An alternative method to the celebrated and well known Doppler technique of remote sensing of motion, based on the Doppler effect of frequency shifted scattered waves is investigated. The prospects of dramatic improvements in the Doppler technique beyond the present state of the art are not promising. It is therefore worthwhile to investigate alternative and complementary methods. A theoretical discussion of such a non-Doppler method is given here. The first order velocity effects in the acoustical wave equation are treated as inhomogeneous (source) terms and the solution is represented in terms of a Green function integral. Hence this solution is the first order of the Born approximation. For simple cases such as channel flows, the integral can be evaluated, either directly or by means of the stationary-phase approximation method. It is then shown that a broadband interrogating pulse (which would be quite useless for Doppler-type remote sensing) produces a scattered signal, the pulse shape of which is directly related to the velocity profile. The extraction of the velocity profile involves time integration, and hence the present method has a feature of noise reduction built into it. However, there are still many open questions regarding the implementation of the present method in the presence of spurious effects, such as turbulence, pressure and temperature gradients, etc. These problems are briefly discussed.