Generalized Doppler effect: Coherent and incoherent spectra

The generalized Doppler effect spectrum produced by particles moving on arbitrary trajectories is analyzed. Scattering by a single particle moving in the farfield of a transmitter and a receiver is considered. It is shown that the Doppler spectrum exciting the particle is due to two factors: The longitudinal (or radial) component of the motion produces a spectrum similar to the conventional Doppler effect, by affecting the phase of the excitation wave. In addition, spectral effects are produced by the particle moving transversely (or angularly) through the spatially modulated radiation pattern of the transmitter. By virtue of the reciprocity properties of transmitters and receivers, each spectral component of the excitation signal again gives rise to spectra induced by the longitudinal and the transversal components of the motion relative to the receiver. The combined Doppler spectrum observed at the receiving transducer or antenna output is a convolution of all four spectra. The behavior of an ensemble of scattering particles is analyzed. The statistics are found by defining the particle and/or the trajectory parameters as random variables. Presently, it is assumed that the particles are identical, their positions are uncorrelated, and multiple scattering is ignored. It is shown that the interference of scattered waves from various particles, manifested in the coherent radiation, gives rise to a spectrum that might be different from that of a single particle. In special cases, the combined spectrum degenerates into a single frequency, and when this is the transmitter's frequency, the Doppler effect completely disappears. This explains the fact that when we have moving media, but the boundary surfaces are at rest, there is no Doppler effect. The results of the present analysis contribute to our understanding of Doppler velocimetry methods using ultrasound or laser radiation in medical and industrial instrumentation.