The modal-synthesis based approach is employed for obtaining the higher-order dynamic 1D finite elements with enhanced convergence properties. The obtained dynamic models are used for modelling short transient waves and wave pulses propagating in elastic or acoustic environments by using rough space steps. Only few nodal points per pulse length are enough for obtaining reasonable simulation results. The main advantage of the createdelements compared with earlier similar approach lies in usage of the diagonal mass matrix of the element. The elements are compatible with conventional finite elements, may be used in branchy non-homogenous structures, as well as enable to implement non-reflecting boundary conditions. The saving the computational resources is estimated.Keywords: wave propagation; finite elements; modal synthesis; phase velocity error.