Download presentation

Presentation is loading. Please wait.

Published byPorter Shere Modified over 3 years ago

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!

Similar presentations

OK

Wednesday, Nov. 19, 2003PHYS 1443-003, Fall 2003 Dr. Jaehoon Yu 1 PHYS 1443 – Section 003 Lecture #21 Wednesday, Nov. 19, 2003 Dr. Mystery Lecturer 1.Fluid.

Wednesday, Nov. 19, 2003PHYS 1443-003, Fall 2003 Dr. Jaehoon Yu 1 PHYS 1443 – Section 003 Lecture #21 Wednesday, Nov. 19, 2003 Dr. Mystery Lecturer 1.Fluid.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on electricity for class 10th Ppt on acute myeloid leukemia Ppt on solar system for class 2 Ppt on ac plant Ppt on festivals around the world Ppt on formal education articles Ppt on lead poisoning Download ppt on oxidation and reduction worksheet Ppt on bluetooth architecture overview Ppt on polynomials for class 8