The synthesis of two types of Localized Wave (L W) pulses is considered; these are the 'Focus Wave Model (FWM) pulse and the X Wave pulse. First, we introduce the modified bidirectional representation where one can select new basis functions resulting in different representations for a solution to the scalar wave equation. Through this new representation, we find a new class of focused X Waves which can be extremely localized. The modified bidirectional decomposition is applied to the nonhomogeneous scalar wave equation, resulting in moving sources generating L W pulses. In this work, we also address the possibility of exciting L W pulses from dynamic apertures, or apertures the effective radius of which is varied with time. Ideal L W pulses cannot be realized because they require infinite time excitation. However, in the case of finite L W pulses, the aperture of excitation is finite and is varied from a time - T to T. We show that the resulting L W pulses are more resistant to decay than classical monochromatic Gaussian pulses occupying the same beam waist. Both types of finite L W pulses, such as the FWM and X Wave pulse, can propagate without significant decay to much greater distances than classical monochromatic pulses. This desirable behavior is attributed to the superior aperture efficiency of the L W pulses, which in turn is attributed to their unique spectral structure. / Ph. D.