Download presentation

Presentation is loading. Please wait.

Published byPorter Shere Modified over 2 years ago

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

2

3

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

5

6
Definition phase S

7
Definition phase-velocity

8
Definition phase S Definition phase-velocity

9

10

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

15

16

17

18

19

20

21

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

23

24

25
SAME as for SOUND WAVES!

26

27

28

29

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 :

33

34

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

45

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

47

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:

52

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

57

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.

61

62

63
Sound waves:

64
Approximate jump conditions: put P 1 = 0!

65

66

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google