1. Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit

4. Central concepts: Phase velocity: velocity with which surfaces of constant phase move Group velocity: velocity with which slow modulations of the wave amplitude move

6. Definition phase S

7. Definition phase-velocity

8. Definition phase S Definition phase-velocity

11. This should vanish for constructive interference!

12. Wave-packet, Fourier Integral

13. Phase factor x effective amplitude

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

22. 1.Incompressible, constant density fluid (like water!) 2.Constant gravitational acceleration in z- direction; 3.Fluid at rest without waves

25. SAME as for SOUND WAVES!

30. 1.At bottom ( z=0) we must have a z = 0:

31. 2. At waters surface we must have P = P atm :

32. 2. At waters surface we must have P = P atm :

35. Shallow lake: Deep lake:

36. shallow lake deep lake

37. Situation in rest frame ship: quasi-stationary

38. wave frequency: wave vector: Ship moves in x -direction with velocity U 1: Wave frequency should vanish in ships rest frame: Doppler:

39. wave frequency: wave vector: Ship moves in x -direction with velocity U 2: Wave phase should be stationary for different wavelengths in ships rest frame:

40. Ship moves in x -direction with velocity U

41. Wave phase in ships frame: Wavenumber:

42. Ship moves in x -direction with velocity U Stationary phase condition for

43. Situation in rest frame ship: quasi-stationary

44. Shocks occur whenever a flow hits an obstacle at a speed larger than the sound speed

46. 1. Shocks are sudden transitions in flow properties such as density, velocity and pressure; 2.In shocks the kinetic energy of the flow is converted into heat, (pressure); 3.Shocks are inevitable if sound waves propagate over long distances; 4.Shocks always occur when a flow hits an obstacle supersonically 5.In shocks, the flow speed along the shock normal changes from supersonic to subsonic

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

49. Go to frame where the `shock is stationary: Incoming marbles: Marbles in stack: 12

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

51. Conclusions: 1. The density increases across the shock 2. The flux of incoming marbles equals the flux of outgoing marbles in the shock rest frame:

53. Generic conservation law:

54. Change of the amount of Q in layer of width 2 e: flux in - flux out

55. Infinitely thin layer: What goes in must come out : F in = F out

56. Infinitely thin layer: What goes in must come out : F in = F out Formal proof: use a limiting process for 0

58. Starting point: 1D ideal fluid equations in conservative form; x is the coordinate along shock normal, velocity V along x -axis! Mass conservation Momentum conservation Energy conservation

59. Mass flux Momentum flux Energy flux Three equations for three unknowns: post-shock state (2) is uniquely determined by pre-shock state (1)! Three conservation laws means three fluxes for flux in = flux out!

60. 1D case: Shocks can only exist if M s >1 ! Weak shocks: M s =1+ with << 1; Strong shocks: M s >> 1.

63. Sound waves:

64. Approximate jump conditions: put P 1 = 0!