# SCIENTIFIC ABSTRACT BABICH, V. M. - BABICH, Y. A.

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CIA-RDP86-00513R000102820017-0

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S

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December 31, 1967

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SCIENCEAB

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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