Real and Complex Doppler Effects in Lossy Media

The theory of the Doppler effect in the presence of lossy media and moving scatterers is investigated. Essentially two kinds of phenomena emerge, which are not distinguishable in the conventional case of lossless media: 1) when the scatterers move uniformly, or the equation of motion involves a slowly varying velocity, it is shown that propagation in lossy media involves complex Doppler effects, i.e., the spatial attenuation of the medium is transformed into the temporal behavior of the received signal, yielding complex Doppler shift frequencies for a real excitation frequency. This phenomenon is closely related to the question of remote-sensing the velocity by means of scattering of waves from moving objects. The complications introduced by the received complex frequency signal are discussed. It is shown that spectrum broadening occurs, and that in certain cases, by judiciously chosen temporal filters the spectral degradation of the received signal can be counteracted. The second class of phenomena is associated with periodic and harmonic motion, when the scatterer is vibrating around a fixed location. In this case the incident wave is frequency modulated by the moving scatterer, giving rise to sidebands at real frequencies. It is shown that in this case the spectral contents of the scattered signal is identical for lossless and lossy media, provided the moving object remains in the vicinity of some location. It is argued that the transformation from the spatial to the temporal regime, caused by the Doppler effect, facilitates the assessment of the medium attenuation from the attributes of the received Doppler shifted temporal signal. Numerical simulations were performed to support and display the aforementioned theoretical arguments.