An efficient time truncation algorithm for transient two-dimensional wave propagation analysis by the boundary element method is proposed. First the complete (without any truncation) formulation is reviewed and subsequently time truncation concepts are discussed. Three applications are presented, where the efficiency of complete and truncation algorithms is compared. The most important conclusions inferred from the numerical analyses are: (i) time truncation errors can be controlled and made as small as required and (ii) time truncation algorithms lead to considerable savings in both CPU time and storage area.