D-R74 652 SUPPRESSION OF FINITE-AMPLITUDE EFFECTS IN SLOSHING 1/1MODES IN CYLINDRICAL CAVITIES(U) NAVAL POSTGRADUATESCHOOL MONTEREY CA H Y SI DEC STUNCLASIFIE F/G12/1i W
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NAVAL POSTGRADUATE SCHOOLMonterey, California
DTI
BTHESISSUPPRESSION OF FINITE-AMPLITUDE EFFECTS
IN SLOSHING MODES IN CYLINDRICAL CAVITIES
'
by
C.-.Si Hwan Yum
December 1.983
C.2
Thesis Advisor: Alan B. Coppens
Approved for public release; distribution unlimited
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4 q B
* * S~~11CURITY CLASSFICATIOIG OF TWOS PAGE (0woa Dos Entoe.E) ______________
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4. TI TLEI (and &ua*I. S. TYlet Of REPORT A PERIOD COVERED
Suppression of Finite-Amplitude Effects MatrsTeiDecember 1983
N?- in Sloshing Modes in Cylindrical Cavitiesi6. PERFORMING ORG. REPORT NUMBER
1AIJTH@NW 11. CONTRACT OR GRANT NUNUER(a)
Si Hwan Yum
SPERFORMUING ORGANIZATION N AMC AND ADDRESS W0. PROGRAM ELEMENT. PROJECT, TASK* . AREA A WOOK UNIT NUMBERS
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I&. SUPPI.EMENTARY NOTIES
Is. KEY WO01104 (CUo an ovovee ie Ifft ae pend sme Utlg , eek number)
non linear wave equation
)A pertubation expansion is formulated for the three dimension-
al, nonlinear, acoustic-wave equation with dissipative termdescribing the viscous and thermal energy losses encountered ina cylindrical cavity. The theoretical results show that non-linear effects in sloshing modes are strongly suppressed.
00 1473 gDo FINVO~ODSLT
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Suppression of Finite-Amplitude Effects in SloshingModes in Cylindrical Cavities
by
Si Hwan YumLieutenant Commander, Republic of Korea NavyB.S., Republic of Korea Naval Academy, 1973
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN ENGINEERING ACOUSTICS
from the
NAVAL POSTGRADUATE SCHOOL-December 1983
Author:
Approved by:SThesis AdviTsor
Cha Egineering Acoustics Academic Committee
1 Dean of Science and Engineering
2
ABSTRACT
A pertubation expansion is formulated for the three-
dimensional, nonlinear, acoustic-wave equation with dissipa-
tive term describing the viscous and thermal energy losses
encountered in a cylindrical cavity. The theoretical results
show that nonlinear effects in sloshing modes are strongly
suppressed.
4%'T
AvilbliyCoe
AvAcein aaor
V Dist Aveil d
.3
TABLE OF CONTENTS
I. INTRODUCTION----------------------- ------------------ 6
II. THE NON-LINEAR WAVE EQUATION-------------------------- 7
A. GENERAL----------------------------------------- 7
B. APPLICATION TO THE CYLINDRICAL CAVITY--------------10
1. Symmetric Modes------------------------------- 11
2. Non-Symmetric Modes ---------------------- 14
JI. METHOD OF SOLUTION------------------------------------ 21
A. POWER SERIES METHOD------------------------------- 23
1. Coefficient --------------------- ------------ 23
2.------------------------------------------------ 27
3. Un(Y)----------------------------------------- 30
IV. CONCLUSIONS ------------------------------------- 314
LIST OF REFERENCES------------------------------------------ 36
INITIAL DISTRIBUTION LIST------------------------------------37
4'4e,
1%
LIST OF SYMBOLS
. phase velocity in the cavity
C1T~j~ C-674
CI effective phase speed associated with a standing wave
at resonance
Cspecific heat at constant pressure
k W/c, propagation constant associated with a standingwave
. peak Mach number of the driven standing wave
=P- P.=- oOIj acoustic pressure
.. instantaneous and equilibrium total pressure in the
field
_T absolute temperature in Kelvin
LA particle velocity
particle speed of the nth-order perturbation solution
infinitesimal-amplitude attenuation constant
=(Y + 1) /2 for a gas
=/?/Cv, ratio of specific heats for a gas
instantaneous and equilibrium densities
velocity potentialI;f, (angular) grequency at which the cavity is driven
(angular) frequency of a resonance
.4r
'"; .' 4 5<' - ..- ,,.'-- -. .. . . . .. . - . -.-- ..- . .
i -#1, " "I ' _ , . , . . \ " '"" ""'
I. INTRODUCTION
The topic of finite amplitude acoustic standing waves in
* sloshing modes of a cylindrical cavity is interesting
theoretically, but development of the subject has not been
extensive. The purpose off this research is to present the
results of a power series perturbation approach to the
problem.
44*~6
II. THE NON-LINEAR WAVE EQUATION
A. GENERAL
It is well known [11 that for Z//4kand M<<, where
measures the fractional loss per wavelength and M is the peak
Mach number of the source, loss terms and nonlinear terms in*2.'-.
the constitutive equations can be separately approximated with
*.. the help of linear, lossless acoustic relations. The force
equation appropriate for acoustical processes in systems for
,',-- which gravitational effects are unimportant is
..%
whereU = U is the particle velocity, Cis the instantaneous
density of the fluid, P is the instantaneous total pressure in
the field, and the operator Lsymbolically describes those
physical processes leading to absorption and dispersion. We
used two additional equations. The first is the equation of
state for a perfect gas
p~CrTK (2-1-a)
where r is a constant whose value depends on the particular
gas involved and TK is the absolute temperature in Kelvin.
The second is the continuity equation
' , ' ' ' - :k.:... .:.;..., .. ,..: ..,. ,..,.v ... ..- ,... , .-.... .... ,7.
*k ."
+ V.L I+S)UJ (2-1-b)
where s = (-)/ois the condensation at any point and
is the equilibrium density of the fluid. If we ignore
rotational effects, then
U~ (2-2)
* where is the velocity potential. Combination of Eas.(2-1)-
(2-2) and the neglect of terms of orders higher then M2 , M(" /z),
and ( / )2 yields a quadratically nonlinear wave equation,
7-C)
where Co2=(.r/:j,) (adiabatic), p=acoustic pressure,(=U.(
-" is the ratio of heat capacities, and
C F1 Lo-7 (2-4)
The left-hand side of Eq.(2-3) is the classical, linear wave
equation with losses pertinent to the system under study.
The right-hand side can be interpreted as a forcing function
consisting of a three-dimensional spatial distribution of phase-
coherent sources.
In a second-order perturbation theory, this volume forcing
function is obtained from the classical(first-order) solution?l of the acoustical problem. The second-order perturbation
solution PI+P 2 describes the non-linearities P2 resulting from
the self interaction of the classical solution Pl.
Higher-order perturbation solutions consider the interaction
of the nonlinear solution with itself, and the forcing function
is composed of products of both classical and non-linearly
generated terms. Thus, if a system is driven at frequency w,
the non-linear term in Eq.(2-3) will force the existence of
all integer multiples 11W of the driving frequeney and the full
solution must contain all harmonics of the input frequency.
In a closed cavity, each of those nonlinearly generated waves
whose frequency lies near the resonance frequency of a standing
wave of the cavity and whose assciated spatial function matches
that of the standing wave can be strongly excited (2)
As far as the author has been able to determine, there has
been only one previous study of this system published in the
open literature. This was by Maslen and Moore in 1956 (3)
Their approach resulted in a series expansion which did not
converge if the relevant nomal mode frequencies were integerally
related. Their interpretation of the quenching of the nonlinear
effect was based on the "scattering effect of the wall".
JK 9
We feel our interpretation based upon nonlinearally generated
volume sources stimulating the allowed standing waves of the
cavity is more accessible and informative. Further, the
mathematical approach developed herein appear to avoid any
difficulties in convergence. Their conclusion that high
amplitude monofreqlzency transverse oscillations can exist is
consistent with our finding.
B. APPLICATION TO THE CYLINDRICAL CAVITY
The circular cylinder is provided with a "point" source of
*, sound. By properly positioning the cylinder with respect to
the sound source it is possible to effectively drive the
enclosed air into various modes of vibration. The rigid-
walled cylinder forces the component of particle velocity
perpendicular to each cavity surface to vanish at the surface.
The resulting steady-state solution to the linear wave equation
in cylindrical coordinates is
P ~ ~ ~ kcm~o a s ) cc65 (ttmtT(2-5)
where Jn are the cylindrical Bessel functions and applicationof the boundary condition to the sides yields
.5L
7'.
where a is the radius of cylinder and J are the argument7".t..
of the exterema of the Bessel function.
The nomal mode frequency is dependent on and
('vrn[' ,- (2- 5- b )
The standing wave well be identified by the ordered integers
(n, m, ) describing its spatial dependence.
1. Symmetric Modes
The simplest waves in cylindrical coordinates are those*. -"
- that depend only on the distance r from Z-axis, the gradient
takes the form
(2-6)
where
* .r (2-6-a)
and k=w/c is the wave number or propagation constant.
The Laplacian becomes
.3 \7 ~ I(2-6-b)
The suitable trial solution appropriate for symmetric modes is
the linear solution(O, m, 0) of frequency w
11
-. - - - .-.•*~- .
The velocity potential for this limit can be defined as
-Uand is
Lit M c ~ ,]J
Because we assume that the surfaces of the davity are rigid
then at r=a the appropriate boundary condition is J,'=O.
Thus, we have -k= j=0"
To generate the second order solution, we first note that
P - zZ o~t
and
Next, the second derivatives of Eqs.(2-10) and (2-11) with
respect to time are
12
T (2-12)
and
Li ~ )Z H ~t - -(2-13)
Substituting Eqs.(2-2) and (2-13) into the first term of the
right-hand side of Eq.(2-3) yields
(7W XS-~ (2-iL )
!'-/ co5, zt T,(-4and the second term of right-hand side of Eq.(2- 3 ) becomes
4-i 3 (2-15)
since
'J or
CO T,(2-17)
With the use of Eqs.(2-14) and (2-15) we can write the right-
13
hand side of Eq.(2-3) as
,~X,
-Cos Z4~ (2-18)
and the appropriate inhomogeneous wave equation for the second
order perturbation solution is
i e,. _ )' M = _z -oj L-Z.C: ]-t X (2u aj 4rT.+ z 2 1 ri
" - +.,. _(Y ",)JO CeSzr 7. (2-19)
2. Non-Symmetric Modes
The non planar waves in cylindrical coordinates are
those that depend on the distance r from Z axis and the angle
" from X-axis. The gradient takes the form
(2-20)7.=- . (± )o
where
14
. . .. .... *. . ... . . .... .. ... ;
I.P 17,
and the Laplacian becomes
V i< + w (2-20-a)
The suitable trial solution appropriate to the forcing function
of frequency w for exciting the (n, m, 0) standing wave is
P, D (2-21)
* where Jn(r) is the Bessel function of the first kind and
rder n. The velocity potential is approximated by
Ja
Tr =-r (L4,) NC TW~~~t (2-22)
W1%
- and t4, is
.?.
Thus,
or
F +t tVCts Z'A (2-25)
15
,...
and
(2-26)
and the second derivative of Eqs.(2-25) and (2-26) with respect
-. -to time is
,IM%]z_ --- - (2-27)
and
± ~ ~ ~ C [iZ2 2J } -/l }2-8)Substituting Eqs.(2-26) and (2-27) into the first term of
Eq.(2-3) yields
16
(2-29
i °. . . .
'4,
- a or
.
IT I±tl -1+C- (2-30)
since
::r ' < % 7- Tr._, r--, )
In 6rder to solve the second term of Eq.(2-3), we can use
C V =x v, (1 I z .. (233
.4.:
mft~ =j; + o (2-35)
17
n .'.e."to solve the.second term of.,'3-" : "..we
can use-.-.:1 2.
j7.
"* since
T(2-36)
andiXTr I'p It,(2-37)
Thus, Substituting Eqs.(2-33) through(2-35) into the secondI
term of Eq.(2-3) yields
C os;C- 7TTtt .., v -7.v IJ + Tl , - TjO ±A) (2-38)
and
CC. + -T14;;5/0-14 TMj4)-D 1 (-9
Ci 18
I ° . o . • -
--C ,L-' .m ',.' " ', " -'" ' " " "d" "" " " ' " " " " " " """ "" " ' """"" " " " "" " "" "" . .
With Eqs.( 2-30), (2-38) and (2-39) we can get the inhomogeeneous
wave equation for the second order perturbat ion as
8X
-5..:i.~~~~ -;r -/ -T l- )
V'..-o
c, El+Tltll-
z +
7- +
/,0 ')T3 i.+ - 1 T1
- FT~Jt TT +(Ni
l,
..w % t, € .. .. . . .. .. .... .... . ..... ... ....... .. .-... .- '. .-
71 - . -7 .- -
If we let n=O in Eq(2-40), it reduces to Eq.(2-1 9 ), as it must.
Substituting n=l into Eq.(2-40) yields the equation with
forcing term resulting from the (1, m, 0) sloshing mode,
4%,
+ M
4.. CL T, ]7
(2-41)
' " ' "2 0
))" o,
" ,' '.'.I - J ._ LV ... ,". . . . '.ZY' .' '' . ) )C, & $'"" . j " - %h ", % "- ' "-,ij
m4.
III. METHOD OF SOLUTION
Recall that the equation with forcing term resulting from
the(l, m, 0) sloshing mode can be written as a function of
frequency,2'.
-- 4 z 0.
Co-_) (3-1)
and the left-hand side of Eq.(2-3) is
-4 V~ +- (3-1-a)
If the harmonics of the frequency at whichi the cavity is driven
are not close to any resonant frequencies, we can ignore the
lossy term. So, Eq.(3-1-A) can be written as
Let us assume that the solution of Eq.(3-1) can be expressed as
P:C -- w UR()+' I '57_ (3-3)
21
, o .21
:."' 'i++ - .
• , . - . . .° . - . . .- .- . *, , . * *- - -- v-,.. .. J , w * .- ~ -,
-. ' - and define
= ~V1 -~)CO C-(n (3-..4)
and
PV' M ,Co t 35
Combination of Eqs.(3-2) and (3-3) yields
I,,
-.. +
and
-p..°'
-- -C (- ' - ) TJ (3-7)
22-.
A. POWER SERIES METHOD
In order to solve the right-hand side of Eq.(3-7), we can
use the definition [4 ]
* ~ ~ ~ T(9T4V K+)~ (-I )9*(4& ztt('~
i ( -Z (3-8)
where3
1. Coefficient
a. Substituting V =L: n+2 into Eq.(3-8) yields
4+:4
or
> (3-10)
where
4 (3-11)
23
'. *, ". . " , - . - - - ,- . --. . - *- , . - . . . .. . . . . . . . .
b. Substituting ' = n+2,,t=n-2 into Eq.(3-8) vyelds
,-:7 :-0: 4:
"E ( . .. .T-~K) = (+i). - 1' Z -.a!!(31
(3-12)
- or
T Z i (3-13)
where
C~= ~-< ~ I(3-14)
c. Substituting Vhn+l,.(t~n-1 into Eq.(3-8) yields
-.. 1
(315
or
[-z. (3 -16 ):. .2(--o
24
,:: . ...::?::.""- ".:'-' " :-::.-,--.:-.': . --. ---.. -:.- .- .- . -- v ..- . .. -.. -..... -. . - . -. .- .. -.-
"-" where
( + (3-17)
d. Substituting V=?t=n into Eq.(3-8) yields
II
" )(V-z (3-18)
or(CC
where
A0m-.2')! (3-20)
e. Substituting V't:n+l into Eq.(3-8) yields
~44
7--A+Z-k - (7-)) -2-k
x (3-21)
-- k
25
%:,
or
where
f. SubstitutingV==n-l into Eq.(3-8) yields
00
.!..'.:4z Iez,.., (-, ka -2&) ! (-kiD
or.t .
where
26
* . * * . . . . . .°- . , . .
g. Substituting V--U-=n-2 into Eq.(3-8) yields
ZT-4!
or
I , where
2. Vn(y)
With the help of Eqs.(3-9) through(3-29) Eq.(3- 7 ) can
be written as
z IJ F\tw&~ j> dl J (3-30)
27
a,."
I--'. J
where! 5*4
* (3-30-a)
and
(3-30-b)
Now, let us assume that
'I0- L Q~1)(3-31).- C- t z
The first derivative with respect to y=:r can be expressed as
_______(3-32)
and the second derivative with respect to y=ir can be expressed
as
_____ V _ (3-33)
Substituting Eqs.(3-31) through(3-33) into the left-hand side
of Eq.(3-30) yields
4V.
28
* ° ~ '- . .° • • . .
-04 z?) (3-34)a
Eq.(3-34) must be equal the right-hand side of Eq.(3-30). Thus,
"- ±-i16"±+ - -
where k 0, 1, 2, 3 .(3-3)
If Eq.(3-35) is substituted into Eq.(3-36), then Eq.(3-35) can
be expressed as
.o,
ctA) t(-L) (iZ-I ) 0C;= - -G 4i DtI + AEbni)
(3-37)
29
"P;-"
"
The normal component of particle velocity is equal to zero on
the boundaries. From the application of the boundary condition
at the rigid boundary at r=a,
I"]" +* -Zj(3-38)
where
Substituting Eq.(3-36) into Eq.(3-38) yields
011 (-L ) (r+ 1 *""r+y -. ) 1 . . - .
0 (3-39)
3. Un(y)
With Eqs.(3-9) through(3-29), Eq.(3-6) can be expressed as
(3-40O)
30
k ' ";' -> t;,,,',.,':,.i:.-;.':'.:'.,,,-.-,;" -.:::.'::,-.><:; .-. :',-,.-,.-:-,- ' .". - ' . - •"- '-. " -." "-".-"'-
. ... ~~~~ ~ ~~ ~ ~~~ , . .. ."E-.' -,,, . -7,0 . . -.-. , ,,..,, . ,
W.4-3 . .7T -. 4 -7 -77.j -;6 - 7 -77,2- 7 .7 7 7
,4 where, as before, A Y /2-1/8 and a =
Now let us assume that
The first derivative with respect to y:4r can be expressed as
U10a DO kj -~ (3-'2
and the second derivative with respect to y=-.rcan be expressed
as
*" Substituting Eqs.(3-41) through(3-43) into the left-hand side of
Eq.(3-40) yields Q!fl4'>1z z( _ )JK i)4=0
Eq.(3-44) must equal the right-hand side of Eq.(3-40).
Thus,
314,:a
S
77:*
'P %
where -:1, 2, 3 .....
If Eq.(3-45) is substituted into Eq.(3-36), then Eq.(3-45) can
be expressed as
16
---z .,+ H,, E,,
,'. '
4.2.
(3-46)
32
.,. . . ....
. . . .
Application of the boundary condition at r=a yields
4 L .'' (3-4.L7)
where
Substituting Eq.(3-36) into Eq.(3-47) yields
+
(3-48)
33
'*--*.r*-..3 -3
t7,..,
.17-7
IV. CONCLUSIONS
Recall that we can compute from Eqs.(3-37) through(3-39)
and Eqs.(3-14), (3-17), (3-20). Thus, with the use of Eq.(3-31),
we can get Vn . From Eqs.(3-46) through(3-48) and Eqs.(3-11),
(3-20), (3-23), (3-26), (3-29), we can compute 11. Thus, with
the use of Eq.(3-14), we can get Un . Therefore we can compute
Vn(Y) and Un(y). Let us calculate the finite amplitude effects
resulting from the nonlinear distortion of a forced radial mode.
If we excite a(O, m, 0) mode and obtain the pressure at the
circumference, then Given this value of use
of Eqs.(3-14), (3-17), (3-20) give the quantities C, D, E.
From Eqs.(3-35) and (3-36) we can compute O . Thus, we can get
Vn(Y) from Eq.(3-31). Let us calculate the finite amplitude
effects resulting from the nonlinear distortion of a forced
sloshing mode. If we excite a(l, m, 0) mode and obtain the pre-
ssure at the circumference, then :-1'4, Given this value
of use of Eqs.(3-ll), (3-20), (3-23), (3-26), (3-29) give the
quantities B. E, G, H, 0. From Eqs.(3-45) we can compute
Thus, we can get Un(y) from Eq.(3-41). Substituting n=0 atradial modes into Vn(Y) and Un(y) yields V0 (y):U0 (y). Therefore,
with the use of Eq.(3-3), we can get the solution of Eq.(3-2).
A. RADIAL MODES(0, m, 0)
0L % MT 0 , (~1 kWt
.34
then
-" so that
B. SLOSHING MODESUl, m, 0)
'..< then
P2 -2 c s zt
so that
",V2
)/ I
..-.-
LIST OF REFERENCES'a..
1. Blackstock, D. T., "Convergence of the Keck-BeyerPerturbation Solution for Plane Waves of Finite Amplitudein a Viscous Fluid", J. Acoust. Soc. Am., Vol. 39, No. 2,pp. 411-413, February 1966.
2. Coppens, A. B., and Sanders, J. V., "Finite-AmplitudeStanding Waves Within Real Cavity", J. Acouft. Soc. Am.,Vol. 58, No. 6, pp. 1133-1140, December 1975.
3. Maslen, S. H., and Moore, F. K., "On Strong TransverseWaves Wi±hout Shocks in a Circular Cylinder", Journal ofthe Aeronautical Sciences, Vol. 23, No. 6, pp. 583-593,June 1956.
4. Abramowitz, M. and Stegun, I. A., "Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables",National Bureau of Standards Applied Mathematics Series,No. 55, pp. 255, 360, December 1965.
'...
-.'
,a-
36
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No. Copies
1. Defense Technical Information Center 2Cameron Station
Alexandria, VA 22314
2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, CA 93943
3. Department Library, Code 61Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943
4. Professor Alan B. Coppens, Code 6lDz 5Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943
5. Professor James V. Sanders, Code 61Sd 5
Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943
6. Chief of Naval Research
800 Quincy St.Arlington, VA 22217
7. LCDR Charles L. Burmaster, Code 6lZrDepartment of PhysicsNaval Postgraduate SchoolMonterey, CA 93943
8. LCDR Chil-Ki Baek 5Republic of Korea Naval AcademyDepartment of PhysicsChin-Hae City, Seoul, Korea
9. The University of Texas at Austin 1Applied Research LaboratoriesAttn: D. T. BlackstockAustin, TX 78712
37
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