Optimization of elastic junctions with regard to transmission of wave energy

Abstract

Transmission of the energy of an incident extensional wave through an elastic junction between two uniform and collinear bars is considered. The junction consists of a finite number of uniform segments with equal transit times. We seek the optimum junction with given transit time, which maximizes the efficiency of energy transmission for given input and output bars and a given incident wave of finite duration. We use a variational approach to derive a necessary condition for optimality in the form of a non-linear system of equations for coefficients from which the characteristic impedances of the optimum junction and the corresponding efficiency of energy transmission can be determined. We show that optimum junctions for a class of incident waves with piece-wise constant amplitudes have a number of plateaus with constant characteristic impedances and equal transit times. These plateaus remain the same if the number of segments of the junction is multiplied by any integer. Finally, we give numerical examples of optimum junctions for different types of incident waves. They illustrate, e.g., that large increases in efficiency can be achieved when the incident wave and the junction have characteristic times of the same order of magnitude. They also illustrate that optimum junctions possess a certain antisymmetry.