Khelladi H., Djelouah H.

For a better understanding of the nonlinear ultrasound interactions with biological media, an analysis based on Goldberg’s number is elaborated in order to estimate the amplitude of the generated harmonics in the case of a finite amplitude plane wave that propagates in a dissipative liquid. Krassilnikov’s experimental works show a close dependence of the absorption coefficient with the intensity [1]. Because of the mathematical formulation complexity, most authors do not take into account this dependence in their numerical modeling. In this study, the absorption coefficient dependence versus the intensity is taken into account. An analysis on the validity domain of the fundamental and the second harmonic analytical expressions established on the quasi-linear approximation is elaborated while being based only on Goldberg’s number. Moreover, an error analysis is made to consider the deviations resulting from the quasi-linear approximation as compared to the numerical solution of Burgers’s equation. This investigation is based on Krassilnikov’s experimental data, which have been chosen for their precision. These experimental data concern water and glycerin that correspond respectively to a weakly dissipative liquid approaching the urine or amniotic fluid characteristics and a strongly dissipative liquid having some similarities with soft tissues. According to this investigation, the analytical expressions of the fundamental and the second harmonic analytical expressions established on the quasi-linear approximation can constitute a good approximation of the numerical solution of Burgers’ equation for a medium characterized by a Goldberg number very low as compared to the unity, otherwise the analytical expressions of the fundamental and the second harmonic already established on the quasi-linear approximation are not checked and must be redefined. [1] V. A. Krassilnikov, V. V. Shklovskaya-Kordy, L. K. Zarembo, “On the Propagation of Ultrasonic Waves of Finite Amplitude in Liquids”, J.Acoust.Soc.Am, vol 29, N°5, May, 1957, pp 642-647.