THE EQUATIONS OF NONLINEAR ACOUSTICS IN A MOVING HETEROGENEOUS MEDIA – BURGERS, KZ & EIKONAL EQUATIONS, NUMERICAL SOLUTIONS AND APPLICATIONS François COULOUVRAT CNRS Research Director Laboratoire de Modélisation en Mécanique Université Pierre et Marie Curie – Paris 6 - France F. Coulouvrat – Lecture introduction – Bangalore – March 2005

Acoustical weak shock waves : natural sources Thunder                            Sonic boom of meteors Volcanoes

Acoustical weak shock waves : artificial sources Cracking whip Brass musical instruments                            Fans of aircraft engines (BSN) Lithotripsy & HIFU Sonic boom Explosions

Weak nonlinearities… Acoustical Mach number or Concorde sonic boom at ground level P0 = 100 Pa – c0 = 340 m/s – r0 = 1,2 kg/m3 Mac = 10-3 For HIFU, Mac can be up to 0.1 F. Coulouvrat – Lecture introduction – Bangalore – March 2005

…but shock waves BoomFile test flights, USAF, Edwards base, 1987 F. Coulouvrat – Lecture introduction – Bangalore – March 2005

Propagation media are heterogeneous and moving Sonic boom propagation BSN Mach up to 0.9 HIFU Human body : tissues, blood, bones Wind speed > 40 m/s F. Coulouvrat – Lecture introduction – Bangalore – March 2005

Road map Part 1 : Equations of weakly nonlinear acoustics - General case : hyperbolic system of 5 equations (p,u,s). - High frequency approximation New single formulation for different cases : geometrical approximation (rays), inviscid Burgers’ equation (single conservation law) KZ equation (paraxial approximation of wave eq.) others… Part 2 : Some particular cases inviscid Burgers’ equation KZ equation : numerical algorithms & applications Part 3 : Eikonal equation – Lagrangian (ray tracing) & Eulerian numerical solutions - application to sonic boom F. Coulouvrat – Lecture introduction – Bangalore – March 2005

Asymptotic high frequency is required Meshing Space : 10 cm Time : 0.3 ms  3 x 1023 dl ! 300 s                            12 km 50 km 100 km