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STAR(,I-,iJIi,',f, '.. I- -,~- ", f-A, tile t---mrprii,.,~ -; rrw. t)011 ttAFI~e I ,~ 5 . j. f . j S"I UK'i"'N' A. F. on thri properties of haidened , enern.met. 8 no.61137-139 (MIRA 18:8) 1. Dnep-oikttzo~-.31hly instItIlt. LITVINOV, I.R.; ZAZHIRKO, Y.N., assiatont; BABIGH V.M., starshiy prepodavatell Slippage-preventive circult of 118 series electric locomotives. Xlek. i tepl. tiaga 4 no. 9:8-11 S 160. (MIRA 13:12) 1. Glavayy inzhener sluzhby lokomotivnogo khozyaystva Tomskoy dorogi (for Litvinov). 2. Tomakly elaktromekbanichoskiy institut inzhenerov zhel.eznodorozhhogo transporta (for Zazhirko.,. Babich). (Ilectric locomotives) M%themtjoal Rev!Lews Vois 15 No* 2 Peb, 19154 AnalysiB jeabV, V. M. On the extension of functions. Uspehi Maivin.-Ta-tik (N.S.) 8, no. 2(54), 111-113 (1953). (Ru! ian) Let be a closed botinded region in E. whose l6ounclary is PICCVWIS(. of clafs 011,and let f (p), p t A. be a function de- fined in A and of class 011 in A, i.e.. f is continuous with all its partial derivatives of otiler k in A. H. Whitney [Trans. Anier. Math. Soc. 36, 63--89 (1934)] and M. R. Hestenes [Duke Math. J. 8. 183-192 (194 1); these Rev. 2. 2191 havel proved various theorems on the extension of functions, in[ partictilar that every ftinction of class 011 in A his an ex- tension -.%-hich is of class C01 in E.. 'rhe author says that a - fnnction f is of class III,(" if the cnntinui~y requirement o the partial derivatives of order k in the definition of cla 01) is replaced by Ly'-integrability. 'I'lien he proves tha every function f of cla%s ff,(') in A has an extension whichi is of class in E,. liestenes'nictliod based on a modified forin of reflection principle is uscd. L Cesad. DABEHI V.M. (Leningrad); RUSAKOVA, N.Ya. (Leningrad) Propagation of Rayleigh waves along the surface of an inhomo- ganeous elastic body of arbitrary shape. Zhur.vych.matA mat. fiz. 2 no.4:652-665 Jl-Ag 162. OMA 15:8) (Elasticity) (Waves) $~O 341 !;k I, o'lfl)j Imio) "r it lHil 01~11." b) vr, SNAP. 91, No. 4, 763 .5 (19.5i) in Rm.,-,,% I nj;?i,jj U.S. Sit. Fjund. NSP 'ir~1.11. it y (N~ci I I. I ha I tlie. :j it,, III; o I I it ifi .11 mi mwlt! x-xc'v'Ime. cot fo, IvIal iot'~Jojjl 1,111. In ltwitC of thiR, III,: d;,Id10,i1ILC 11.111; dly J%,:1t%:I1JI,', It".0 niti;; Pl,! i-mill'i M,,01i'dw 0,1110 I'i~vf ).N :111k: III,: ul'i-Ic $if Il..: I;, A VOlve, Ilic pot'l, it i~ 'cv!;cJ 1~10. IIN'till pajir"fliv, 6 01c :It T.c 'it,Z%'Iy of "'-Alu'tA ~'p, Aiua tA ji ~Z-im%d 111z't !I"! %d-di, . it, it.,: 'tk)VC i-ril kl-v W~ 1,-),r it, v';oal "1-J loNttr Ilim 11'.tj .11 1~!'" It i'. 1"),11A obtairica unnor N: C.XpLAiuJ dicorics ba,.,cd or jvonxtriWal op6ci and it is coi - Ot4ded that futum proiu--i in (lih field of geophysics depcnds on lic ful! &iopllc~ttion of the dyi.arnlcal thcory of claWcity. A. C. %VIUMN USSRkathematics - Theory of Elasticity Card : 1/1 Autbors Babich, V. 14. Title I Solution of the Cauchy problem for the system of equations of the theo,-y of elasticity of a heterogeneous elastic medium Periodical Dokl. AN SSSR, 96~ Ed. 6, 1125 - 1128, June 1954 Abstract I Author recommends the S. L. Sobolev method for the derivation of a solu- tion.of the Cauchy problem for the system-of equations of the elasticity thaory. A study of generalized functions showed that the basis of the Sobolev method is exactly the same as thnt or the Adamar method. The formulation of a fundamental solution which would be accurate for the case of constant equation coefficients is also applicable in the case of variable but sufficiently smooth coefficients. Six references. Institution - The A. A. Zhdanov State University, Leningrad Presented by : Academician V. I. Smirnov, April 5, 1954 V USO/ Mathematics Theory of Elasticity Card Authors t Babich, V. M. Title t On equations of motion of a nonlinear elastic medium Periodical t Dokj. AN SSSR., Vol. 97p Ed,~ 1,, 41 - 44, -July 1954 Abstract I The author prorases a solution of Henke-Schmidt equations, relative to the motion of a consolidated plastic medium, basing the method of their solution on the theory of coincident kinematic and dynamic conditions. A detailed examination is made of the form and speed of the waves propagated as a result of the motion (maximur, medium and minimum waves). Five references; four of thesej, USSR referencest of which the last one is of 1951. Institution : The A. A'. Zhdanoy State University of Leningrad Presented by : Academicianp V. I. Smirnov, April 1954 SUBJECT USSR~MATHEMATICS/Differential equations CARD 1/1 PG - 655 AUTHOR BIBI V.M. TITLZ On some papers of V.M. Panferov on the domain of the theory of elasto-plastio deformations. PESIODICAL Priklad.MateMech. 20, 767-771 (1956) reviewed 3/1957 The author criticizes sharply a series of Panferov's publications during -the years 1949-1952 ~Priklad.Mat.Mech. jjL 11 ibid. 162 21 ibid. I6jL 31 Ve8tnik MGU 8L (1952 ). The author detects many inexactnesses and in- correct conclusions, among others an unwarranted applioation of the methods of Ritz and Galerkin. INSTITUTIORt Leningrad. SLOBODZTSKIY, L.N.; BABICH, Y.M. Doundeduess of the.DirichlOt Integral. Dokl,AN SSSR 106 no*4,. 604-606 7 156. (NLRL 9:6) LLeningradskiy pedagogichaskiy Institut. Predstavlano akade- mikom V.I.Smirnov3rm. (IntegriLls) BABICHI V. M. "A Radial Method for Computing the Intensity of Wave Fronts," by V..M. Babich, Leningrad State University imeni Zhdanov, presented by Academician V. 1. Smirnov, Doklady Akademii Nau)t SSSR, Vol 110, No 3, 1956, PP 355 - 357 For a full picture of dynamic seismology in the case of a hetero- geneous wave, the writer considers it necessary to analyze separately longitudinal and transverse wave intensities. Such computations are facilitated by analyzing the correlations formed by characteristic mani- fold equations expressing undulatory processes. For the solution of these equations the writer introduces equations of elasticity theory expressing a wave front with a discontinuity. Me propagation direction of longitudinal waves the writer ca:ils "rays." In the case of small dimensions, the assumption is made in the first approximation that the heterogeneous medium is homogeneous and the curvilinear front is recti- linear, which simplifies the solution. Sum 1219 BABIGH, V. M. "Ray Theory of Wave Front Intensity,, 11 pa,:)or presented at 4th All-Union Acoultics Conf., 26 1-lay 4 Olin 5,,, Moscow. it )-1-2/16 AUT11ORS:Babich V.M*. and Alekseyev, A.S. TITLE: On the Ray Method of Calculating the Intensity of Wave- fronts (0 luchevom metode vychisleniya intensivnosti volnovykh frontov) PERIODICAL: Izvestiya Akademii Nauk SSSR, Seriya Geofizicheskaya, 1958v Nr lt pp'.17-31 (USSR) ABSTRACT: The growth of dyna ic seismology leads to the necessity of calculating the intensity of longitudinal and trans- verse waves in inhomogeneous media at the reflection of the waves from curvilinear boundaries. Such calculations can be carried out by considering the relations obtaining on the characteristic manifolds of the equations describ- ing the wave processes. Analogous considerations lie at the basis of the methods of Hadamard (Ref.1) and Sobolev (Ref.?) for the solution of the Cauchy problem for hyper- bolic equations. The method described in this paper has previously been applied to Maxwell's equations (Refs-3-5) and to the wave equation (Refs.6-9). Levin and Rytov (Ref.10)p and Zvolinskiy and Skuridin (Refs.11 and 12) have applied ray considerations to the equations of the Card 1/9 /I J-1,1/1U On the Ray Method of Calculating the Intensity of Wavefronts. theory of elasticity, but in none of these papers are to be found the equations 4.?f 4-3P 4.5 and 4.7, which are at the basis of the method described. The method of des- cribing the function f(m , a ) for a concentrated source, which is an important pari of2the method, is also new. Let t = Z(x, y, z) be the equation of the wavefront at time t . Let the wave process under consideration be described by the scalar or vector fanction U(xt y9 z9 t) where it is assumed that U(XIYIZtt) :-- Uo(xpytz)fo (t + U1(XIYIZ)fl (t +0 (f 9 (t (Eq.1.1) in which f, (t) = fl(t)p f1l(t) = fo(t) 2 It is assumed that in some sense the fanction f 2(t) can be neglected in comparison with its derivative. If Eq.(1.1) is substituted into 1 Card 2/9 Uxx + Uyy - C2(x, Y) Utt 0 (Eq.2.1) On the Ray Method of Calculating the Intensity of Wavefronts. and the coefficient of V equated to zero, there results 0 grad t grad U 0+ U 0A*V = 0 (Eq.Z.4) which is studied in some detail. Equations analogous to Eq.(?.4) for the case of an inhomogeneous elastic medium are derived by substituting the expression for the vector U(x, y, t) from Eq.(1.1) into the two-dimensional differ- "gntial equations of motion of an inhomogeneous elastic medium. Thus we have - (X + )u) (grad tUo )grad -C - pUo (grad 1C) 2 + pUo = 0, (Eq. 2.6 (?L + )a) (grad *tU,)grad -r - PU, (grad t)2 + PUI = 0 (Eq.2.7) where Card 3/9 On the Ray Method of Calculatin6 the Intensity of Wavefronts. g(U (X + )a) ~(div U I + grad (Rograd + _O)grad + ju FU 6T + 9 (grad t/oxgrad T) i + 2 (grad Ugrad L17-o 0y + grad X No grad Z) + (grad pLJO)grad 'C + (gradu grad r)Po (Eq.2.8) and U i, j are unit vectors in the U0 t ox, uoy~ directions of x and y respectively. Eq.(2.6) is a system of two homogeneous equations in the two unknowns Uox and U 0y P and it can be shown that the determinant of this system only vanishes in two cases. These are: (a) when Igrad -rJ2 P (longitudinal wave) in x + 2P which case we shall write T~ for 'U; and 7- - - (b) lgradtJ2 - l - P (transverse wave) in which case b IU we shall write tb for T In the first case it can be Card 4/9 shown that: 4)-l-2/16 On the Ray Method of Calculating the Intensity of Wavefronts. U0 = Vograd Za (Eq.3-1) where go is a scalar function of position. Eq.(2.7) can also be regarded as an algebraic system of equations for the unknown compon- ents of the vector U Again there are twe, conditions =1 for solution, the first of which can be written in the f0 rm : 1Z (U0 a)grad Ta= 0 (Eq.3-3) If Eq.(3-1) is substituted into Eq.(3-3), after some simplification 2 690 + [a2AtL - OL + 2)1) rad 1 grad ~ T,a (9 Zaj go = 0 (Eq .3.6 I P is obtained. In Eq.(3.6) the derivative is calculated Card 5/9 along the ray of the lor4;itudinul wave. If U n and U 10 On the Ray Method of Calculating the Intensity of Wavefronts. are the components of the U0 along the normal and the binormal to a ray of the transverse wave, then the condit- ion for the solubility of the system ?.7 can be written in the form: 1.3 + 9-ThU + (b2Ar, + grad u grad blub D P 'tb)Un = 0 auo - - 9-Thu n+ (b Atb + grad p grad T b) U V = 0 alrlb Suppose that a point on the ray is characterized by the quantity and the ray itself by)~he parameter a and I et x x (a, T ) 9 y = y ((x, r , ort in vector form, X = X(a,t Eq. (,?.4) can be written in the form 2 au0 U0 -7- +- - = 0 Card 6/9 0 6'r cjjj 6r c jj .)-l-2/1,5 On the Ray Method of Calculating the Intensity of Wavefronts. and it can be shown that by a method analogous to that used by Umov (Ref.?3, pp.161-163) this equation has the s o lut. i o n: Uo C f (a) (Eq.4.1) where f (a) is an arbitrary function of the parameter a. In a similar manner, from Eq.(3.6) we obtain: U01 - 1 f V[%; 7pa (Fq.4.?) where a characterizes a ray from the longitudinal wave. Similar considerations lead to the expression for the intensity of transverse waves: Card 7/9 49-1-2/16 On the Ray Method of Calculating the Intensity of Wavefronts. luol = V# I - f (0) ;I pb (Eq.4'.3) In the three-dimensional case a ray is characterized by the two parameters a, and a. and Eqs.(4'.2) and R .3) have their analogies in: IU01 - -1 :~ f (CL1, a 2) (Eq.4-5) I -Vja?a and; U01 f (Ply P2) (P-q.4'.7) Yjb pb where J The authors conclude by consider- ing three examples: (1) The reflection of waves from a curvilinear boundary; (2) Media whose inhomogeneity de- Card 8/9 pends on 1 coordinate; (3) The diffraction of a cylinctri- ljj-1-2/16 On the Ray Method of Calculating the Intensity of Wavefronts. cal wave at a wedge. There are 3 figures and 28 references, ?l of which are Slavic. ASSOCIATION: Ac~, of Sciences of the USSR, Leningrad Branch of the Mathematical Institute imeni V.A.Steklov (Akademiya nauk SSSR, Leningradskoye otdeleniye Matematicheskogo instituta im. V.A. Steklova) SUBMITTED: July 29, 1956'. AVAIIABLE: Library of Congress. Card 9/9 BABICH, V.M. Propagation of no.246t228-26o uneta lonary waves 158, 1. Leningrndskiy go 'retvanMy ve motion, v 4 and tho enuetle, Uch.zsp. LGU (xrRA 12-2) universitst. Theory of) ia 88871 S/044/60/000/007/025/058 C111/C222 AUTHORt Babich, V.M. TITLEt The propagation of instationary waves and the caustic PERIODICALs Referativnyy zhurnal. Matematika, no-7, 1960, 119. Abstract no-7718. Uch.zap.LGU, 1958, no.246, 228-26o TEM Let a wave process be described by the equation 1 02(XVY ) utt-uxx- -U 0 with the variable velocity c(x,y), which is sufficiently YY dr smooth and satisfies the condition 0