Use of an analytic signal to model interaction between an acoustic wave and a moving target with a time-dependent velocity

The formalism of the analytic signal and its complex envelope is used to model the interaction between an acoustic wave and a moving target in applications involving the Doppler effect. Specifically, it is shown that modeling by means of analytic signal and related concepts is suitable for the interaction between a probing acoustic wave and the scatters included in the investigated medium through which the wave propagates. When such a scatterer is a moving target, the interaction can be viewed as an angular modulation: phase modulation or frequency modulation, depending on which parameter is used as the carried message. The act of demodulating by means of a two-channel lock-in amplifier follows this theoretical modeling closely because this device simply extracts the complex envelope from the modulated signal. In the case of a target with time-dependent velocity, this modeling is very useful because the experimental results are actually the instantaneous phase and the instantaneous frequency of the complex envelope associated with the modulated signal. In this sense the physical phenomenon of interaction between an acoustical wave and a moving target (Doppler effect) can be depicted easily and clearly. Experimental results obtained with a continuous probing wave interacting with a single bubble moving in a fluid at rest are given to illustrate the proposed approach.