SOME INVESTIGATIONS RELATING TO THE ...Some Investigations Relating to the Elastostatics of a Tapered Tube by Barry Bernstein 1. Introduction I The problem of elastostatics of a tapered

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Some ~ n v e s t i g d t i o r i s Re la t i ng t o t h e E l a s t o s t a t i c s o f a Tapered Tube

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Tapered Tube B i h a n o n i c Equations Hollow Tapered Cy l inder Method o f Charac te r i s t i cs E l a s t i c i t y Poisson I n t e g r a l Non-Separable Coordinate Systems Successive Approximations

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Separation o f va r iab les f o r the harmonic and biharmonic i s n o t poss ib le f o r the coordinate system desc r ib ing a tapered hol low c y l i n d e r so a l t e r n a t i v e methods are required. The method o f cha rac te r i s t i cs , Poisson i n t e g r a l repre- sentat ions, se r ies i n two coordinate var iab les , and the method o f successive approximations are discussed i n some d e t a i l . The on ly method t o show promise f o r f u t u r e work i s t h e method o f successive approximations used by A. Zak t o i n v e s t i g a t e s i n g u l a r i t y st resses a t the end o f a cy l i nder .

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TABLE OF CONTENTS

Page

. . .

INTRODUCTION AND SPECIAL L I S T .OF SYMBOLS . . . . . . . . . . . . 5

THE NATURE OF THE PROBLEM . . . . . . . . . . . . . . . . . . 8

THE WORK OF ZAK . . . . . . . . . . . . . . . . . . . . . . . . 10

INTEGRAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . 13

THE WORK OF NIVEN . . . . . . . . . . . . . . . . . . . . . . . . . 1 6

AN INTEGRAL REPRESENTATION OF THE SOLUTION TO LAPLACE'S EQUATION I N NIVEN'S COORDINATES . . . . . . . . . . 2 0

SOME POLYNOMIAL AND SERIES SOLUTIONS . . . . . . . . . . . . . 25

SHORT ANNOTATED BIBLIOGRAPHY . . . . . . . . . . . . . . . . . 3 0

. . . . . . . . . . . . . . . . . . . . . . . . . . CONCLUSION 3 5

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . 3 6

. . . . . . . . . . . . . . . . . . . . . . . . . DISTRIBUTION 3 7

Some Inves t iga t ions Relat ing t o

t he E l a s t o s t a t i c s of a Tapered Tube

by

Barry Bernstein

1. Introduction I

The problem of e l a s t o s t a t i c s of a tapered tube i s one f o r

which one searches t h e l i t e r a t u r e i n vain. The problem seems

t o be c lose t o t h a t of a cy l ind r i ca l tube, bu t this appearance

i s q u i t e deceptive. Several s i n g u l a r i t i e s appear i n t h e

tapered tube which do no t i n t he cy l ind r i ca l tube. Furthermore,

i n a coordinate system appropr ia te t o t h e cy l ind r i ca l tube,

separat ion of var iab les i s possible. Not so w i t h t he tapered

tube. I n this r epo r t we s h a l l explore some approaches t o t he

problem. No approach t h a t we s h a l l present has a t t h i s time

shown i t s e l f t o be t h e c l e a r way t o proceed. However, some

of them may have some promise.

After an in t roduct ion t o t h e problem, we s h a l l d iscuss

some methods found i n t h e l i t e r a t u r e . Then we s h a l l present

some new exploratory r e s u l t s . Although we cannot be conclusive

a t this time, we hope t h a t we have opened some p o s s i b i l i t i e s

f o r f u t u r e development.

References a r e i n t he annotated bibliography, Section 8.

Special L i s t of Symbols:

Because we a r e quoting from d i f f e r e n t sources which

use t he same symbols i n d i f f e r e n t ways, and s ince we wish,

with only, perhaps, reasonable modification t h a t t he reader

be ab l e t o recognize t h e symbols i n t he quoted sources, we

cannot be completely cons i s ten t i n using a symbol i n only one

way i n t h i s report . For this reason, we have compiled a l is t

of symbols here with t h e d i f f e r e n t uses of t he same symbol

explained. Symbols a r e l i s t e d roughly i n t h e order i n which

they appear i n t he t e x t , except t h a t a l l l i s t i n g s of d i f f e r e n t

uses of t h e same symbol appear together. If t h e reader w i l l

r e f e r t o this l i s t , confusion w i l l be avoided.

Svmbol . Uses - V p o t e n t i a l f b c t i o n ( sec t ion 2)

V a so lu t ion of (28) with m = l ( sec t ion 7)

C0,C1,C2,C3 constants ( sec t ion 2)

vrn Fourier component of p o t e n t i a l ( sec t ion 2)

R,@ ,Z cy l ind r i ca l coordinates

m an i n t e g e r

d" reduced po ten t i a l ( sec t ion 2)

~2 def.ined by equation ( 4 )

r={+iy coordinates f o r t he tapered tube ( sec t ion 2 ) ( e s s e n t i a l l y and 7 of sec t ion 2 a r e t he Niven coordinates and 0 of s ec t ion 5) P

9 7 c h a r a c t e r i s t i c coordinates ( sec t ions 5 ,6)

t h e dis tance of t h e i n t e r sec t ion of t h e inner and ou te r surface of t h e tapered tube from the a x i s

separat ion funct ions (equation 6 )

some a r b i t r a r y function ( sec t ion 6 )

t he dis tance from t h e s ingu la r i t y i n Zakvs coordinate system (sec t ion 3) except t h a t Zak takes c = 1, this i s t h e same a s t h e Niven coordinate r of sec t ion 5

a Niven coordinate (sections 5,6,7) in which p = dnr

Zak's angular coordinate (section 3). The - same as the Niven coordinate 8 (sections 5,6,7

Southwell potentials (section 3)

gamma function (section 4)

a curve (section 6)

a separation function (section 3)

a constant (section 3)

the taper angle (see figure)

a function (equation 7)

Cartesian coordinates

defined by equation (23) (section 6)

same as vm, but in Hein1s notation

self explanatory - (equation 9) Niven coordinates (section 5)

n/2 - '8 some special coordinates (equation 15 and following equation)

some constant coefficients (equation 17)

defined by equation (18)

,characteristic function (section 6)

Niven coordinates of a given point (section 6)

characteristic coordinates of a given point (section 6)

hypergeometric function (section 6)

points of intersection of characteristics with curve

as defined in the equation following (25)

a s defined i n equation (27)

a s i n equation (28) - same a s urn funct ions t o be determined (equation 29)

an index ( sec t ion 7 )

constants t o be determined ( sec t ion 7 )

an in t ege r (equation 34)

t he g r e a t e s t i n t ege r i n n/2 ( sec t ion 7)

coe f f i c i en t s ' t o be determined (equation 39)

2. The Nature of t he Problem

That t h e problem of e l a s t o s t a t i c s hangs on t he study of

Laplace 's equation i s wel l known. A review of so lu t ions of

such problems i n terms of p o t e n t i a l funct ions is given by

Green aqd Zerna, sec t ion ( 5 , 6 ) [13]. If one could handle

Laplace 's equation f o r t he tapered tube, then e l a s t o s t a t i c

problems would be acces s ib l e .

The first e f f o r t , ' then, t h a t seems reasonable i s t o see

i f separat ion of va r i ab l e s is possible. We tu rn , then, t o t he

work by Snow [I]. We consider here chapter I X , p. 228 of this

work.

In cy l ind r i ca l coordinates R, Z , @ , we have f o r Laplace 's

equation f o r a p o t e n t i a l V

From equation (I), we may immediately separa te out t h e

angular coordinate @ by wr i t i ng V i n a Four ier s e r i e s i n @.

Indeed, Snow wr i t e s

where Co, C1, C2, C and 0, a r e constants. The coe f f i c i en t 3 P s a t i s f i e s

O r , pu t t ing 7

i n equation (2 ) one obta ins an equation f o r t he reduced

p o t e n t i a l urn, namely

where

Now i f one looks a t t he R-Z plane one sees t h a t t h e t r a c e

on this plane of a tapered region can be represented a s a wedge,

one s ide of.which is p a r a l l e l t o t he Z-axis a t some d is tance ,

say, c , from the a x i s , and crossing the other s ide a t , say,

t h e Z-axis a t an angle a ( s e e : f i g u r e ) , which we c a l l t h e , I

t ape r angle. A conformal mapping, then,' from the Z + i R plane

i n t o t he plane of r = b + i q given by

r = ~ o g ( ~ + R - iC)

gi'ves f o r ( 3 )

Now equation (5) is an equation in 4 , T , which are natural coordinates for the tapered tube. Indeed, in this

coordinate system, the surfaces of the tube become r = 0 and

51 = a. One may say more: This is essentially the only ortho-

gonal coordinite system in which the surfaces of the tube

become coordinate surfaces for any value of a.

We now ask the question whether or not one may find

solutions of the form

where T. is to be found. The answer is given by Snow (pp. 252-

253). It appears that the answer is no, since

a2 e 3 t C 6e cosq(l+e cosv)

a t a? '(a + e' sin? l2 = C . 2 (a+e sin?) + 0

which, by application of Snow's result to our'equation implies

that separation of variables, even to within a known factor T,

is not possible for the coordinate system ( 4 , ?) . Techniques of separation,of variables, with all their

ramifications, then fail. Other techniques must then be sought.

And a look at some of these is then our task.

3. The Work of Zak

Here we shall discuss a technique used by Zak for solving

a problem of a cylinder with stress singularities. The method

happens to involve the Southwell potentials [14], but the

essential feature of it is the method of obtaining a sequence

of functions which approach a solution.

Zak used a coordinate system which is essentially that

developed in the previous section and, indeed, is equivalent

to Nivents coordinates (section 5). If we replace 5 by the

letter p and by the letter 8 , we shall have the coordinate system which he uses. In this section we shall adhere to Zakls

notation. HoGever, the same letter p will be used differently elsewhere in this report, so caution on the part of the reader

is urged. Please refer to the list of symbols. If referred

to the tapered tube, Zak's coordinates are the Niven coordinates

normalized so that c=l.

The Southwell potentials as modified by Zak satisfy the

equations

in cylindrical coordinates. After expressing these equations

in terms of his p and 8, Zak seeks a solution for, say, fL in

the form

and obtains a sequence of equations

so t h a t each funct ion F depends on t h e previous ones. A P

s imi l a r technique i s appl ied t o r. It is not d i f f i c u l t t o see t h a t Zakls technique could

r ead i ly be appl ied t o t he tapered tube problem: The proper

coordinate form and t h e technique a r e a l ready developed.

A t t h e time of wr i t ing of this repor t , we f e e l t h a t Zakls

method may be the most promisingwhere it can be applied. It

appears t o have two disadvantages. Zak expands a term a s

which has a s i ts domain of convergence a region near f = O .

(This region was of i n t e r e s t f o r t h e study of a s ingu la r i t y

For t h e tapered tube it may be of i n t e r e s t i f one l i m i t s

oneself t o regions where I f s inbl < 1, but t h i s means t h a t I

t h e r a d i a l length allowed i s l imi ted by the angle of taper.

For example, f o r a t ape r angle of Z O , t h e expansion i s

va l id f o r up t o about 28 (dis tance from s i n g u l a r i t y about P 28 times the quant i ty C i n Niven coordinates) and convergence

would probably be slow i f were near 28. P Although we do n o t see how t o do it a t p resen t , it may

be 'pos s ib l e t o apply Zakts technique t o a f a r away region.

But the trouble at the moment is that as one goes toward

larger P there are points closer and closer to the surface $= 0 at which (1 - f sin$)-' becomes infinite.

The second disadvantage which may be minor is that one

does not deal with a sequence of exact solutions. However,

this would not necessarily impair its usefulness where con-

vergence is rapid enough. Nevertheless, in the broad study of

the question, .a search for exact solutions should be made. If -...-.l. -,.1..+<-..- --.. 1 2 I... - - - - - I . , -A <-A- +L.- -,.-,..+<-.. -0 Q YUUl l ~ U I U C L U I l O C U U L U "G ~ Y ~ C U V L C U L l l C U C11G Y U I U C L U I I U I C1

problem, they might or might not provide a better method than

that of Zak in some given situation. In sections 6 and 7 we

report on a search for such solutions.

4. Integral Equations

The method of integral equations rests on the representa-

tion of the solution of Laplace's equation as an integral. A

review of such integral' representations is given by Temple [ 7 ] ,

who contends that the culmination of this work is in Whittaker's

result that potential functions which are replar near the

Basically the method of integral equations consists of setting

up equations for the unknown :function in an integral expression

such as (7). !These equations are based on the boundary

conditions.

A review of the use of the method of integral equations

is given by Heins [2], who makes use of the Poisson integral

representation: For a function Sn(R,Z) satisfying

which is the equation(*) satisfied by a Fourier component

of an harmonic function S, one obtains

where

Now the validity of the Poisson Integral Representation

(10) hangs upon the regularity of the solution on the Z axis.

Indeed the assumption of such'regularity is stated explicitly

by Heins (p. 789) and the problems solved (e.g. a charged

disc, or a lens, with axis along the Z-axis) do not violate

* Note Equation (8) is the same as equation (2) using Heins1 notation instead of Snow's.

t h i s condition. Other work which we have found so f a r [ 9 , 10,

11, 121 does no t seem t o v i o l a t e this condition.

We must caution t h a t we have no t a t th; time of wr i t ing

of t h i s repor t f u l l y digested t h e question of whether r e g u l a r i t y

on t h e Z-axis i s absolute ly c r u c i a l t o whether o r n o t t h e

problem of t he tapered tube i s amenable t o a Poisson In t eg ra l

type analysis . However, t he Z-axis i s outs ide t he domain of

required v a l i d i t y of so lu t ions t o such problems. So the re is

no reason t o expect h a t t he Poisson In t eg ra l w i l l give t he

answer. On t h e o ther hand, n e i t h e r can one a s s e r t a t this

po in t t h a t it w i l l n o t f i g u r e i n a method of solving the tapered

tube problem. Indeed, perhaps we need a so lu t ion v a l i d outs ide

the inner surface a s well a s a so lu t ion va l id i n s i d e t h e outer

surface of t he tapered tube, so t h a t t h e i r region of common

v a l i d i t y w i l l be a s desired.

Another method which we f e e l needs f u r t h e r exploration

i s t h a t of Snow [I], Chapter IX. Again, we f e e l a t t he time

of wr i t ing of t h i s r epo r t t h a t we have no t y e t seen through

t h e method well enough t o be c e r t a i n t n a t it w i l l appiy i n

whole o r i n p a r t t o t h e tapered tube. The d i f f i c u l t y a t t h e

a x i s a r i s e s i n t ry ing t o map the R-S plane i n t o t he wedge-

region which i s t h a t of t he tapered tube on the R-Z plane

without ge t t i ng i n t o t he same type of d i f f i c u l t i e s with t h e

mjs: Howeveri f o r re;lonF; s imi l a r t o +.hne s t a t e d i n con-

nection w i t h t he Poisson In t eg ra l , we f e e l t h a t t he matter i s

no t a t a l l s e t t l e d a t t h i s time and t h a t we should, indeed,

l i k e t o consider it fur ther .

Nevertheless, in order to seek integral equation solutions

appropriate to the tapered tube, it would be nice to have an

integral representation which is tailored to hold in the proper

region. To this end, we have carried out an investigation

based on 'the theory of characteristics. It may seem odd to

do this today, but in nineteenth century work, the relation of

the wave equation to Laplace's equation through the use of

complex characteristics was well accepted. We shall present

these results as soon as we have discussed the work of Niven.

5. The Work of Niven

A coordinate system appropriate to the tapered tube was

treated by Niven [ 4 ] . Indeed, he defines a coordinate system w

r, 9, $.by w

x = (C + r cos9)cos$

y = (C + r cosz)sin$

z = r sin 9

(where the tilda is our notation). ,

We find it more convenient to deal with the complement

of 9. Thus, we shall interchange sin9 and cos9 to write

x = (C + r sing)cos$

y = (C + r sine)sin$

z = r cos9

and, since these differ so trivially from Niven's coordinates,

we shall call these Niven coordinates also. It is clear then

that in the Niven coordinates the surfaces of the tapered tube

with taper angle a are simply 8 = 0 and 9=a.

The t r a c e o f the o u t e r and inner surfaces o f t h e tapered tube on the R-Z plane. The taper angle i s a. The Niven coordinates ( r , ~ ) a r e shown.

Niven then presented a form of lap lace,'^ equation i n

these coordinates, namely

+ r aZv " 7 = 0 . + r C O S ~ a$

w

It i s c l e a r t h a t by making t h e subs t i t u t i on 8 = n/2 - 8, one -

g e t s ins tead

a av av -r (c + r s i n e ) - + a ( c + r s i n e ) - a r ar ae ae

+ 7 = ~ . aZv c + r s i n e a$

Niven then wr i t e s t h i s equation i n terms of f = L n r f o r

t he case where V i s independent of $. We s h a l l w r i t e ins tead

t h e equation i n t h e general case , namely

These Niven coordinates have, i n f a c t , been met before

i n this repor t and indeed and 8 a r e respec t ive ly t he 2 and P 7 of sec t ion 2. In this sec t ion , however, we s h a l l use C

and f o r o ther coordinates.

Niven wr i t e s Laplace 's equation f o r t h e case where V i s

independent of gf i n terms of ( c h a r a c t e r i s t i c ) coordinates

To him t h i s i s j u s t a s t e p i n a process which i n f a c t

does n o t seem t o lead us very far. Indeed t h e next s t e p i s

t o pu t

t + ? = e V (15a)

6-,q = t (15b)

and a r r i v e a t

which does indeed admit separat ion of var iab les solutions.

However, t h e simple observation t h a t

shows t h a t indeed we a r e merely back i n a s l i g h t l y unfamiliar

form t o cy l ind r i ca l coordinates R and Z and t h a t t he separat ion

of var iab les w i l l y i e ld t h e f ami l i a r Bessel functions.

The r e s t of Niven's r e s u l t s a r e of mathematical i n t e r e s t ,

bu t do no t help us with our tapered tube problem. He does

indeed obtain some closed form so lu t ions , bu t they do no t have

the proper equipotent ia l surfaces appropr ia te t o t he tapered

tube problem. Indeed, h i s method is t o seek so lu t ions of t he

f orm

where m depends on n i n some way, and then c lever ly t o choose

some such dependence which allows the s e r i e s t o y i e ld a solut ion.

The method is interesting, but it probably lacks the generality

which we need in order to obtain enough solutions that we can

handle the tapered tube problem. Indeed, in section 7 below

we shall present solutions containing types of terms not found

in (16).

In closing, we point out that these type of coordinates

were used by Riemann [7] to solve a problem of an anchor ring.

Although Riemann's paper is very pretty, again it contains no

hint how to seek solutions with equipotential surfaces appro-

priate to the tapered tube.

6. An Integral Representation of the Solution to Laplace's Equation in Niven's Coordinates

Consider, now, the form of the equation for the potential

v(~)(R,z) in polar coordinates (R, @ ) as given by equation (2).

We shall obtain formally an integral representation of the

solution to this equation in the Niven coordinates (p,e), where

P R = c + e sine

z = ef'cose.

To begin with, let us put

v(m) = Rmw,

(where, of course, W will also depend on m, but we find it

notationally simpler not to write it explicitly). We obtain,

then for (2)

Now let us change to coordinates , q , where

(Note that (20a) and (20b) follow from (17) and (13). )

We then obtain for (19)

Now (20)-is in characteristic form, albeit the char-

acteristics are complex, either in the R, Z coordinates or the

p, 8 coordinates. It is clear, then, that the characteristics

are .given by

p + ie = constant and

. P - ie = constant.

Now equation (20) is of the form given in Sommerfeldls

book [15], section 11, in relation to a hydrodynamic example

+-^-+^a h.. D i .-,mc.nn m r r m s + h r r a irr.r*l r r s c c, "-1 I . + < rrn ,, ,,,,, ,, ,,,,,,,.. ,,,, ,,,,,, ,,.,,,.,, c. ,,,,,,,.. tc the

adjoint equation

- - a w l

for each point P ~ ( ~ ~ , B ~ ) such that H = l at P and

and 7 m r l as - - H = O

a ? 2(4.+y)

on the characteristics = const and 5 = const respectively

which pass through fO, €IO. Riemann solved this problem. The

so lu t ion f o r our case i s

where

z = - ( Z - t0)(-q - To)

( t + ? ) ( t o + 'Zo)'

F(a , $; Y , z ) i s the hypergeometric function

To ob ta in an i n t e g r a l form of t h e so lu t ion of (20), then

we apply t h e r e l a t i o n (7) p. (54) of Sommerfeld [15], which we

wr i t e .in t he form

where r is a por t ion of a curve on which da ta a r e given,

P1 and P2 a r e t h e i n t e r sec t ions of t h e curve with t h e respect ive

c h a r a c t e r i s t i c s through (po,BO), t h e i n t eg ra t ion is taken t o be

along r from P1 t o P2. (Here we must be care fu l t h a t P1 be

the po in t where = y o i n t e r s e c t s r and P2 i s t h e po in t

where = to i n t e r s e c t s r . ) Also

a t HW.

We wish now to evaluate the right hand side of (25),

where we shall take for . r the curve 8 = 0, which corresponds

to that surface of the tapered tube which is parallel to the

axis. We calmlate first for any function u

Then we obtain from (25) and the relations following it

where

Now (26) and (27) give an integral representation of

the solution. The limits p0 - i e0 and + i e0 are of course Po the (complex) values of P at which the characteristics (21a) and (21b) through (fo,BO) intersect 8 = 0.

We have been proceeding formally, but it i s c lear tha t

we must assume analytic boundary data, f o r otherwise the

integrand w i l l be path dependent.

We sha l l argue t h a t the integrand Q ( p , O ) i s r ea l f o r rea l

values of , Assume t h i s f o r the moment. P , If we make the change of variable

p = p, + i$

i n the int'egrand, we get f o r the in tegra l term

By putting - $ f o r $ i n the first in tegra l on the r ight

hand side, we get fo r the integral term

which, under our assumptions, becomes

where t h e integrand is evaluated a t ( + i ,O). This i s a form P a of t he i n t e g r a l representa t ion of t h e solut ion.

Now l e t us look a t t h e assumptions on Q. F i r s t of a l l ,

i n order t o be sure t h a t our operations a r e l eg i t ima te , we

must ass ign ana ly t i c boundary da t a on W( ,0 ) and aW/de a t P 0 These must be assigned so t h a t they a r e r e a l when P i s rea l . Next look a t H. There i s no problem of a n a l y t i c i t y

of H. However, we s h a l l check t h a t H i s r e a l when e and P are r e a l , so t h a t it w i l l a l s o follow t h a t aH/aB w i l l be r e a l a t

r e a l (p,O).

We have from (20) and (24) t h a t

Now the l as t equation is the product of an expression with

t h a t of i ts complex conjugate and i s thus r ea l . From (23) , it

then follows t h a t z i s r e a l f o r r e a l ( p , e ) , and hence, from

(22) and (23) we see t h a t H i s r e a l f o r r e a l ( ? , e l .

This es t ab l i shes , a t l e a s t formally, t he i n t e g r a l repre-

senta t ion.

7. Some Polynomial and Ser ies Solutions

Although it i s n o t poss ible t o separate va r i ab l e s i n t h e

Niven coordinates, o r i n an equivalent coordinate system, it

i s possible t o obta in some exact so lu t ions of more gene ra l i t y

than those given by Niven. To t h i s end l e t u s s t a r t with

equation ( 3 ) f o r t h e reduced po ten t i a l i n cy l ind r i ca l

coordinates. We drop the superscript m for notational con-

venience. We have, then

2 2 1 2 R V U + ( 4 - m ) ~ = ~ . (28)

We substitute for U the series

where the index p and the functions fk(Z) are to be determined.

Substitution of (28) into (29) yields

Setting the coefficient of f0(z) to zero, we get the

indicial equation

I p = = +m.

i We choose the larger root p = ~ + r n of the indicial equation. (As with Bessel functions, the smaller root will give nothing new. ) We obtain by substitution into (30)

(2m+2)fl(Z) = O (31)

Equations (31) and (32) taken together tell us that

fk(Z) = 0 for odd k

and that for even k, k = 2j, we have

This gives us a method of obtaining solutions, since we

can now pick fo(@) as we wish and make use of the recursion

relation (32). We shall explore some of the consequences of

this procedure.

We see, in particular, from (33) that if we choose fo to

be a polynomial in Z, then the number of terms in the expansion

(29) will be finite. The solution will be a square root of R

times a polynomial in R and Z. We shall investigate the case

n ,fo(Z) = Z , n=0,1,2 ,.... (34)

From these one can, in fact, by addition and multiplication

by constants, arrive at the results for any choice of a poly-

nomial for fo.

Now (34) and (33) tell us that

for some c to be determined. If we put (35) into (33) we 3 obtain

The solution to the recursion relation (36) is

where j = O,l,. . . ,n/2 if n is even and j = O,l,. . . , (n-1)/2 if

n is odd. /

I f we take con!m! = 1, we obtain then a so lu t ion

where h ( n ) = n/2 if n i s even and h(n) = (n-1)/2 i f n is odd.

Equation (38) gives , then, a square r o o t of R times a /

polynomial i n R and 2. Now re turn ing t o Niven coordinates,

we obtain

m+ 1- -113 u = ( c + r s i n e ) '(r C O S ~ ) ~ h p ) 2( ($sece+tane) 23.

j = O 2 J(n-2j)! j! (n+j ) !

Since there a r e f i n i t e l y many terms, t he re i s no problem

of convergence.

Another s e t of exact so lu t ions can be generated by wr i t ing

f o e ) = 2-n

where n is an integer. ' We obta in , then, by the same process

According t o We r a t i o t e s t , t h i s s e r i e s w i l l converge

R f o r IZ1 < 1.

In the Niven coordinates, then we obtain

u = r-"(c+r s i n e ) w ( % e c e + t a n e ) 2 j j = O 2 J j! (j+m)! r

which converges when

c + r s e c e < r c o s e ( 1 s t quadrant)

o r C I- > cose - s i n e '

Now l e t us consider a t ape r angle a < $, so t h a t we a r e

concerned w i t h , say,

o < e < a .

Then case- s i n e i s decreasing w i t h 8, s ince

Thus

d a 8 ( ~ ~ s B - s i n e ) = - s ine - case.

C < C COSB - sine' \ cosa - s ina ' O < e < a .

Therefore convergence i n t h e wedge region w i l l occur f o r

C > cosa - s ina

which means t h a t t h e so lu t ion w i l l be va l id away from the

s ingu la s i t y where t he ou te r and inner surface of t he tapered

tube meet. This i s the s o r t of region t h a t f i gu re s i n a gun

tube.

For t he case of m = O , we have a l s o succeeded i n f inding

an addi t iona l so lu t ion involving a logarithm and a polynomial.

We seek so lu t ions t o (28) i n t he form

where V s a t i s f i e s (28) with m = 0. We obtain

We take f o r V t he so lu t ion (38) which we w r i t e a s

where c i s given by (37). After some s t ra ightforward manipu- 3 l a t i o n we obta in t h e recurs ion r e l a t i o n

We note t h a t s i nce t he re a r e only a f i n i t e number of non-zero

c and s ince t he coe f f i c i en t of b . i n (40) becomes zero f o r 3' J-1 j = (n+2)/2 o r j = (n+l)/2 (depending on whether n i s even o r

odd) there a r e only a f i n i t e number of non-zero b . and so both J

V and the summation term i n (39) become polynomials.

These solutions. , then, i n Niven coordinates become

Some thought must be given t o t h e bes t way t o assemble

t he se so lu t ions i n order t o s a t i s f y given boundary conditions.

8. Short Annotated Bibliography

The annotat ions here w i l l a l s o be b r i e f s ince t he informa-

t i o n t h a t they convey is a l s o i n t h e t e x t , o f ten i n more d e t a i l .

[l] Snow, C. Hypergeometric and Legendre Functions with

Application t o I n t e g r a l Equations of Po t en t i a l Theory,

U.S. Department of Commerce, National Bureau of

Standards, Applied Mathematics Se r i e s 19 (1952).

This work seems very r i c h and may well hold t he c lue t o

a procedure f o r a t t ack ing the tapered tube problem. It does

e s t ab l i sh c r i t e r i a f o r whether one may separate va r i ab l e s f o r

Laplace's equation a f t e r transforming the po la r coordinates

R , Z t o another s e t by a conformal mapping. However, t h e

relev;m+ chapter which we f e e l rnrjJd s t i l l contain some c l u p

i s Chapter I X , Some I n t e g r a l Equations of Po ten t i a l Theorya.

We advocate some f u r t h e r study t o see i f t he method can be

adapted t o t h e tapered tube. However, it may take some c lever

i n s i g h t t o implement it.

[2] Heins, A. E. l 1 Axially-Symmetric Boundary Value Problems

Bull. Amer. Math. Soc. 71, pp. 787-808 (1965). - This i s a review of t h e use of t h e Poisson I n t e g r a l

formulation t o solve problems of po ten t i a l theory by t h e means

of i n t e g r a l equations. A f ea tu re of t he method is t h a t t he

c n l ~ + < n m is represented by &y in tegral ,Arhe~e in tegr lnd inyoj ,v~s ""A- "AU*.

t h e values of t he so lu t ion on t h e Z axis . Another r e l a t i o n ,

such as the Helmholtz represen ta t ion , f o r example, is then

used t o s e t up a r e l a t i o n , ac tua l ly an i n t e g r a l equation, with

which t o determine the unknown integrand i n t he Poisson in t eg ra l

When this r e l a t i o n can be solved, then one has t he so lu t ion

t o t h e problem represented a s an in tegra l . Problems given as

examples inclu'de t he e l e c t r o s t a t i c f i e l d s about d i s c s and

c i r c u l a r l enses with axes on the Z axis .

It i s no t c l e a r a t t h e time of wr i t ing of t h i s r epo r t

whether o r n o t t he method can be e f f ec t ive ly used f o r t h e

tapered tube s ince 1) the method i s bas i ca l ly designed f o r

functions which are regular on the Z axis (or at least a

portion of it) and 2) its application seems to depend on its

proper use in an appropriate coordinate system. Whether and

how to use it in whole or in part for the tapered tube problem

deserves further study.

[3] Zak, A. R. "Elastic Analysis of Cylindrical Configurations

with Stress Singularitiesw , J. Applied Mech. 39 ( E ) HZ

pp. 501-506 (1972).

As the title implies, the author is interested here in a

study of stress singularities in a cylindrical tube. However,

the coordinate system which he adopts is essentially the same as

the Niven coordinates. The method does not use a sequence of

exact solutions, but it does obtain a series which converges to

an exact solution where it converges. As the method is applied,

it is limited by a singular curve (P sin$= I), a function which

occurs in the equatio~s (1 -p sin$)-'. As the distance p from p = o ( which would be the intersection of the inner and outer surfaces in the tapered tube problem) gets larger, the location

of the singular curve gets closer and closer to where sin$=O,

which limits the region in which Zakts method could be used, at

least without so-se modification. We calculated that for a taper

angle of 2- one is limited to'a distance less than about 28 times

the distance from the axis to the intersection of the inner and

outer surfaces of the tapered tube.

[4] Niven, D. W. It On a Special Form of Laplace's Equation"

Messenger of Mathematics X, pp. 114-117 (1881).

Except for the paper by Riemann [7], this work of Niven

is the only one we found which treated the coordinate system

appropriate to the tapered tube. Some special solutions were

found, but do not apply to the tapered tube.

[5] Temple, G. Whittaker's Work in the Integral Repre-

sentations of Harmonic Functions" Proc. Edinburgh

Math. SOC. - 11, pp. 11-24 (1958). This is a nice review article which goes into the history

of integral representations of potential functions. It con-

tains a derivation of the Poisson Integral and developments

leading 'to the Whittaker integral, as well as comments on its

applicability.

[6] Whittaker, E. T. . " On the Partial Differential Equations of Mathematical Physics" Mathematische Annalen 57, ....= pp. 333-355 (1903).

This paper is of interest because of the development of

the Whittaker integral and a demonstration of its use in an

example problem (the potential of a prolate spheroid).

[7] Riemann, B. Ueber das Potential eines Ringes",

Gesammelte Werke, Chapter XXIV.

This is of .interest because coordinates appropriate to

the tapered tube are used. However, the method of solution

0f.a problem, namely that of an anchor ring, to which the

paper is devoted, does not in any clear way indicate how to

attack the tapered tube.

[8] Weinstein, A. Generalized Axially Symmetric Potential

Theoryn, Bull. Amer. Math. Soc. 59, pp. 20-38 (1953). - Weinstein develops a method of attacking problems by use

of a generalization of potential theory. This subject merits

some further study to see if some relation can be found to the

tapered tube problem.

In addition we mention some papers of W. D. Collins, who

uses the method described by Heins. These are of interest for

the detail which they provide on the method.

193 Collins, W. D. It On the Solution of Some Axisymmetric

Boundary Value Problems by Means of Integral Equations

1,Some Problems for a Spherical CapM , Quart. J. Mech.

Appl. Math 12, - pp. 232-241 (1959). [lo] Collins, W. D. On the Solution of Sose Axisymmetric

Boundary Value Problems by Means of Integral Equations

11, Further Problems for a Circular Disc and a

Spherical Capvt, Mathematika 6, pp. 120-133 (1959). - [ll] Collins, W. D. On the Solution of Some Axisymmetric

Boundary Value Problems by Means of Integral Equations

IV, The Electrostatic Potential Due to a Spherical

Cap Between Two Infinite Conducting Planes", Proc.

Edinburgh Math.Soc.(2) 12, pp. 95-106 (1960/61). ....-

[12] Col l ins , W. D. " On t h e Solution of Some Axisymmetric

Bounds'* value Problems by Means of In t eg ra l Equations

V I I , The E l e c t r o s t a t i c Po t en t i a l Due t o a Spherical

Cap S i tua ted Ins ide a Ci rcu la r Cyl inderw, Proc.

Edinburgh Math. Soc. (2) 13, pp. 13-23 (1962). - [13] Green and Zerna, " Theoret ica l E l a s t i c i t y n , Oxford U.

Press , Oxford (1968).

[14] Allen, D. N. Relaxation Methodsw,McGraw-Hill, New

York (1954).

[15] Sommerfeld, A. ~ a s t i a l D i f f e r en t i a l Equations i n

Physics1, , Academic Press , New York (1949).

9. Conclusion

The problem of t he bes t way t o proceed t o obta in a n a l y t i c a l

so lu t ions of t h e tapered tube problem remains unresolved a t t he

moment. We f e e l that 'some progress' has been made i n developing

exact so lu t ions i n t h e appropr ia te coordinate system. Perhaps

a method of assembling these so lu t ions t o f i t boundary data

would be f r u i t f u l . Perhaps a formulation i n i n t e g r a l equation

form, would be t h e way t o proceed. Again we f e e l t h a t some

progress has been made by our i n t e g r a l representa t ion of

s ec t i on 6 . And of course Zakls bas ic idea could be developed

fu r the r . Indeed, f o r small t a p e r angles it could be u se fu l

n o t too f a r from the i n t e r sec t i on of t h e inner and ou t e r sur-

f ace s a s it stands. But it would be b e t t e r t o t r y t o develop

the idea of t he method f o r po in t s f a r from t h i s in te r sec t ion .

we have, indeed, made a s t a b a t t he problem and hope t h a t our

contributions will be the first step in the solution. However,

at this point, we feel that much remains to be done.

Acknowledgments

This work was conducted at the Ballistics Research Labora-

tory at Aberdeen Proving Ground, Maryland, under Contract

DAAG29-76-GO100 with Battelle Columbus Laboratories under the

Laboratory Coopexative Research Program. We are grateful to

Mr. Alexander S. Elder of the-Ballistics Research Laboratory

for his help and supervision in this project.

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