R. von Mises - Mathematical theory of compressible fluid flow, страница 99

T h e case 1) isdenoted unambiguously by q —•o o or by a—•o o ; case 2) by q—» 0. If some limit resultholds, say for λ—•— o o or for Μ —> 0, then it permits two different interpretations(which may not both be of interest).m14. T h e region of convergence thus obtained differs somewhat from that in thepaper of R. v. M I S E S and M . SCHIFFER, cit N o t e 12 (p.

262), which is Θ < 3 λ .Bergman has shown that the function ψ can be evaluated outside this triangularregion of convergence by means of Borel's summation method, and in the paper" S o m e methods for solutions of boundary value problems of linear partial differentialequations", Proc. Symp. Appl.

Math. {A.M.S.)6 (1956), pp. 11-29, he shows that thismethod gives a representation which is valid in the whole subsonic region.In the paper S. BERGMAN, " O n solutions of linear partial differential equationsof mixed t y p e " , Am. J. Math. 74 (1952), pp. 444-474, he derives a representation validfor 0 > 3 λ . T h e problem of connecting this representation with that for Θ < 3 λ ,however, still meets with difficulties.15. T h e following simple way has been indicated by Schiffer (in a recent letter toH . G . ) .

T h e actual F of E q . (26) is replaced by a very near-by function F for whichthe estimate (41), with $ given by (40), can be asserted for all negative values of λ.This function F is defined as follows:22222F(\)= 0for λ < —A,andF(\)=F(\)for- ASince F(\) tends rapidly to zero for λ—•— o o , the functions F(\)arbitrarily close for large enough A.T h e equations (290, (29*) are then similarly modified. W e put:Go = 1 ;5» = 0forλ S -A,Gn+i = 5« + FGnfor2^ λ < 0.and F(λ)—A < λ,will beη > 0.From the proof in M I S E S and SCHIFFER, cit.

N o t e 12, p. 261, it follows that such an F(\)satisfies(41), and it is then easily seen that the estimate (43) holds for theGnCHAPTER493VArticle 21for all negative λ-values. W i t h this modified function F(X) all statements in the textare correct, and we obtain the results which follow E q . (45).I t should be kept in mind that the G (\) are computed numerically from (29'),and that one starts always from a finite (however large) negative λ-value. Thismeans in fact that in the actual computation F is replaced by F and the G by theG .

T h a t the series (35), formed by means of the & , approximates arbitrarily thetheoretical solution [with the F of (26)], follows from the stability of the solution ofthe partial differential equation (8) under a slight change of the coefficient-functionF{\). Without such a property of stability any numerical approach would be inad­missible.nnnn16. Beginning in 1937 (see N o t e 9) and subsequently in many publications, Berg­man showed how to transform an arbitrary analytic function f(z) of one complexvariable ζ = λ + ιθ into a solution u(z, z) of a differential equation of the type ofE q .

(21.8), where the variable s is replaced by λ. His formula isand Ε has to satisfy a partial differential equation in three independent variablesand a boundary condition (see e.g. BERGMAN and SCHIFFER [1], p. 287). H e alsoshowed that a particular choice of Ε leads back to the kernel G of E q .

(35) [withΖ replaced by z\.In 1954 J . B . D I A Z and G. S. S. LUDFORD [ ' O n two methods of generating solutionsof linear partial differential equations by means of definite i n t e g r a l s " , Quart.Appl.Math. 12 (1955), pp. 422-427] pointed out that an integral representation such as (49)was already contained in a paper of 1895 by L E R O U X and they gave a formula for Κin terms of E. [ J . L E R O U X , "Sur les integrates des equations lineaires aux deriveespartielles du second ordre a deux variables independantes", Ann. sci. ecole norm,sup., Ser.

3, 12 (1895), pp. 227-316.] Compare also J . B . D I A Z and G. S. S. LUDFORD,"On the integration methods of Bergman and L e R o u x " , Quart. Appl. Math. 14(1957), pp. 428-432. W e note, however, that L e Roux considers hyperbolic differentialequations and that there is no indication that he intended to use his representation inconnection with multivalued solutions and solutions possessing singularities the wayBergman and Lighthill have done.

On the other hand, the use of the generator K, ascompared to Bergman's use of Ε provides in the present case definite simplificationsin Bergman's theory. T h e equation which Κ satisfies is now the same as that for u, cf.(51) and (51'), and is simpler than that for E. Also, with Κ as generator, the choice ofa suitable f u n c t i o n / ( 0 [namely (53)] becomes very simple. T h e relation between Κand Ε was also worked out independently by W .

Gibson.A t any rate the general representation in Sec. 7 is merely a frame which receivesits content when appropriate particular generators are defined, as was done by Berg­man and Lighthill.17. Τ . M . Cherry has also compared the method of Cherry and Lighthill withthat of Bergman: Τ . M . CHERRY, " R e l a t i o n between Bergman's and Chaplygin'smethods of solving the hodograph e q u a t i o n " , Quart.

Appl. Math. 9 (1951), pp. 92-94.There is a misleading statement about this paper by Υ . H. K u o in [31], p. 531 lastlines.18. S. BERGMAN, " L i n e a r operators in the theory of partial differential equations",Trans. Am. Math. Soc. 53 (1943), pp. 130-155 (in particular p. 140 ff.); and "Certainclasses of analytic functions of two real variables and their properties", ibid. 57(1945), pp. 299-331.r494NOTES A N DADDENDAArticle 2219.

See N o t e 111.39.20. For more details of this type of phenomenon, see the papers cited in N o t e111.40.21. See Notes 111.42,45 for nonperfect gas.22. A combination of the treatments given in Sec. 14.2 and the present sectionleads to the necessary conditions for an abrupt transition in the general case ofnonsteady three-dimensional motion. Thus, we find that Eqs. (3a) through (3d)hold with u replaced by u'', the component of relative velocity normal to the movingdiscontinuity, and with υ now the (vector) component of velocity tangential to thediscontinuity surface.

In addition, the inequality (16) holds.23. For a discussion of the origins of shock theory, see N o t e 111.44. T h e shockconditions for steady plane flow were first treated by T . M E Y E R , cit. N o t e IV.25.Some of the algebraic simplifications introduced in the following sections have beennoted by G. BIRKHOFF and J . W . W A L S H , " N o t e on the maximum shock deflection",Quart. Appl. Math.

12 (1954), pp. 83-86, and F. SCHUBERT, " Z u r Theorie des stationaren Verdichtungsstosses", Z. angew. Math. Mech. 23 (1943), pp. 129-138.24. This is the basis of an approximation method which originated with J . A C K ERET, "Luftkrafte auf Flugel, die mit grosserer als Schallgeschwindigkeit bewegtw e r d e n " , Z. Flugtech. Motorluftschiffahrt,16 (1925), pp. 72-74 [included in [20],translation: Ν AC A Tech. Mem. 317 (1925)]. Ackeret's method is equivalent to thesmall perturbation or linearized theory, see N o t e 1.20; compare also N o t e V.64. T h emethod was extended by A . BUSEMANN in a series of papers culminating in " A e r o dynamischer Auftrieb bei Uberschallgeschwindigkeit", Luftfahrtforschung12 (1935),pp.

210-220 [included in [20], translation: British Ministryof Supply, Reports andTechnical Publications 2844 (1937)]. M o r e recent work has been done by K . O. FRIED­RICHS, cit. N o t e I I I . 4 6 . For an extensive discussion and bibliography see the article" H i g h e r approximations" by M . J . LIGHTHILL in [31].25. This result is due to L. PRANDTL, " B e i t r a g e zur Theorie der Dampfstromungdurch D u s e n " , Z. Ver.

deut. Ingr. 48 (1904), pp. 348-350, while the more general rela­tion (20) was obtained by T . M E Y E R , cit. N o t e IV.25.26. T h e relations between £, η, M,M(and also £, η, Μι, Μ ' in the one-dimen­sional nonsteady case) are the same as those between £, η, Μι ,M for a normal shock,tables for the latter being given for example in [37]. These are complemented by tablesdetermining the inclination σι of the shock once Mi and the deflection δ are known(see the next section).27. A .

BUSEMANN, "Verdichtungsstosse in ebenen Gasstromungen", in A . GILLES,L. HOPF, and T . v. KXRMAN (editors), Vortrage aus dem Gebiete der Aerodynamikundverwandter Gebiete (Aachen 1929), Berlin: J . Springer, 1930, pp. 162-169 (translation:Ν AC A Tech. Mem. 1199 (1949)). A full discussion is also given in [19], §27.in2n2228. This is a zero order approximation.

T o the first order in (1 — q /qi) the shockbisects the angle between either the two C or the two C~ at each point, as may be seenfrom the osculation property given next in the text. For, in the first order, the chordQiQ of the shock polar is equally inclined to the tangents at its ends, and, in the sameorder, the tangent at Q coincides with the tangent to the Γ " through Q . This re­sult determines the position of the shock in the first order approximation theory ofN o t e 24.

A similar result holds in one-dimensional flow.2+22229. N o t e that these arguments still hold for a nonperfect gas and that P is thepoint of maximum entropy (minimum p ) on the corresponding ray through P i inthe tangent plane. Discussion of the pressure-hill relations was given by A . BUSE­MANN, cit. N o t e 27.30. There is no acceptable criterion for making a choice between these two shocksin cases where the remaining boundary conditions cannot be fully taken into account2aCHAPTER495VArticle 22(see A r t . 23). Reasons for rejecting all shocks which are attached to a profile in auniform stream and have subsonic conditions behind them have been advanced byΤ . Y .