PHYSICS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80-00809A000600200333-1
• Release Decision: RIPPUB
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• Document Page Count: 17
• Document Creation Date: December 22, 2016
• Document Release Date: June 29, 2011
• Sequence Number: 333
• Publication Date: October 20, 1948
• Content Type: REPORT

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FOD
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20'Oct 1948
THIS IS UNEVALUATED INFORMATION FOR THE RESEARCH
USF OF TRAINED INTELLIGENCE ANALYSTS
SOURCE Russian periodical, Z4urpa7 Ekeperimup ntal?no,T i Teoretioheekoy Fiziki. No 7,
1946, (FDE Per Abe 40T93 -- Translation specifically requested T7.
P OUSTICAL RADIATIUlt OF i7SCILUTIlgG BODIES IN A COMPRESSIBLE FIJID
M.D. )haeklnd
NOTE: Numbers in bracket' refer to the bibliograph17
In this work Is carried out a theoretical investigation of the problem
of the acoustical radiation of oacillaLinp, bodies In a compressible fluid
for the most general case of harmonic oscillation of a solid and deformable
body.
STATE
ARMY
AIR
DISTRIBUTION
FOR OFfiCiAL USE KY
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During the oscillation of the solid body, the hydrodynamic forces acting
upon it may be divided into inertial and damping forcer..
The inertial forces are expressed linearly through acceleration. In
determining the lax of motion of the vacillating body, the coefficients of
the Inertial forces ectin' on the bccy play the part of additional masses,
of moments o' inertia, etc. Hence the coefficients of the Inertial forces
may be called connected masses, this being a generalization of the exacting
concept of a connected mass for an infinite and tncaapreseible fluid.
The properties of symmetry of connected masses, occurring in incaepreneible
fluids, remain valid also in the examined case, The numerical. values of the
generalized connected masses are related,to the frequency of oscillation.
The damping forces accounting for continuous exp?sndittxe of energy
on the formtic.A o,' acoustical waves are linearly related to the velocities.
The ease properties of uymmetry hold true for the coefficients of damping
as for the generalization of the connected masses. On the basis of the
formu2,i obtained for the generalized connected --no" -and coefficients of
damping, eccsrate and approximate calculations may be made for several con-
Orate cases.
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Ie. BABIC ERUATIOHS
A body oscillating in a fluid produces around itself a periodical
compression and vacuum in the fluid and, hence, leads to the formation
of acoustical waves.
The general case of the problem of acoustical radiation of a body
of arbitrary fora producing small oscillations in an ideal canpressible
fluid can be reduced to the deteraination of the potential of velocities
? (x,y,e,t) fulfilling the conditions. On the surface of the body 8
the etres*line condition
obtains, where M is the applied point of the ,urface E, and v the normal
component of velocity of any point on the surface 3. We shall also oon-
eider the normal to the surface S to be directed into the liquid. The
function vn is related to the forward and angular velocities and to the
velocity of deformation of the body. We have:
is the vector of the velocity of the origin of the oocrdinatee;.Oa (extLyt(Ji;
In(Rr~,t)?Y?n +(I1xrn)n+Vpia nV+(r,Xn)i.+nV1, (i.2)
where ZlaU1ex+U'2'ey+U,e.Z
In the total mat of fluid the potential of velocities 0 fnlfi]ls
B, and r, = ON is the radius vector of point St:
-AU
f'd-3+ a'xB- (la3)
where S is the velocity of sound determined ty the pressure p and the
density a from the formula:
In addition to these conditions, the potential of velocitiea $
satisfies at infinity the principle of radiation in that the radiated
acoustical naves diverge in all directions from the oscillating body,
. e., at large distances from the body, radiated waves develop into
diverging spherical waves.
For all further cases I *t us examine a case of simple harmonic
oscillation of a body resulting from a frequency k
V=Vb ki /l we? jV =v1e iklU
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d .. ay az c
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I
Ears, and later in complex expression containing the multiple e L i t
lZ
it is necessary to examine only the real part.
Considering the disturbed oscillating motion of the liquid as constant,
let us assume that
`lo determine the function 9 (x,y,z) we have the conditions
+Ai** +)/'p-akeycndS(vak1c) (1>7)
6Xi dY1 dz2
11 (-A,yf,z t)'i (x.y,a)eIAt.
a n.Y f (-?e)fn),wtn.vg onS/,
R-P 40
1M
For the function-4? (x,y,z) we have an equation giving a generalization
where the points P ( ,rj,fb) move along the surface S and the external
direction of the norms is lloved.
On the basis of formula 1.10 it is easy to determine the asymptotic nature
of the disturbed motion of the fluid with larger values of R. For this purpose,
let us introduce the spherical coordinates of R, t9. with the center at the
origin of the coordinates;, 1. e., r
x:Psis 8 cos y, y R ~s~s slc z Jkas 8
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l
~r sl n ~ cos ~ + ~js~n ~!.)t LOs 8 (1.14)
~11
and ? (Q)denotes terms of tae order of smalleee ZL.
Thus from formula 1.10 we obtain the asymjtoticfformuia
S~eavx1il - iv so rr;h?t40s(0, co.st-
d~- 1-- (1.16)
TCso5l7l~l~l Sin ?iT V,-%O7, ~ ! ?. --.s1 r
aMao wavAa/ "--'
on a sphere having retina R with the center at the origin of the coordinates.
N= Jdr~p R ,4'L s;n ids,
where p is the pressure exerted by the oscillations of the body,
2. If
(UV). : ,,, u - d t a W 7r + it R i = (1.19 )
-P x7r f I -q
ehero the line over a Letter denotes the usual transition to a compla:: con-
illsalov na those values and proceeding to the limit where k-+ao , we
n.n? 1
(1.2C)
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The function Q (V,8, ~r) ie linearly relatad to the cceplex amplitude
of the forward and angular velocities and to the amplitude of the velocity
of deformation. Therefore, the energy of the radiation is quadratic in form
with reference to these amplitudes.
II, aMWALI= CO !I!CTo MA83i5 A!) CFICI$IiT9 OF McHMATIOI
It has been established above that the problem of the determination of
the disturbed motion of a liquid resolves itself into a determination of the
function 9 which satisfies the conditions (1.7 and 1.9). In
view of the linearity of the problem, it is possible'to obtain
where the erectors ? and have projections on theta rem of of the the boor uaates
In the whole mans of fluid each of the funatacne'lnlestdefies the wave
equations
.a& Von a''. m tax em V1 (2.3)
and, finally, these functions satisfy the prinoip13 of radiation
4~ /Ata +iLyon1 =e. (2.4)
4-0Go ( a R
From the general analysis abo4e it follows that with large values of R,
the asymptotic character of the function 9'(x,9,?) is determined by -he
formulas
s fix' y'
? "' qua (Y' e' r) ' 7-74~" t. C1 1
1 reiyJ&(;. -7Vfg,,tsndes(o. )&i.iT
s (2.6)
+e:esr n~s,?r1tws(r"4)COS 91)ds.
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The functions lP?z(lL,rq,Z)e'ki(niil, 31101, potentials of the velocity
of the disturbed section of the fluid during oscillations of the solid body,
the o?" ente of the velocities of which have one amplitude, and the function
9pn 9 is the potential of the velooitibe of the disturbed movement of the
fluid during purely deformatory oscillations of the body.
The functions ?, (x,y,z) are related to the form of the surface S and
the parameter ar . The symmetry of the surface S accounts for the correspond
img symmetry in the structure of the functions P. (x, y, x) L without relation
to the parameter Y . In fact, If the surface 9 is symmetrical in relation
to the plane Oxz, it follows from their border conditions (2.2) that
a , e) 9~s (a r, 3, r)
dA IM ~bn /M' '...
()b-(),1, 6)
where K and M' are points on the surface R which are ey?etrical in relation
to the plane Oxs. From these relations and from formula (1.10) it follows
that for those parts of the plans Oxt inside the liquid, the following
equations are correct:
4.01=0 (, .,,3,S). 9m(X,y,7-)=0(7-ts~. d~. (2.9'
y
At points symmetrical with reference to the plane Ozz, we have
x)m (X, 2) S) (2.10)
F,,, (x,1,a)=.'-50", (x,- ~) Z)(rn=2, .44.).
If, in addition, the surface S is symeastrical with reference to the
plane Oyz, we shall have:
(2.122)
\~ M~1oa -!des r, 9 rx,~f .z). ~p?i(-X,J-~z N,.i 2.13)
a.~
J4 )
where P esmi P` oa a points as the surface S which are eymetrie:al with
reference to the plane Oys.
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l
JM i"; k1t% -,pff 5' d.~~~,7nao~2...9X2.14)
.5
It Is obvious that these coefficients are related to the geometric pro-
perties of the surface S and are functions of the parameter Y . A matrix
of the 6th order, composed of these coefficients, is symmetrical, i.e.,
In fact, applying Green's formula to the region D, outside the surface
~{++, where J,` is a sphere of radius R, we obtain
OR i-ez - 5P Saw -fff (,p,,d 90,A' J)d-r.
5 t-2
But the right-hand part of this equation, due to equation 2.3, becomes
zero. On the other hand, on the oasis of the asymptotic formulas 2.1+ we
have:
Hence after the maximum change R - co , we ehail hase
SS(~? - 9:-M )oe
s
and this demonstrates the eyamrtry of the matrix of the coefficients crj nv:
If the plane Oxz is the plane of symmetry of the surface S, out cf the
21 constants dotermining the matrix of the coefficients ~s, ,, only 12 coe'-
fioients will differ from zero, i.e., when j s 1,3,3 and as 2,4,6, we have
"w Q . If, in addition, the plans Oyc is the plane of symmetry of the
su ddrface S. '".en, ee:,ept for the diagonal ooefficiente, only G15 and O1& will
differ from zero. Finally, if the surface S has three mutually perpendicular
planes of ermsetry, only the diagonal coefficients Gjj will differ from zero.
Let no carry on the analysis of the hydrodynamic forces acting or. an
oaaillat'ng body. Using F to denote the principle vector of the hydrodynamic
'forces acting upon the body and B. for the principle moment of these forces in
relation to the origin of the. coordinates, we shall have the usual formulae;
Vke }(f, v +4, a,+ gpe ).
F=- Spflds, !'. a-~Spfiexn)dS,
,1s s
(2.17)
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F and Ng are bydrodynemio forces and the moment caused by the oscillations
o the body, investigated as a solid body, and
rutting the expression (2.18) into'the formula
F 1H FO 4- rd, p I' ? a MG + Md
Bare F. and lit, are b'drodynaaic forces canned by the purely defarmstory
oscillations of the body, and
F--pike10 fid S' 'M? _! ke (,r.)( n ) d5 ; (2.20)
J "a
Fd=pk lof (O' .V t #vw) dS,
Mdapi$ "fff (.1 fi W
(2.21)
Uploylog the expression (2.11) for the force. Fd and the aamant Md,
ve find thatt
E rI)m A& .
n-1
Anm (q = /, Z, s,,) (5) (2.22).
Vhere , B2, Z are projections of the principal vector of the hydrodynamic
force. 1d, and 14,, Z5, E6 are projections of the principle moment Md of these
forces.
Thus, the hydiradyneaic forces pr.,duoed by the oscillation of the solid
body are divided into inertial forces;
A , (2.23)
d dt
Un . (2.2k)
a-I
lb determining the law of, action of an oscillating body, the coefficient.
I4a1 of the inertial forces acting on the body play the part of additional
sasses, of tents of inertia, etc., and the gtmsetitiea~n are coefficients
of deformation.
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Therefore, the ooefficiente}A4oin of inertial forces may be called
connected masses, this being a generalization of the existing concept of
connected masses for an infinite and inoonpressible fluid. As stated above,
the coefficients pnn,,,,a,nd),,,,,are related to the geometric properties of the
body and are tunotiohs of the parameter V , more precisely, of LJA , where
.t901w is the length of the radiated acoustical waves and L I. the oharacteriatic
linear dimension of the body.
In the limited case when '1= /1 , i.e.,-when there are very long radiated
wavee(X.' L), the functions & m in the vicinity of the body satiefy the
laplace equation: `Y
d% , In ... a`` m + = 0'
d X?- a Is- y
and, consequently, the coeffirtents (,r* are real.
(2.25)
GJm f'jr""-rf~J dOmd 5, AJm =0.' (2.26)
In this case the limited value of the generalized connected masses 4tj m
coincides with the values of the connected sasses during the notion of the
body in the incompressible fluid.
In other limited cases when .I as, i.e., when the radiated waves are very.
short () L), the acausticai radiation proceeds according to the law of
geometrio acoustics. Saoh eieeent of the surface 8 radiates a plane wave in
which the velocity of the fluid equals merely the normal -ampanent velocity
of the given element. of the surface Be
Proceeding from ooneiderationsof energy, let no express the coefficients
of deformation by the energy of radiation carried off by the waves in unit
time. Let B be th+ mechanical energy i.e., the kinetic and potential energy
of the vrlume of the liquid in the region D, Included between the surfaces S
and . Then, applying the theory of energy of this volume and confining
ourselves, for simplification of the radiation, to oscillations of a solid
body, we have
where N is the energy carried off by the waves over the surface of the sphere
during unit time.
The total energy E of the volume of liquid +:nder examination during
oscillation is a periodic function of the time; therefore, the average value
of the force dr/dt for the period of the oscillation Is reduced to zero.
Fu ther ore, it is easy to see that the work of the inertial forces dosing
the period of oscillation to equal to zero. Thnc we obtain
ft'
E n mAnm Re O n '-rn cp. (2.2E)
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f
Here the summation in the second sum extends to all values n.$ a without
recurrence. From equation 2.28 we find the formula for An,? at all values
of n and mt
>,nm a C np(? , if t)fr/n~oncl 2,ifn = n), (2-29)
During oscillations of a solid body the function 0 (1/)?,*), responsible
for the disturbed motion of the fluid, may be expressed linearly by the function
Qn C!, 19,4x)
Un Q?
n..r
Coneenuently,'after substituting in (l.2C) we obtain
.Ram= ~ `~ ~ Qn or,, ei ;l Q`~'~'' )S-n?d ~
x.1T Tr
so
The whole statement given above refers to the, spatial problem of the
acoustical radiation of an oscillating body. In the case of the plane pro-
blem of the radiation of the bogndary S the problem reeoivee into the dster-
aination of the functioniq(x,y)g,"fulfillIng the conditions:
The last of thepe conditions exprc.;aee mathematically that, at great
distances from the boundary the radiated waves are converted into cylindrical,
eaves.
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The form the function Q
$ (H 0 (yr) a'? - 6?J H?2)P))d5(2.36) tj. an
whare Hv ,f-y)r) is HankeVe function of the second type,.r is the distance
between point (x,y) and point (yF 1) of the boundary a, i.e.,
r= d( ._ ~) #(y_n) U
as previously, obtain for the function Ot (x,y) the asymptotic formula
(F (X,y) 8n R1 e {~~4`+
where R and 0 are polar coordinates, and
fe '05, (2.39)
R, and with the center at the origin of the coordinates, we obtain the follow-
ing feirauia for the average magnitude of the energy of the radiation carried
It is possible quite analogeu dly to carry on an analysis of the division of
the h1drodynaeio forces acting on the oscillating boundary a. Considerestions
of energy lead to the following formula for coefficients of deformation during
Ncp~ ~PP- 7rI'/Q?fr (9)~' 10 (2.40)
i/
+7r
x/Ym a s Qn (r 9)Qerm (- 0) d e,
Q" (Y' (9)=
k "1COS9+nsine; 6Qm tre$? S(n,4)cos 0+
+i o5(fjn)Sig,pj1 1-1
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QD(y, 4) a
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1
and the functions (x,y) (m 2,3) fulf'ill`the conditions (2.3'!) and (2.35)
the following conditions on the boundary a.
and
III. MPI,ES
Let us examine several examples of calcu]ation for generalized connected
masses and coefficients of deformation, Let us first carry out an approximate
calculation for ooefficientsof deformation based on the application of the
energy formulas (2.31 add 2.,41).Exsmining the mall values of the parameter d
according to the degrees of
Let us take a eymmstrical body with reference to the coordinate planes,
oscillating progressively with three degrees of freedom; then, from formulae
Q,miJsin 19 cos Qat; )3in e
AA. j-3 ) ,
Q3=% r Cos a ( y+ CO)
where V I a the. volume of the body, and/. 1.1 (0), mow, P2_(0) and)A 33 (0) are
According to formula 2.32 we find:
X. mpKV03 r'( . - AA-An (0)
nn /~21T I P
Here the curved mark (^+) over the letter denotes a valu4 01 given
magnitude at mall values of the parameter Y
Analogous formulae bold-true in the rotary oscillations of a body.
In the particular base of the arbitrary configuration of a disk during its
--i vcos &rf e "'?'
lie
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where ie the value of the functlona'~lL on the upper aide of the disk,
and Qrt is the value of these functions on the lover side of the disk. Re-
taining in formula 3.3 only the terms containing .) r ,, we find:
As an application of formula 2.41 let us examine a plane lamina of '
'Infinite span effecting vertical and rotary oscillations. Proceeding as in
the previous examples, for the functions and Q' ve obtain-
Q10 00 ~, Co5 a 2) p~ . -~-1~~s~orP (0)
Employing the veil-known expression for connected maeeec of a plane-
let us now examine some examples of ex of calculation for generalized
cormected naesee and coefficients of deformation during acoustical radiation
of oscillating bodies in a compressible liquit.
An a first exempla, let us investigate the acoustical sadiation produced
by a globe of radius a during its vertical oscilzatfon. The fuuction fi(x,y,z)
corresponding to the oscillations of the globe takes the form:
where r to the distance between the point P (x,y,z) and the center of the
globe and 0 it the angle Between the direction of radius-r and the axis Oz.;
According to formula 2.14 we have:
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and, conee'fuently, ve find the p x reepions~. (~S) and 7 ('), represented
in Figure is
4 +2 PV-
4
v-- R
'?? = QA (0)=3~p~~~~y 1rPR Y R~~ (3.7)
In a like manner it is possible to find the exact value of the connected
mane and of the coefficient of deformation during oscillations of a cylinder
in the direction of axis Oy. For the function Gj (x,y), responsible for the
oscillations of the cylinder, we find that:
i a We 1) (d r )
(3.8)
2i' HON c ('/r) Sirs O (x vA),
is l-
and, according to formula 2.14, we obtain the expressions for).A (I) and?e?(d),
represented in Figure 2i
}. (y) - ,t (o) h ( J)11_ -nde + ((W-x No
t N,)L No)
L f,-X.xNo) (3.9)
71'1'' M - xt/P j''+ (At, --K No j
(j. (o)=pir R , ,
Her+ JD (x) is Vessel's function and Np,(x) is Nc.nmann'e function.
1+4+ 11-44-711+41-
1 z 3 4 /R 6
Figure 1. The Connected Mass and the Coefficient of Deformation During
Oscillations of a Globe
If the wave functions of tame and Mathieu ansA the function of a parabolic
cylinder are employed, it Is possible to investigate several more general prob-
lems, for exampla, the problem of acoustical radiation arising during the
oscillation of an ellipsoid, an elliptic or parabolic cylinder.
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I
.-I I I J H+1 J--FJ1fffl#1
/ 3
eimpleet example of acoustical radiation produced by a plane lamina of
infinite span. In this cage the function Q (r,y), responsible for the
vertical and rotary oscillations of the lemma, is determined aacordirg to
An an application of the above-mentioned functions let us examine the
tjgure i. The Connected Mesa and the Coefficient of Deformation of a Cylinder
0 (x.iy)= -Y-d?, (X,y)+.l
angular velocity and the functions Q, and Qifulfi3l-the vave ogtation and
Introducing the,0lliptioal coordinates sc. ? Zy--QC.h (9 + 'L V) ), ye
The border conditions (3.111 in the ooordinatee f and 7 viil take the forms
a . n sirs n dQ ~.Ln~ei.n O? g,)hi^r? s O /9 ,>>
~, s ~,. O eyher~ =C~ / % 13^ ex (~ D. jr). (3.1k)
let as represent the funotiane Q,a,odt%in terns of dissociation in 0" 1"
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r
- 16 -
41,E / k) rJ4? 111 {I~l?
Aare sezotl (n) and5e an( r1) tare tie periodic odd functions of Mathieu with
the following normalisation
By employing.the orthogana1Mathieu function anti conditions (3.13), we obtain
the following expression for the coefficients iAoi and 0,,L4, :
which fulfill condition 2.;5), and constente Bye,-, and p,, Ar are the coefficients
of dlesooiation, which are to be determined.
It is easy to see that the dissociations (3.15) fulfill condition (3.14).
~` Se -( i) 1 s'r1
5/ it
%n,.it4)and Sie ) are the corresponding Ftankel Mwthieu's functions
where Fsi and B2 are the first coefficients of dieaociation in the Fourier
series corresponding to Mathieu'e function Seri (f4) and -tcs,, (I) )
("'o3 designation) Al.
For the connected masses and coefficiente of deformation we have the
formless
w.
4,
din Ifd i i~'w~~~LJ/wyl-~3.17)
STAT
Sanitized Copy Approved for Release 2011/06/29: CIA-RDP80-00809A000600200333-1
 Sanitized Copy Approved for Release 2011/06/29: CIA-RDP80-00809A000600200333-1
I
M --- A - - -137 se- E!
The oooffioients B and B2 deorease very rapidly with an increase in
the ntaber n; ocasequenfly, to oalcubste ^-y, and, ,,,g, it is possible to
take a small nastier of terms in the dissociations (3.15) and (3.19).
1. H. Iamb, &Jdrodynemios. 1932.
2. K. A. 0 Strett, Low and Mathieu Iunctic s and Fnnstions Related to
t in Phseioe o]ogtt. , r9 y.
Sanitized Copy Approved for Release 2011/06/29: CIA-RDP80-00809A000600200333-1