Gas Dynamics, Lecture 6 (Waves & shocks) see: astro.ru.nl/~achterb/

1. Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 6(Waves & shocks)see: www.astro.ru.nl/~achterb/

4. Phase- and group velocity Central concepts: Phase velocity: velocity with which surfaces of constant phasemove Group velocity: velocity with which slow modulations of the wave amplitude move

6. Phase velocity Definition phase S

7. Phase velocity Definition phase S Definition phase-velocity

8. Phase velocity Definition phase S Definition phase-velocity

9. Group velocity: the case of a “narrow” wave packet

10. Group velocity: the case of a “narrow” wave packet

11. Group velocity: the case of a “narrow” wave packet (cntd) This should vanish for constructive interference!

12. Group Velocity Wave-packet, Fourier Integral

13. Group Velocity Wave-packet, Fourier Integral Phase factor x effective amplitude

14. Group Velocity Wave-packet, Fourier Integral Phase factor x effective amplitude Constructive interference in integral when

15. Summary and example: sound waves in a moving fluid

16. Summary and example: sound waves

17. Summary and example: sound waves

19. Application: Kelvin Ship Waves

21. Waves in a lake of constant depth

22. Fundamental equations: Incompressible, constant density fluid (like water!) Constant gravitational acceleration in z-direction; Fluid at rest without waves

23. Unperturbed state without waves:

24. Small perturbations:

25. Equation of motion small perturbations: SAME as for SOUND WAVES!

26. Solve for pressure perturbation first!

27. Solution for pressure perturbation:

28. Solve equation of motion:

29. Solve equation of motion:

30. There are boundary conditions: #1 At bottom (z=0)we must have az = 0:

31. There are boundary conditions: #2 2. At water’s surface we must have P = Patm:

32. There are boundary conditions: #2 2. At water’s surface we must have P = Patm:

33. Dispersion relation from boundary conditions:

34. Dispersion relation from boundary conditions:

35. Limits of SHALLOW and DEEP lake Shallow lake: Deep lake:

36. Universal form using dimensionless variables for frequency and wavenumber: deep lake shallow lake

37. Finally: ship waves Situation in rest frame ship: quasi-stationary

38. Case of a deep lake wave frequency: wave vector: Ship moves in x-direction with velocity U 1: Wave frequency should vanish in ship’s rest frame: Doppler:

39. Case of a deep lake (2) wave frequency: wave vector: Ship moves in x-direction with velocity U 2: Wave phase should be stationary for different wavelengths in ship’s rest frame:

40. Case of a deep lake (3) Ship moves in x-direction with velocity U

41. Case of a deep lake (4) Ship moves in x-direction with velocity U Wave phase in ship’s frame: Wavenumber:

42. Case of a deep lake (5) Ship moves in x-direction with velocity U Stationary phase condition for

43. Kelvin Ship Waves Situation in rest frame ship: quasi-stationary

44. Shocks: non-linear fluid structures Shocks occur whenever a flow hits an obstacle at a speed larger than the sound speed

46. Shock properties • Shocks are sudden transitions in flow properties • such as density, velocity and pressure; • In shocks the kinetic energy of the flow is converted • into heat, (pressure); • Shocks are inevitable if sound waves propagate over • long distances; • Shocks always occur when a flow hits an obstacle • supersonically • In shocks, the flow speed along the shock normal • changes from supersonic to subsonic

47. The marble-tube analogy for shocks

48. Time between two `collisions’ `Shock speed’ = growth velocity of the stack.

49. 1 2 Go to frame where the `shock’ is stationary: Incoming marbles: Marbles in stack:

50. 2 1 Flux = density x velocity Incoming flux: Outgoing flux: