Optimum transmission of elastic waves through joints

An elastic wave transmitting system (e.g. a nonuniform elastic rod) consisting of an input section with constant impedance, a joint with variable impedance, and an output section with constant impedance is considered. The efficiency of energy transmission (the ratio of transmitted to incident elastic wave energy) is determined and the following optimization problem is solved for two sample cases: Given the properties of the wave transmitting system and a limited duration of the incident wave, determine the shape of this wave such that the efficiency of energy transmission is maximized. In the first case (a joint with constant impedance) the optimization problem leads to a matrix eigenvalue problem. In the second case (a joint with concentrated mass) it leads to an eigenvalue problem for an integral equation. The efficiencies obtained for the optimum incident waves are compared with those obtained for rectagular incident waves an it is found that the differences are generally small. The results, which are discussed with particular reference to the transmission of elastic wave energy through drill rod joints, can also be interpreted for e.g. shallow water waves and electromagnetic waves in a transmission line.