Gas Dynamics, Lecture 5 (Waves) see: astro.ru.nl/~achterb/

1. Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 5(Waves)see: www.astro.ru.nl/~achterb/

2. Waves

3. Simple (linear) waves • Properties: • Small perturbations of velocity, density and pressure • Periodic behavior (“sines and cosines”) in space and time • No effect of boundary conditions

4. Small amplitude waves (1) 1. Wave amplitude is small: unperturbed position small displacement

5. Small amplitude waves (2) Sinusoidal behavior in space and time: plane wave representation Complex amplitude vector Phase factor

6. Waves, wavelength and the wave vector

7. Small amplitude waves (3) 3. Wave amplitude is small in the following sense: • |a| is much smaller than the wavelength λ; • |a| is much smaller than gradient scale of the flow; • Density and pressure variations remain small:

8. Mathematical technique: • Perturbation analysis: • Expand fundamental equations in displacement ξ(x,t); • Neglect all terms of order ξ2and higher! • Express density and pressure variations in terms of • ξ(x,t); Neglect all terms of order ξ2and higher! • Find equation of motion for ξ(x,t) where only terms • linear in ξ(x,t) appear; • - Substitute plane wave assumption.

9. Perturbation analysis: simple mechanical example • Small-amplitude • motion; • Valid in the vicinity • of an equilibrium • position;

10. Perturbation analysis:fundamental equations Equilibrium position:

11. Perturbation analysis:motion near x=0 Taylor expansion: Near equilibrium position x=0:

12. Perturbation analysis:motion near x=0 Equation of motion near x=0:

13. Perturbation analysis:motion near x=0 Solutions:

14. Who measures what?Lagrangian and Eulerian variations Two fundamental types of observer in fluid mechanics: Observer fixed to coordinate system measures the Eulerian perturbation: Observer moving with the flow Measured the Lagrangian perturbation

15. Lagrangianlabels: useful mathematical concept Lagrangian Labels are carried along by the flow

16. Lagrangian labels: useful mathematical concept Conventional choice: position x0of a fluid-element at some fixed reference time t0 As always:

17. Re-interpretation of time-derivatives: At a fixed position Comovingwith the flow

18. Re-interpretation of time-derivatives +Re-interpretation of perturbations: At a fixed position Comovingwith the flow Lagrangian and Eulerian perturbations: Lagrangian: Eulerian:

19. Important consequence:Commutation Relations for derivatives!

20. Relation between Lagrangian and Eulerian perturbations Stay at old position! Follow the fluid to new position!

21. Relation between Lagrangian and Eulerian perturbations (2)

22. Relation between Lagrangian and Eulerian perturbations (3)

23. Relation between Lagrangian and Eulerian perturbations (4)

24. Final result for small perturbations: Small change induced by ξ in Q at fixed position Effect of position shift ξ

25. Almost trivial example of these rules (1):

26. Almost trivial example of these rules (2): Formal calculation:

27. Perturbation analysis: general approach

28. Application: velocity perturbation due to small-amplitude wave (1) Commutation Rules

29. Application: velocity perturbation due to small-amplitude wave (2) Commutation Rules Definition of the comoving derivative:

30. Eulerian and Lagrangian velocity perturbations: General relation between the two kinds of perturbations:

31. Summary: velocity perturbations (1) Special simple case: stationary unperturbed fluid that has V = 0:

32. Summary: velocity perturbations (2) Another specialcase: unperturbedfluid has uniform velocity V ≠ 0:

33. Density perturbation: 1D case Mass conservation:

34. Density perturbation (2)

35. Density perturbation (2)

36. Density perturbation (3)

37. Density perturbation (4)

38. Generalization results from 1D to 3D: One dimension: Three dimensions:

39. Pressure perturbation: Adiabatic flow: From general relation between Lagrangian and Eulerian perturbations:

40. Summary: changes induced by a small fluid displacement ξ(x,t):

41. Perturbation analysis: general approach

42. Linear sound waves in a homogeneous, stationary gas • Main assumptions: • Unperturbed gas is uniform: no gradients in density, • pressure or temperature; • Unperturbed gas is stationary: without the presence • of waves the velocity vanishes; • The velocity, density and pressure perturbations • associated with the waves are small

43. Immediate consequence:perturbations are “simple”: Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave:

44. Immediate consequence:perturbations are “simple”: Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave: This is KINEMATICS, not DYNAMICS!

45. Aim: to derive the DYNAMICS of the problem! To derive a linearequation of motion for the displacement vector (x,t) by linearizing the equation of motion for the gas. Method: Take the Lagrangian variation of the equation of motion.

46. Perturbing the Equation of Motion To find the equation of motion governing small perturbations you have to perturb the equation of motion!

47. Unperturbed gas is uniform and at rest: Apply a small displacement

48. Unperturbed gas is uniform and at rest: Apply a small displacement Because the unperturbed state is so simple, the linear perturbations in density, pressure and velocity are also simple!

49. Effect of linear perturbations on the equation of motion: fluid acceleration Use commutation rules again:

50. Effect of linear perturbations on the equation of motion: pressure force Use commutation rules again: I have used: