Second-Order Correction to the Classical Doppler and Eco-Doppler Formulas

The acoustic Doppler effect has been analytically investigated in the time domain for point source (S) and receiver (R) arbitrarily moving within a 3D homogeneous and isotropic medium. A general Doppler equation was derived expanding the position vector of S in series of powers of Δt at time t, and that of R in series of powers of Δt′ at time t′, as the signal emitted by S between the times t and t + Δt is received by R between the times t′ and t′ + Δt′. In the first-order approximation (retaining only terms of first-order in Δt and Δt′), this equation yields the classical Doppler and eco-Doppler formulas. In the second-order approximation (neglecting third and higher order terms in Δt and Δt′), the analytical solution of the equation was derived and further expanded in series of powers to determine the analytical expression of the second-order correction factor to the classical Doppler and eco-Doppler formulas. This correction factor depends on both the transversal velocity and the longitudinal acceleration of S (at time t) and R (at time t′), with respect to the line direction joining the position of S at time t and that of R at time t′. The results indicate that the use of first-order approximation is fully justified in eco-Doppler ultrasound imaging, and highlight the conditions on the relevant physical parameters for which the second-order correction factor becomes considerable.