Reflection, transmission and dissipation of the energy of an incident extensional wave at a linearly viscoelastic junction between two uniform and collinear linearly elastic bars are considered. The junction consists of a finite number of uniform segments of the same material and length. The optimum shape of a junction with given material, length and number of segments which maximizes the energy transmission for given input and output bars and a given incident wave of finite duration is determined numerically with the use of a quasi-Newton method. Results are presented for rectangular incident waves of different durations and 40-segment junctions of standard linear solid material. In the special case of linearly elastic material, the optimum junctions have piece-wise constant characteristic impedances with a certain number of plateaux of equal lengths. These plateaux are independent of the number of segments provided that this number is an integral multiple of the number of plateaux. The optimum viscoelastic junctions have the appearance of deformed and displaced versions of their elastic counterparts. Thus, the plateaux of the elastic junctions are increasingly deformed and displaced with increased damping and, less markedly, with decreased response time of the material. The transitions between these plateaux of a junction appear to be discontinuous, similarly as in the case of elastic material. The apparent discontinuities become less notable with increased damping of the material.