Some fundamental transmission properties of impedance transitions (Q2265850)

The authors consider the reflection and transmission of waves in a one- dimensional lossless medium, using a set of equations abstracted from the problem of the propagation of extensional waves in a linearly elastic bar. The system treated is \(f_ x=Z(x)v_ t\), \(v_ x=(1/Z(x))f_ t\). Here f and v are functions of x and t, and Z(x) is the characteristic impedance of the medium; subscripts denote partial differentiation. The class of functions treated here are those for which \(Z(x)=Z_{IN}\) (a constant), for x non-positive, \(Z(x)=Z_{OUT}\) (also constant) for x greater than a positive constant d, and which are piecewise constant on N subintervals of equal length d/N in (0,d). The restriction of Z(x) to [0,d] is termed an impedance transition. Use of the Fourier transform on the time variable t converts problems for impedance transitions into questions concerning functions of a complex variable; the authors exploit this method to explore the transmission of momentum and energy through an impedance transition, and to determine conditions for transitions to be equivalent in their transmission properties. In many cases the proofs are sketched or omitted, reference being made to a technical report by the same authors; examples are given. Typical results are: (1) Relative momentum transmission depends only on \(Z_{IN}/Z_{OUT}\), not on the detailed behavior of Z(x). (2) There are at most, and generally exactly, \(2^ N\) impedance transitions with the same transmission properties as a given transition. (3) For a given incident wave with finite duration and energy, there is an optimal impedance transition which maximizes the efficiency of energy transmission.