INVISCID BURGERS’ EQUATION

1. INVISCID BURGERS’ EQUATION •  1) sound speed in nonlinear acoustics • 2) nonlinear distorsion of the wave profile • 3) weak shock theory • 4) the acoustical potential and the numerical solver

2. Linear and nonlinear sound speed Linear sound speed Nonlinear sound speed

3. Nonlinear distorsion of the wave profile

4. Inviscid Burgers’ equation 1D wave equation Nonlinear sound speed Burgers’ eq.

5. Poisson’ solution Implicit solution + Demonstration Valid as long as or

6. Generation of harmonics – Fubini’s solution Harmonic signal Harmonic cascade …

7. Weak shock theory Weak formulation of Burgers’ equation Shock at time Weak shock theory

8. Law of equal areas (Landau) Weak shock theory Law of equal areas A+=A-

9. Example 1: the « N » wave Initial signal Poisson’ s solution Shock position Signal energy Energy is dissipated through shocks (second principle [s]>0 [P]>0 : compression shocks)

10. Example 1: the « N » wave

11. Example 2: the sine wave Initial signal Poisson’s solution Poisson’s solution remains periodic and antisymmetric Weak shock

12. Example 2: the sine wave Saturation : received energy is independant of emitted one !

13. Nonlinear attenuation The rate of energy dissipation depends on the temporal waveform of the initial signal

14. The acoustical potential Acoustical potential Poisson’s solution with Potential is continuous + compression shocks • Shocks are at the intersection of branches + • 2) Fmono = max Fmulti

15. The acoustical potential (II)

16. Evolution of complex waveforms Distorsion  HF Shock formation N wave  Shock evolution Shock fusion  BF

17. Conclusion The dominant phenomenon in nonlinear acoustics is the dependancy of sound speed with instanteneous wave amplitude. The evolution of the temporal waveform of a signal as a function of the propagation distance is modelled at 1D by the inviscid Burgers’ equation. Poisson’s solution provides an implicit solution that maybe multivalued. Starting from an initial profile, non-linearities lead to the waveform distorsion, up to the shock formation distance. This is associated to a shift of the frequency spectrum towards high frequencies. Shocks are necessarily compression ones (to satisfy the 2nd principle of thermodynamics). They are determined according to the weak shock theory, or law of equal areas. This implies dissipation of energy, the rate of which depends on the temporal waveform of the signal. Beyond the shock formation distance, waveforms evolve rapidly towards linear profiles (simple waves) matching moving shocks. This generally implies a slow shift of the frequency spectrum towards low frequencies. An efficient way to compute any wave profile at any distance is to use the potential. For potential, the physically admissible solution (that satisfies the entropy condition) is the largest value among the multivalued ones.