A finite element method is developed for calculation of fields excited by focused radiators and scattered by moving particles, and the associated Doppler spectra. The algorithm was tested for known cases, e.g., the field along the axis of rectangular and circular aperture transducers. In all cases the algorithm generated data which adequately approximated solutions of simple canonical problems, and results published in the literature. The known phenomenon of Doppler spectra broadening in the presence of disturbed flow is simulated here. It is shown that broadening and ripple are related to non-lineal motion of particles within the sampling volume. Moreover, the results show that Doppler spectra broadening occurs around the Doppler central frequency and that the spectrum shape depends on the particle's motional characteristics and the sampling density. The results show that broadened spectra attributed to "flow reversal" are similar to spectra resulting from a disturbed flow within the sampling volume. In Doppler systems it is necessary to derive the sign of the Doppler shift frequency, which depends on the particle's motion along the transducer's axis. This problem is usually solved by using a quadrature scheme. Accordingly, the received signal is split into two channels and multiplied by in phase and quadrature sinusoidal signals at the carrier frequency. It is shown that in the presence method, the quadrature is unnecessary. Consequently, time consuming calculations are obviated, and instead of sampling at the high frequency (RF), we deal directly with the Doppler shift frequencies. The method described in this paper can be used as a research and design tool for focused radiators incorporating different aperture shapes, and for understanding die physical phenomena associated with field calculations and particle movements in different media. The high speed of the present algorithm facilitates its future incorporation into real time systems. Moreover, the algorithm is very flexible, hence relevant parameters can be varied over a wide range.