1. 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

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

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

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

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

6. 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

7. 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

8. 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