R. von Mises - Mathematical theory of compressible fluid flow, страница 94

R . v . M I S E S , cit. N o t e 20. This problem was also considered by A . H . T A U B ,"Interaction of progressive rarefaction w a v e s " , Ann. Math. 47, (1946), pp. 811-828;see also Sec. 13.4. A special case arising in ballistics was discussed by A . E. LOVEand F. B . PIDDUCK, cit. N o t e 19. Of course the same results can be obtained by R i e ­mann's method, but no so directly.26. Only three of these conditions are independent. T h e effect of choosing otherfu 9i,fi> g", c and k which satisfy E q s .

(53) is to add a term a + βξ + τ £ + δ£ to/ and a term — a -f βη —+ δη to g, where α, β, y, δ are constants. This merelyadds a constant to V in E q . (42) and hence has no effect on χ and t as functions of uand ν in Eqs. (47). W e could take C = 0, but in the example of Sec. 13.5 it is moreconvenient to take C ^ 0.27. Only five of these equations are independent. T h e effect of choosing othervalues for the ten constants is to add a term a + βξ + τ £ + δξ + E£ to / and aterm — a + βη — yy -f- δη — E?; to g.

As before α, β, y, δ have no effect on thefinal result, while the E merely shifts the origin of x.28. R. v . M I S E S , cit. N o t e 20. This problem was also considered by K . BECHERT," Z u r Theorie ebener Storungen in reibungsfreien G a s e n " , Ann. Physik, Ser. 5, 37U940), pp. 89-123, and I I ibid. 38 (1940), pp. 1-25.29. T h e general solution of the homogeneous equation corresponding to (61) is/ = a + (B £ + Co* 4- £>o£ , g = - α+ ®OT? - Co*? + S W , but the addition ofterms £ 3D[z(£)] and η $)[χ(η)]in (62) has no effect on the final answer in the presentapproach.

A n equivalent but less elegant result is obtained by the usual method ofvariation of parameters where one would set Yi = F = 0; instead we put 3D = 0and Fi = F .23220032432β3322343CHAPTERIII479Article 1230. Such a problem arises in the treatment of interstellar gas clouds, see for exam­ple Ε. T . COPSON, cit.

N o t e 20, and D . C . P A C K , " A note on the unsteady motion ofa compressible fluid", Proc.CambridgePhil.Soc. 49 (1953), pp. 493-497.Article 1331. T h e solutions now known as simple (or progressive) waves were found by S. D .POISSON ["Mernoire sur la thoorie du s o n " , J. ecole polytech., Ser.

1, 14 (1808), pp. 319392] for the special case κ = 1 (isothermal). Later S. EARNSHAW ["On the mathe­matical theory of sound", Phil. Trans. Roy. Soc. London 150 (1860), pp. 133-148] ex­tended the study to a general elastic fluid. For a discussion see LORD R A Y L E I G H ,N o t e 3. R I E M A N N , cit.

N o t e 11.23, showed that an arbitrary limited disturbance ofan unlimited gas at rest eventually separates into two simple waves.32. Geometrical ideas such as the envelope Ε appear in a paper to which we shallneed to refer later: H . HUGONIOT, " M o m o i r e sur la propagation du movement dansles corps, et spocialement dans les gaz p a r f a i t s " , J. ecole polytech., Ser. 1, 57 (1887),pp. 1-97, and 58 (1889), pp.

1-125.33. Centered waves were used by Riemann, cit. N o t e 11.23, in his discussion ofinitial discontinuities.34. T h e eventual breakdown of such a simple wave was first pointed out by G. G.STOKES, " O n a difficulty in the theory of sound", Phil. Mag. Ser. 3, 33 (1848), pp.349-356, or Mathematical and Scientific Papers, Vol. 2, London and N e w Y o r k : Cam­bridge U n i v . Press, 1883, pp.

51-55. Stokes treated the isothermal case. T h e resultwas extended to the general elastic case by B . R I E M A N N , cit. N o t e 11.23. For a purelyanalytical treatment of the change in type of a simple wave, see S. EARNSHAW, cit.N o t e 31, and COURANT and FRIEDRICHS [21], pp. 96-97.35. T h e case in which the path of the piston is prescribed was treated first by S.EARNSHAW, cit. N o t e 31, and then by H .

HUGONIOT, cit. N o t e 32, LORD R A Y L E I G H ,cit. N o t e 3, and A . F. P I L L O W , " T h e formation and growth of shock waves in the onedimensional motion of a g a s " , Proc. Cambridge Phil. Soc. 45 (1949), pp. 558-586. Inthe famous problem of ballistics first studied by J. L.

LAGRANGE in 1793, the path ofthe piston must be determined: S. D . POISSON, "Formules relatives au movement duboulet dans l'intorieur du canon, extraites des manuscrits de L a g r a n g e " , / . ecolepolytech., Ser. 1, 21 (1832), pp. 187-204, or Oeuvres de Lagrange, V o l . 7, Paris: GauthierVillars, 1877, pp. 603-615; for extensive discussions see Α .

Ε. H . LOVE and F. Β.PIDDUCK, cit. N o t e 19, and M . C. PLATRIER, " A n a l y s e du probleme balistique deL a g r a n g e " , Mem. artillerie frang. 15 (1936), pp. 431-477.36. Thus according to (13.11') we have37. T h e problem of interaction of symmetric waves is equivalent to that of the re­flection of one of them at a fixed wall (the line of s y m m e t r y ) . T h e latter occurs in La­grange's problem, see N o t e 35. Penetration of general simple waves was discussed byA . H . T A U B , cit.

N o t e 25.38. C. D E P R I M A has shown that there is a connection between this function t andthe Riemann function of the ^-equation [(12.34) with η = —3], see [21], pp. 194-196.480NOTES A N D A D D E N D AArticle 13Thus for general κ, the solution in the penetration region ist = ίο Ω(ι>ο,*>ο ;£,»?),where Ω(£,ΐ7£1,171) is given in N o t e 22, with η = - (κ + 1)/2(κ - 1 ) , see E q . (12.33).Knowingthe corresponding function χ can be determined from (12.23) by integra­tion. Nonsymmetric waves, whose centers have the same l v a l u e , can be made sym­metric by superimposing a suitable constant velocity on the whole flow.Article 1439. I t should be emphasized that this is a direct appeal to experience.

T o reach asimilar contradiction in the case of steady plane flow a more careful formulation ofthe experimental evidence is necessary (see Sec. 2 2 . 1 ) .40. Mathematically, the order of the system of differential equations governing themotion is reduced on setting μ = 0. Such systems lead naturally to so-called asymp­totic phenomena (see end of Sec. 2 ) , which in fluid dynamics first appeared in the"boundary layer t h e o r y " of L.

PRANDTL, " U b e r Flussigkeitsbewegung bei sehr kleinerR e i b u n g " , Verhandl.III.internal.math.-Kongresses,HeidelbergThis paper is included in L. PRANDTL and A . B E T Z , Vier(1904), pp. 484-491.AbhandlungenzurHydro-dynamik und Aerodynamik, Gottingen: Kaiser Wilhelm-Institut fur Stromungsforschung, 1927, pp. 1-8 (reprinted 1943,.Ann Arbor: E d w a r d s ) .

T h e asymptotic charac­ter of the boundary layer was later pointed out by T . v. K X R M X N , " U b e r laminareund turbulente R e i b u n g " , Z. angew.Math.Mech.MTSES, "Bemerkungen zur H y d r o d y n a m i k " , ibid.1 ( 1 9 2 1 ) , p p . 233-252 and R .v.7 ( 1 9 2 7 ) , pp. 4 2 5 - 4 3 1 .

F o r a surveyof the many facets of asymptotic phenomena see K . O. FRIEDRICHS, " A s y m p t o t i cphenomena in mathematical p h y s i c s " , Bull.Am. Math.Soc. 61 ( 1 9 5 5 ) , p p . 485-504.A technique for obtaining uniformly valid approximations in such problems has beendeveloped by M . J . Lighthill and others, see H . S. T S I E N , " T h e Poincaro-LighthillK u o m e t h o d " , Advances41.in Appl.Mech.4 ( 1 9 5 6 ) , p p . 281-349. Cf. also Sees.

24.5, 6.See for example [ 2 4 ] , pp. 477-756.42. F o r a nonperfect gas yp/(y — l ) p must be replaced by the enthalpyI(p,p)of the gas, see Eq. ( 2 . 2 3 ' ) · T h e corresponding changes in A r t . 11 which lead to this re­sult are obtained by retaining U in Eq. (11.5) instead of replacing it by gRT/(y— 1).For a full discussion see D .

GILBARG, cit. N o t e 3. Similar changes must then be madein the remainder of this section.43. Such flow patterns are also called weak solutions of the differential equationsof ideal fluid theory, and can be alternatively defined by certain integral conditions,see for example P. D . L A X , " I n i t i a l value problems for non-linear hyperbolic equa­t i o n s " , Contractpp.Nonr58804, Dept.of Math.,Univ.of Kansas,Tech. Kept.14(1955),13-57.44.

B . R I E M A N N cit. N o t e 11.23, W . J . M . R A N K I N E cit. N o t e 3, and H . HUGONIOTcit. N o t e 32. Hugoniot also took the occurrence of these discontinuities for granted.Rankine justified his results on the basis of heat conduction alone, but failed to seethat this case is singular. Neither Riemann, Rankine, nor Hugoniot included theinequality (14.9) and hence considered rarefaction shocks as well as condensationshocks (see next section).

LORD RAYLEIGH, cit. N o t e 3, gave a survey of the questionand in particular justified the Rankine-Hugoniot shock conditions on the basis ofviscosity. See also R. RUDENBERG, " U b e r die Fortpflanzungsgeschwindigkeit undImpulsstiirke von Verdichtungsstossen", ArtilleristischeMonatsh. Nos. 113 and114 ( 1 9 1 6 ) , pp. 237-265 and 285-316, and M . J . L I G H T H I L L , cit. N o t e 3.45. See N o t e 42. T h e results in the following sections have been extended to theCHAPTER481IIIArticle 14case of a nonperfect gas; see P. D U H E M , "Sur la propagation des ondes de choc ausein des fluides", Z. physik. Chem. 69 (1909), pp.