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

2. Main assumptions: 1. Unperturbed gas is uniform: no gradients in density, pressure or temperature; 2.Unperturbed gas is stationary: without the presence of waves the velocity vanishes; 3.The velocity, density and pressure perturbations associated with the waves are small

3. Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave: This is KINEMATICS, not DYNAMICS!

9. An algebraic relation between vectors! Plane wave assumption converts a differential equation into an algebraic equation!

10. In matrix notation for k in x-y plane:

11. Algebraic wave equation: three coupled linear equations

12. Algebraic wave equation: three coupled linear equations Solution condition: vanishing determinant, an equation for ω given k

13. Dispersion relation

15. H L H L L

17. Sound waves in a uniform medium: Wave equation: Plane-wave solution:

18. If you have a wave equation for, then the plane wave assumption corresponds to “simple substitution”:

19. Medium with uniform velocity V: Comoving time derivative In unperturbed flow

20. Use comoving derivative

21. Wave equation Use comoving derivative

22. Wave equation Use comoving derivative Doppler-shifted frequency Plane wave assumption

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

26. Definition phase S

27. Definition phase-velocity

28. Definition phase S Definition phase-velocity

31. This should vanish for constructive interference!

32. Wave-packet, Fourier Integral

33. Phase factor x effective amplitude

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

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

45. SAME as for SOUND WAVES!

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

51. 2. At water’s surface we must have P = P atm :

52. 2. At water’s surface we must have P = P atm :

55. Shallow lake: Deep lake:

56. shallow lake deep lake

57. Situation in rest frame ship: quasi-stationary

58. wave frequency: wave vector: Ship moves in x -direction with velocity U 1: Wave frequency should vanish in ship’s rest frame: Doppler:

59. 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:

60. Ship moves in x -direction with velocity U

61. Wave phase in ship’s frame: Wavenumber:

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

63. Situation in rest frame ship: quasi-stationary