an integer M and functions 6, a0, ..., such that (2) is a solution of (1) then we may generate other solutions of (l) of the same form.…”

Average acoustic pulse dispersion length in condensed matter channels

“…stress pulse as it propagates through the channel. But, distortion free attenuation is also possible -see references [2][3][4]9]. .…”