In this thesis, a step-by-step time marching solution is developed to solve scalar wave propagation problems in structures composed of unidimensional elements (truss elements, in particular). The method is capable of solving problems without the need for subdivision of the elements. The underlying idea is to use some pre-integration relations along with equilibrium equation. In this method, the initial conditions are satisfied precisely and the equilibrium equation is satisfied by using a time-weighted residual method. The boundary conditions at both ends of each element are simultaneously satisfied on the boundry of problem and at the end of each step time. The most important feature of this method is storing the information for each step on the coefficients of the basis functions (here exponential ones) such that the solution progresses over time just by using appropriate recursive relations for updating the coefficients and of course without the need for selecting domain points. In this study, attempts have been made, first, to explore the limitations of the available methods, i.e. the finite element method and the spectral element method, and then, to examine the efficiency of the proposed method against these two methods. For this purpose, first, axial wave propagation in the structure and its elements is studied by considering the axial stiffness only. Next, by considering the shearing stiffness alone, the formulation is extended for considering the effects of shear wave propagation. Finally, by considering the bending stiffness, combined with axial stiffness, simultaneous propagation of axial and bending waves is investigated. The results of this study may be used in the analysis and design of structures under various dynamic loads such as moving loads or those induced by impact or earthquake. Furthermore, by evaluating the precise dynamic response of the structures due to impact loads, there will be a real possibility of conveniently diagnosing defects in the structural elements. Ultimately, to demonstrate the capability of the proposed method, along with its ability in speeding up the solution process, a variety of problems is solved thorough which the claim for the accuracy of the results is supported. Key words : Step-by-Step Time Solution, Scalar Wave Propagation Equation, Exponential Basis Functions, Standard Finite Element Method, Spectral Finite Element Method, Dynamic Response.