Sound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8

Transcription

1 References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that can be understood using the fundamental equations of gas dynamics that we have developed so far. Astrophysical Gas Dynamics: Sound waves 1/5

2 1.1 Perturbation equations Simplest case: fluid equations without magnetic field and gravity (Euler equations): t v i t + vi 0 x i + v i v j x j p x i (1) Astrophysical Gas Dynamics: Sound waves /5

3 Consider small perturbations to state 0, and v i v 0 i. Without loss of generality we may take. Expand to first order: 0 + v i v 0 i + v i v 0 i 0 v i 0 + v i 0 v i + O () v i v j O x j (3) Astrophysical Gas Dynamics: Sound waves 3/5

4 The symbol O means second order in the quantities, v i, i.e. terms such as, v i v j, v i etc. The perturbation to the pressure is determined by the perturbations to the density and entropy:. Define the pa- If we take the flow to be adiabatic, then rameter by: p p p s p p + s s s 0 p p c s p 0 0 s Astrophysical Gas Dynamics: Sound waves 4/5 (4) (5)

5 The perturbation equations become: t 0 vi t + 0 vi 0 x i c + s 0 xi (6) Time derivative of first equation and divergence of second equation: 0 t v tx i 0 i v x i t i c + s x i x i (7) Astrophysical Gas Dynamics: Sound waves 5/5

6 Subtract > t c s x i x i (8) This the wave equation for a disturbance moving at velocity. Speed of sound (9) Astrophysical Gas Dynamics: Sound waves 6/5

7 1. Plane-wave solutions Take Aexpik x t t k x i x i (10) c s t k c s k Astrophysical Gas Dynamics: Sound waves 7/5

8 What does this plane wave represent? z y n x Consider a surface of constant phase k -- k x t 0 (11) k From the expression for the density, this represents a surface of constant density whose location evolves with time. This surface is planar and for our purposes is best represented in the form: k -- x k ---t k k (1) Astrophysical Gas Dynamics: Sound waves 8/5

9 From the above equation, we read off the normal to the plane: and the perpendicular distance of the plane from the origin is D n ---t k (13) (14) Therefore, the speed at which the plane is moving away from the origin is: k -- k --- c k s k (15) Astrophysical Gas Dynamics: Sound waves 9/5

10 Thus the solution represents a wave moving at a speed direction of the wave vector Solution for velocity in the The velocity perturbations are derived by considering perturbations of the form: (16) Note the appearance of a vector A i for the amplitude since v i is a vector. k v i A i expik x t Astrophysical Gas Dynamics: Sound waves 10/5

11 The time and spatial derivatives of v i are: vi t ia i expik x t x i iki A exp i k x t (17) Substitute into perturbation equation for velocity: 0 vi t c + s 0 xi (18) Astrophysical Gas Dynamics: Sound waves 11/5

12 i 0 A i + c s ik i A 0 A i c k i s A c k i s --- k 0 k A 0 (19) k i A k 0 The amplitude of the velocity perturbation A A i c s (0) Astrophysical Gas Dynamics: Sound waves 1/5

13 i.e. Amplitude of velocity perturbation Relative amplitude density perturbation (1) What does small mean? i.e perturbation velocities much less than the speed of sound. Nature of sound wave Since «1 v i «() A i k i A k 0 (3) Astrophysical Gas Dynamics: Sound waves 13/5

14 then the velocity of the oscillating elements of gas are in the direction of the wave-vector, i.e. the wave is longitudinal. 1.3 Numerical value of the speed of sound Take the equation of state p Ks p s Ks 1 p kt m (4) Astrophysical Gas Dynamics: Sound waves 14/5

15 Therefore: kt m (5) N.B. The speed of sound in an ideal gas only depends upon temperature. e.g. air at sea level on a warm day 1.4 T 300 K m/s (6) T T 10 7 K atmosphere in an elliptical galaxy 10 7 K, 0.6, km/s (7) Astrophysical Gas Dynamics: Sound waves 15/5

16 Subsonic and supersonic flow Disturbances in a gas travel at the speed of sound relative to the fluid. i.e. information in the gas propagates at the sound speed Sound wave v V Astrophysical Gas Dynamics: Sound waves 16/5

17 A wave travelling at a velocity of v wrt to the fluid cannot send a signal backwards to warn of an impending obstacle. This leads to the formation of shocks (to be considered later). Therefore the speed V is a critical one in gas dynamics. Supersonic Flow v Subsonic Flow v (8) Mach Number M v ---- (9) Nobody ever heard the bullet that killed him -Th. von Karman Astrophysical Gas Dynamics: Sound waves 17/5

18 3 Energy & momentum in sound waves 3.1 Expressions for energy density and energy flux Energy density 1 tot --v Expand out the quantities in this equation to second order + (30) s s s s ( ) + 1 s + -- s s s (31) Astrophysical Gas Dynamics: Sound waves 18/5

19 Since we are only considering adiabatic fluctuations, then s 0 and (3) We now evaluate these terms using thermodynamic identities. Since, then s ktds s 1 --d h --d (33) d hd + ktds s h (34) Astrophysical Gas Dynamics: Sound waves 19/5

20 and s h s + p p p (35) h -- h Astrophysical Gas Dynamics: Sound waves 0/5

21 Hence the total (kinetic + internal) energy is to second order: 1 tot v h (36) The term h 0 is associated with a change in energy in a given volume due to a change in mass and eventually disappears. 0 Energy flux F E i 1 v i --v + h 0 h 0 v i + 0 hv i + h 0 v i + O 3 (37) Astrophysical Gas Dynamics: Sound waves 1/5

22 Note that the kinetic energy term is of third order. We need to relate variables. Since then and dh h to the variations in other thermodynamic ktds 1 --dp h + dh ktds 1 --p 1 --dp h p s (38) (39) (40) Astrophysical Gas Dynamics: Sound waves /5

23 Hence, using the previous expression for : F E i F Ei 0 h 0 v i + h 0 v i + 0 hv i (41) 0 h 0 v i + h 0 v i + c v s i We combine this with the expression for the energy density: tot 0 h v Because of the energy equation, we have the conservation law: 0 (4) tot t F Ei + 0 x i (43) Astrophysical Gas Dynamics: Sound waves 3/5

24 Some of the terms in these expressions can be simplified. First, the leading red term in, 0 is constant and is just the background energy density. Now examine the continuity equation written out to second order: t + 0 v i + v i 0 x i (44) Multiplying this by h 0 gives: h0 t + h0 x v + h v 0 0 i 0 i i (45) Astrophysical Gas Dynamics: Sound waves 4/5

25 so that when we insert the red and blue terms in the energy conservation equation, they drop out and we are left with 1 -- t 0 v cs vi 0 x i (46) Hence, we take for the energy density and its associated energy flux sw v Fsw Ei c s v i pv i (47) Astrophysical Gas Dynamics: Sound waves 5/5

26 3. Energy density and energy flux in a plane wave Express and v i in terms of travelling waves and take the real part: Aexpik x t Acosk x t v i k i A expik x t k 0 (48) k i A cosk x t k 0 Astrophysical Gas Dynamics: Sound waves 6/5

27 Hence, sw v F Ei A cos k x 0 t sw c c3 s s v i -----A k i ---cos k k x t swk i --- k o (49) Average the energy density and flux over a period of the wave using: T T cos T k x t dt 1 -- (50) Astrophysical Gas Dynamics: Sound waves 7/5

28 so that the average energy density and flux are: F Ei sw A A k i --- c k s sw k i --- k o (51) Notes The energy flux is in the direction of the wave. and is equal to the sound speed times the energy density. Astrophysical Gas Dynamics: Sound waves 8/5

29 4 Jeans mass - sound waves with self gravity 4.1 Physical motivation There are numerous sites in the interstellar medium of our own galaxy that are the birthplaces of stars. These are cold dense clouds in which stars can form as a result of the process of gravitational collapse. Why is cold and dense important? Surprisingly, we can get some idea of this by looking at the propagation of sound waves in a molecular cloud. Astrophysical Gas Dynamics: Sound waves 9/5

30 The Horsehead nebula is a dark molecular cloud that is the site of ongoing star formation. The image at the left was obtained at the Kitt Peak National Observatory in Arizona The image at the right was obtained using the Hubble Space Telescope. Astrophysical Gas Dynamics: Sound waves 30/5

31 4. Mathematical treatment of gravitational instability Continuity, momentum and Poisson equations: t + vi 0 x i v i t v i 1 v j x j -- p x i x i (5) In order to model self-gravity of the gas we have included Poisson s equation for the gravitational potential. 4G Astrophysical Gas Dynamics: Sound waves 31/5

32 Perturbations: constant v i v i (53) 0 + Astrophysical Gas Dynamics: Sound waves 3/5

33 Perturbation equations: 0 t + + x i 0 v i t + ( vi ) + 0 x i 0vi x i v i 1 p 0 1 p p t 0 xi 0 xi xi x 0 i x i (54) 0 + 4G 0 + Zeroth order terms are shown in blue. The term in red is associated with what is sometimes unkindly known as the Jeans Swindle. Jeans neglected this term in his derivation of the Jeans Mass. Astrophysical Gas Dynamics: Sound waves 33/5

34 When can we neglect the pressure gradient term? 1 p 0 Compare p with Let us suppose that the back- xi xi ground pressure gradient has a length scale L, and that the perturbations have a length scale l i.e. and p xi 0 p l (55) p xi p L 0 (56) Astrophysical Gas Dynamics: Sound waves 34/5

35 The pressure gradient term associated with the perturbations is larger than the background pressure gradient term if p l p 0 l p » «----- c L L p 0 c 0 (57) So, for perturbation length scales much less than the background lenght scale in the cloud, we can neglect the background pressure gradient. Astrophysical Gas Dynamics: Sound waves 35/5

36 Now use the zeroth order equations: 1 p xi x 0 4G 0 i (58) Astrophysical Gas Dynamics: Sound waves 36/5

37 This gives: t 0vi + 0 ( vi ) + 0 x i x i V i t 1 p p xi xi x 0 i (59) Simplest case: initial density constant and neglect variation of potential. 0 constant 4G (60) 0 constant (61) Astrophysical Gas Dynamics: Sound waves 37/5

38 The assumption means that the red term above is neglected. The perturbation equations simplify to: t + ( vi ) 0 0 x i v i t Plane wave solution: p + 0 xi 0 Astrophysical Gas Dynamics: Sound waves 38/5 x i Aexpik x t v i A i expik x t Bexpik x t 4G (6) (63)

39 Substitute into perturbation equations: ia + 0 ik i A i 0 ia i ik i A+ ik i B 0 0 (64) k B 4GA Astrophysical Gas Dynamics: Sound waves 39/5

40 We can arrange these equations into the following set of linear equations: A 0 k i A i k i A 0 A i + k i B 0 (65) 4GA + k B 0 Astrophysical Gas Dynamics: Sound waves 40/5

41 There are a number of ways of dealing with this set of linear algebraic equations for the quantities, AA i and B. One of the best is simply to write them as a set of 5 equations as follows: Astrophysical Gas Dynamics: Sound waves 41/5

42 0 k 1 0 k 0 k k k k 0 0 k -----k k 3 0 A A 1 A A 3 B (66) 4G k Astrophysical Gas Dynamics: Sound waves 4/5

43 This is a homogeneous set of equations that only has a non-trivial solution if the determinant is zero. It is possible to work the determinant out by hand, since it contains a number of zeroes. However, Maple/Mathematica also readily provides the following expression for the determinant: k + c s k 4G 0 (67) Apart from the trivial solutions ( or k 0) to 0, the only non-trivial solutions are described by: c s k 4G 0 (68) giving c s k 4G 0 (69) Astrophysical Gas Dynamics: Sound waves 43/5

44 This constitutes an interesting and fundamental difference from normal sound waves because of the additional term relating to self-gravity. Define the Jeans wave number k J by c s k J 4G 0 c s k k J (70) Then if k k J 0 i g say (71) Astrophysical Gas Dynamics: Sound waves 44/5

45 For these imaginary solutions the perturbation in each variable is proportional to expit exp g t. (7) One of the imaginary roots corresponds to decaying modes, the other to growing modes, i.e. instability. Jeans Length J k J G 0 (73) Astrophysical Gas Dynamics: Sound waves 45/5

46 Jeans length in a molecular cloud J n 10 9 m 3 T 10 K 1 kt / 0.9 km/s m kt mg 0 / kt / pc m Gn (74) Astrophysical Gas Dynamics: Sound waves 46/5

47 Jeans mass The Jeans mass, M J is defined to be the mass in the region defined by the reciprocal of the Jeans wave number: M J 0 k 3 J c s 3 / G 0 kt Gm13 / 0 3 / (75) 0.4solar masses Astrophysical Gas Dynamics: Sound waves 47/5

48 for the above parameters. Star formation - the modern approach The above discussion is the standard treatment for gravitational instability without the influence of a magnetic field and gives interesting sizes for the initial collapsing region and the mass of the collapsing object. The current attitude to the Jeans mass is that the physics of star formation is more complicated and that magnetic fields and turbulence are involved. However, it is thought that the Jeans Mass is relevant to the masses of molecular cloud cores. Astrophysical Gas Dynamics: Sound waves 48/5

49 Physics of the Jean mass M R Consider a gas cloud of mass M and radius R and neglect pressure forces. Then the equation of motion of the cloud is: dv i dt GM x i R --- R Astrophysical Gas Dynamics: Sound waves 49/5 (76)

50 Dimensionally, this equation is: R --- t GM R Free-fall time t 3 ff / G GM 1 / R Now consider the sound crossing time: t s R ---- (77) (78) Astrophysical Gas Dynamics: Sound waves 50/5

51 If the sound-crossing time is less then the free-fall time, then the collapsing region is able to produce enough pressure to halt the collapse since signals have the time to go from one side of the region to the other. Thus the condition for collapse is t s t ff Another criterion: R ---- G c 1 / s Euler s equations with pressure and gravity: R c s 1 / J G (79) dv i dt 1p -- xi x i (80) Astrophysical Gas Dynamics: Sound waves 51/5

52 Collapse will occur if gravitational forces overwhelm the pressure forces. Since p x i p x i c s xi (81) then Euler s equations are: dv i dt xi x i For collapse the gravitational forces have to win, so that, GM R R R R -- GM R c s 1 / G R J GM R (8) (83) Astrophysical Gas Dynamics: Sound waves 5/5