Some hydronautic inertia coefficients

Sorne JIqtI odjjaanucal inertia Coe/fleient s. 161

at P is due entirely to the

singularities of within the

sphere S.

Use is made of this tandiiig splie to obtain further verifications of tite extended F'ouinuli by identifying the value of

at a point P at time t

with that obtained from the surface integrals ill \vlIiCh tue lion dory values of and its derivatives are inserted, its application to the function

f(T)/[KR], which includes L5 11rticttlar cases, the scalar

:utd vector retarleti Potemttiais and tue function ¡(ci -gives the desired valties, vlticli afford strong evidence that the formula dcduccd can he taken as the Kirchlioff formula

exteutled to a moving surface.

In conclusion, I wish to express ruy thanks to Prof. G. A. Schott, F'.1.S., Ficad of the Mathematics Department in the

University College of Wales, Abcrystwvth, for sugestiuig this problem, and for his a'sistance and advice hile tue work wis in progress, atti tu FI. Bateinan, U'. U.S., Protéssor of Matlienatics, Theoretical Physics a id Aeronautics at the California Institute oE Technology, Pasadena, for the

evalua-tion of integral (2).

Noyunu Brid.ro Lilairttory of Physics,

(Jal i f rnia In .t it lite,

1tsadiia. C:tlit,,iiia.

.pi'il S, I 929.

XV. Sottie 1!it1rorIyaaìnical Inertia Coe/jicient.s'. .1111. j. Loci-\VOOD TAYLtUi.. D.Sc;, JÏiitillions Cotti in ¡tice Research Feliu ir,

Utnrersttìj of lAverpool .

r lT-lE question of tite "virtual inertia" (If abody immersed

- itt fluid lias acquired some additional interest from the

i n flticnce which it appears to have on tite na turai frequency of vibration of a ship ,anti most of tite results which follow

have been obtained with this problem iii view. Part I. gives solutions for motion in two dimensions due to tite transla-tion of cylinders having various cross-sectransla-tions, and discusses

the application of the results to tite ship iroidem.

The

results cati be applied i nitnediatelv to the caso of vertical

* Communicated. Lv the Author.

t Niehofls, Trans. lust, of Naval Architects, p. 141 (924) ; Moullin5 Proc. Cambridge Phil. Soc. xxiv. p. 400 (1927-2).

162 l). J. L. Tavlur on swne

vibrations, since the sprcial bounda rv c(n(ht iOn appropriate

to the free surface is auíonaticallv fulfilled at an axis of

svmtnetr perpendicular io the direction of iloaion. The

free surface condition for ilw ase of horizontal vibration

requires special coiderañon

Part Ii.). Part III. deals

with the effect of rigid boundaries, e.g., as in shliow ater or canals, and Part iV. the effect of tite abandonment of the restriction to two-dimensional morioti, and of conì1rcssihility of the fluid in a particular casethat of tite circular cylinder. The inertia is generally less, in tie two-dimensional case, for

motion pprallcl to the free surface

thati for motion in a

perpendicuir direction, when the 1iriadth of the cylinder is

greater than the depth ;

a rigid boundary increases the inertia, as also does cotn1)resibi]itv to a small extnt, but

freedom of the fluid to move in three dimensions naturally hs the effect of reducing hie energy.

PAM I.

The prohlemn of fluid motion due to the translation in a direction per1endicular to the generators of cylinders having the following -ections is considered.

Sections symmetrical about two perpendicular axes, bounded by

Two circular arcs intersecting at any angle. Two parabolic arcs intersect ing orthogonally. Four equal straight lines.

(1) Four seinicircies.

A circle with projectng lamime. A square with rounded corners.

(1) Section boa ncled by Circula r Ares.

in terms of complex variables, z, w, and aim intermediate variable, t, tlm equations

= cot t, = in cot nf, = n coseC nf,

represent a uniform stream of unit velocity flowing past a

sectiom as described, in a di rection pm rai lei a itil l)C1P dicular to time conimon chord, respectively. 'l'ue correspondence of the various planes is as in fig. 1, the exterior angle of

inter-section of the arcs being 2ir/n n being positive and greater

Hydrodynarnical Inertia Coe7lciens. 163

can be solved by the method of images). Expanding the

trigonometrical functions, we have, when z, w are large,

I (n2-1)

w=z+i_

+...,

and

i (n'+2)

4

z 6 Fig. 1. jc B D A D

s

K JF' L c A (n,) 8 C

1f A is the area of the section. aud C the "entrained area," corresponding to the kinetic ellergy of the fluid (1/2 pC 02

for density p, and velocity U) when the uniform stream is

annulled,

(A+(1)(n-1).

(A+02)=

(n2+9) M 2

K.J

_D K _A

164 Dr. J. L. Taylor on some by a result of Leathem's Also

-Ií

7\

.T

A =

_nl

cosce2 + cot

-n

n)

so that C1, C2 may be found by subtraction. The table

shows the ¶alues of C, b (the withi, of the section measured perpendicular to the direction of motion), and C/I12.

It is

seen that the latter coefficient does not vary very greatly,

the extreme values being 2/ir and 'mr/2(w2/6 1), pr OE636 and 1013 respectively. The latter value corresponds to the limìting case of w2 when n is large, i. e., for two circles in contact, moving perpendicular to the common tangent. The solution for this case may be written alternatively

W2 = r cosèc ?T/Z. Similarly, w1=in cot irfz,

C1b2= 7T/8 (w2/3 1) = .900

represents tnotion parallel to the common tangent, or may

be regarded as the solution for a single circle, of unit

radiuE. in

contact with a

_{idane boundary.} The cases a = 1, 2 correspond to the known solutions for the lamina

and circle respectively.

(2) Section boumled b1 Parabolic Arcs. For motion parallel to the chord (flg. 2),

1t \

_t=:j

_(w2-1).ilw.

\Vlien w, z are large, this gives

:=wl/tw+

...

(A +()=ir/3.

* Leathem, Phil. Trans. A, ccxv. p. 453 (1915).

n 1 _6/3 4/3 3/2 2 3 4 5 6 2ir/n 3600 300 270 240 180 120 00 60 0 Radius 9 141 NS I US 141 2 Width, b. 0 536 ,, bi 2 2 o, O 197 828 2 488 115 9 90 2 2 314 346 231 100 4S3 283 200 74; 4 480 02 314 2S8 282 281 314 478 742 154 C1/b1 (636) 65.5 709 135 185 835 857 878 (.900) .021112 75 119 704 703 185 806 928 902 (lOi)

- -k-

L 1

Iigth'odynamzaal Inertia Coecient8. 165 The length of the choid i

1/4{

i

(1_w2y.dw=ir2/2.K2,

K being the complete elliptic integral of modulus 1/

Fig. 2. J

0

B J Be uce A

=

x 4.x ) = ir/12 . K4 =OE687,

C=7r/3-0687 = ft360.

C/t2 0360±ft515 = (J98.

For motion perpeudiculiir to the chord (fig. 3),

r _w

166 _{Dr. J. L. Taylor on} sorne giving

z=w-1

3.w+

(A+C)=2r/3,

-C

2,r;S 0687

=

I 4O7,

/b2=14072061 =0683.

The coefficients C/b2 _{agree fairly closely with those for}

the section consisting of orthogonally intersecting circular

whence Fig. 3. C A i-D L ' 702

j ()

.clw, D A J C A

arcs, viz.

_{0709 and 0704 respectively,}

_{as. would be} anticipa ted.

(3) Quadrilateral witJ equal SU/CS.

The angle between the side and the diagonal parallel to the

direction of motion being n . ir, _{Schsarz's method gives} (fig. 4)

B J

,

The va'ues oE the cuetheient C/b2 are tabulated below or several values oE the anglo n. ir :

'l'he value for n+0 is obtained as follows F( 1/2+

n) = */ 1 n(y + log 1)),

F(iìz)(1-l-n.7),

to the first order, when n is small, y being Eulcr' constant.

nr 900 t0 15 30 -*0

Cb2 . -785 -60 591 543 141

l/'ìiroiyiìaincd Inerf ja Coe//icients. 1t.i7

when z, w are large, so that

(A + (2) =2 . -ir . n.

The length oE a side of the quadrilateral is given by

t"í

w2

'"

F(1/2+n). F(1n)

iz)

(W

21'(3/2)

and

A=s

sin 2. nit

168 Dr. J. L. Taylor rm some IleiÌC

s=(ln. log 4),

A=

2. wir(1-2 . nlog4,

C=1.it2-1o4.

Also.

b2

.I117 CJt2 .

log 4=ft44l.

For the irtict1ar ease of the square

(i =

lj'4), ihe entrained area is obviously the saine for two directions

at right anales and is therefore the saine for iiiotion in any direction.

The present result is therefore directly

coni1arable with the particular case of 11iahuuchinkv's

soinrion* for a rectangle moving paraUel to a side, viz.,

4K2

b Legend re's relation E, K being complete elliptic integrals of ILodulus 1/ *Jz. This is to be compared with

giving a known relation.

2

I

2P(3/2)

5'

-(4) Section bounded by Four Scm icireles.

The solution is derived from the representation of tite interior of a square oh that of a circle, viz.,

Ç _dt

by putting

j (1

')'

w=t+1/t.

(hg. 5.) Hence, when w, z are large a id , t small,

z=

l/t (1/i) +

(AC)=2.7r.

IIydrodivaniicai Inertia Coe/fleivnts. i k39

In the plane of ,

IA ='

(4)

= K1/2,

so that the hall-diagonal of the square ABCD in the

Fig. 5.

0

z-plane is v'/K.

This gives as the total area of' the section

(1±a-/2) 4/I2=2.9, so that

C=

299=?29=O9t58b2,

w ere b is the extreme wi t o t io section

K

the entrained area, C, is the sìme for motion in the perpen-dicular (lirection, it ma)- also Le expressed in ternis of the diagonal 4/K=b say, viz., C=0707 b!, corresponding to the case of motion at 450 to the aies.

8

D

170 _{Dr. J. L. Taylor oit some} (5) Circle wit/i f'rojectin2 Lamine.

The solution is derived from that. for a single lamina,

by adding a second similar term, so that

2 1/2(V'(w2_c) + /(w2_4_c2)

reliresents the flow past a circle of unit radius with a lamina

projecting at either etreiiiity of tite diameter perpendicular to the direction of flow, tite total width 2b being given by

c=(b2-1)/b.

When w is large, expansion gives

Hei) ce

(A + C) = ir (2 +e2),

(J/W = r/4(1 - 1/b2 + 1/b4).

The coeflicient is equal to ir/4 when b=1, and also when b is large, as would he aiiticipited, SIOCO the influence of tite circle of unit radius is then negligible. The minimum valua, for b = V2, i5371-/16 or 0589.

(6) Square with rounded corners.

By adding, in a similar mnunnet to the above, two terms

corresponding to the flow last rectangles of different pro portions, the resulting section is a rectangle with the corners rounded off. - Itiaboncimiusky * has gis-en the solution for a

single rectangle, which may be somewhat

more simply expressed as

w2

cos dw.

w2 1

Titis gives (A+C)

irsin,

A = b x d,

=4(E ces2

.

K)(E' sin2

c.

E, K heimig the elliptic integrals of modulus sin , and E', K those of the coimiplemnentary modulas _cos . The

IIq(irod/Jua inical Inert ja Guefl'icients. 17 1 effect of writing sin for cos is to interchange the breadth ahi depth of the rectangle so that the addition o two terms

2

('{(v/w2cos

+ (v/'u'2_sin2 c)} . dw

-

J

/w2_i

gives a qmmare of side 1/2(b-j-d), the sides being connected by an arc of length ¿s = CO _{ç(1J )2} + ((4i)2 } ( (1W \dW

i t°(o' 2_w2+w2_sjn2a)f

-

_{l,in - - (i..w2)* --} . t W. This gives

s=/cos 2(ir/2--2).

The exact area ot the section may be calculated by

quadrature, but if

is not less than say r/6, it is given

very nearly l)y

(b +d)2

16/30. s2. Also (A +C)=7r/2(sin2c+ cosc) =ir/2.

For

=ir/6,

b=2(1468-0'75 x 1'GSfl) =02035 x 2,

d=2(1'211-0'25 x 2157)=0'672 x 2,

s=1/2 'X7r/6=0'1S5,

A = (0875í)2 0018=0'749,

U =42 0'749 = 0'822 = i'07 b2,

b being the width of the sction, i. e., the side of the square. Comparing this with the corresponding figure for a complete square, l'i 8, i t a h1pear that the red uction of' t h0 area, by rounding oft tu corners, by about 2 per cent, lias reduced the entrained area by 10 per cent.

Application of the abore results.

If cos a't is the velocity potential of a fluid inotioui due to the si mnple-harmonic vibrations of a n im merseil body, this usual condition at a free surface is

172 _{Dr. J. L Taylor on onze}

If the frequency is sufficiently large, . i evidently small, and the cjndition reduces to =0 * The order of

the neglected term is

y . v/a2, where r is the maximum

velocity, of the vibration, while

_{in the vicini;)' of the}

Section is of the order (b . r), l being one of the dimensions

of tue vibrating body. _{The ratio, o/ba2 or g . T2/47r2 .b,}

is very small t'or periods (T) of less than one second, as in practice, b being of die order of, say, 50 feet.

The condition

_{=O is evidently fulfilled at an axis of}

SV ininetry perpendicu] ar tu the direction of motion, e. g., the ii-axis in all time foregoing examples, which _{may therefore be} regarded as a free surface. _{(onersely, for any given section} partly immersed, the section is to be completed by adding its

reexion in the fre surface, and the problem solved for the section so formed. _{'Whemi the boundary of the section cuts} the free surface at an acute angle the velocity becomes infinite at the corner, so that the approximation to the fulfi]ment of the surface condition is locally inadequate,-but the effect on the motion as im whole is probably small. _{The fact that the} theoretically infinite velocity _{at- a sharp corner dot-S not}

affect the practical applicability of the results appears to

have been established by Moullin t in ti-me case of a totally submerged section. _{11e fInds that for a rectangular section} of lireadth twice the depth,an approxitnation to the entrained area is that of the circumscribing circle, i.

_{e., 5/16 . b or}

0°98 b', while interpolation between Riabou'hinskv's results

gives 105 b' in fhir agreement. _{For deeper sections the} approximation does not hold ; thus for a square

(corre-sponding to a partly immersed section of depth equal to

half the breadth) the respective figures are 151 b' and

1188 b'..

It appears more reasonable to base tite approximation on the mnaximnum width, making _{some allo;; ance for the shape} of section, and while no empirical foriitula is likely to give

very accurate results for a great variety of sections, the

expression (b' x A/h . ¿1) or (A X bld) is reasonably accurate for sections (1), (2). (-fI, and (e), and is exact for an ellipse. The area t'or a partly imnmmiersed section. _{vibrating verticali)',} becomes (A' x b/2d') whore A' and d' are the actual area and

depth respectively, but this makes no allowance for any departure froni tWO-(litliefl sional motion (Part _IV.).

C'f. Rayleigh, Collected Papers,' ii._{p. 08.} j' Loe. cit. Loe. cit.

and when

4) = 4) COS1I _o A, . sin 2n. The entrai ned area is given iw

cosh2 ¿ S A,, sin 2ni cos i

-

.1i2

- CO, _e)

ir(1n2 -= (21/ir) cosh2

= (2/ir)b2,

since cosh =b, the major semiui-axis, perpenl.icuuar to tue

direction of motion and to flic free surface. The result is

the same when motion takes picc parallel to the major axis,

and the semicircle and lamina are particular cases. 'l'ue coimi parative figure for the semicircle moving perpendicul any

t tue surface is (ir/2)b2,

o that tuo virtual inertia for

horizontal motion

is only (4/ir2) times that for vertical

motion. For other sections file ratio depends, of Course, Oli the ratio of tile axes.

Mo ullin * carried out exoniments on a flat bar, the most hirect. comparison possible being that. for time bar in. thick

Loe. &it.

Ïlqd roll ijuain zeal Ioertia CoJficients. 173 PART fi.

When the free surface is para lid to the direction of motion the fulfilment of tue condition cp=O is not quite so easy, but a solution cali be obtained tor a semi-cui itical section with the axis in the free surface.

Lu t t i mi g

.v4-iq= similI (E+iq), the boon lary condition,

ke=y=cos1iEsin n

can be f ulfilled for values of i between O and r by

k= cosh A,

') co.n

where 1

-'rhe corresponding value of 4)is

174 _{Dr. J. L. Taylor on some}

with its edge submerged 2 in.

_{Fron the ratio of the}

fre-quency in this condition to that in air, it

is possible to

calculate that the added mass of water is about

058 ¿ per unit length as against 063G, as calculated for a lamina. A soluijún can also be obtained for tIte case of a circular cylind(r, with horizontal axis, submerged toany depth below tho surface. E mplov ¡ng the coordinates

sinei. smb

= cot_{1/2(E + =}

coshcose

we way take -q==O as the free surface and

_{a as the}

boundary of the circular section, of radins

_{cosecha, the} centre being at a dejth coth _{below the surface.}

The boundary condition r=

_{g+constant, for}

_can be satisfied by

= 2 _{-cosnE.coshìui,}

which nmakes

= 2 _{sin nE.sinhnj,}

and this vanishes at the free surface =O. The kinetic energy integral

(7=2)

=

tanli ax. sinli sin a. sinEdE

i _.

(coshcosj

=

n. e2' . tanh ita.

When a is large tite series _{converges rapidly, and} expansion of tite first three terms gives

4ii-{e2+ (e) 4

_.

Expressing titis in terms of

r = cósech a = 2e'(1 + 2a

ve have _{- 2e}

_{+ ...)}

= n'2(1r72c2)

to titis order, putting c = coth a_{= I nearly, for the depth of} immersion to tite centre of tite circle. The factor (1 r2/2c2)

liows tite reduction in tite kinetic_{energy due to the free} surface, when the ratio rJc is fairly small.

!Iy(lrodqnam icai fn il a Cae/fi cient.. 175 When a is in;iil we have the integral

X

n . e - .taub na dn

io

whose vaine is 1/4(7r2/6 1), so that the entrained area

is

a(/6 1) or ir2(2/6 1), r being equal

to i/a.

This result, corresponding to the case of relati ely

m:ii1 immersion, agrees with the result of Part I. (1) for motion

perpendicular to the surface (y=O), i.e., the common tangent to the two circles in the example quoted. The solution just given can in fact be adapted to this case, the expression for

the entrained area being identica], which shows that the

direction of motion is iminaferia).

PART I[I.-EFFECr OF RIGID BOUNDARIES.

(1) Circle with Plane J3oundary.

With the sanie coordinates as in the last example, the

expression

= 2 cos n sin]1 nj sinhna

satisfies the boundary condition .4r

_{= -y}

+&onstant for

=a,

and also makes

=u for

=O, as is required for a boundary

at y==O. The expression foi' the entrained area

_$c.ds= _$c.dE

= 2

e"cothna

4u-

n.e 2.cotlina.

Tn time same wty as before this gives, for a large,

_J.ds= 7r(l+),

indicating the effect of a distant boundary, and for small,

C

.'ir2

\

which again agrees with the example (1) of Part I. It can

be showu that the direction of motion is immaterial in this ease also.

Ç " sin n

sin smb

a

176 Dr. J. L. Taylor on some

(2) &?nicircle with Ba ndaìy and Free Surface. The example just considered, while of interest as giving a comparison with the resulTs in Part IL, (loes not satisfy

the condition

= O at a

i

lane through the axis of the

cylinder, parallel to the fixed boundary, aS would be required for a cylinder of semicircular section in a limited depth of wate r.

Putting w = C .cosec:/2,

9(

sin.r/.coshy/2

-cosh,cos.v

= c

cosx/2.iiihy/2

_{coshy cos}

represents a douLlet at

hie origin, with axis O.t, with

Loundaries r=O at .e= ±ir, ail'1 may also be regarded as representing the motion due to a small circle at the origin,

moving in the direction Ox.

It the velocity is uniti, the

coli di tion

=0

°'4'(1 cosa/21

(f.c

1cosz

gives

,

1c()Sx

,-U = = - x4/48 +..., cos .r/2

being small, which expresses (i in terms of the radius of the circle a on putting x a. Similarly,

= O, r = C/sinhg/2 = y, C = y2/2 +y4/4S +...

verifies, on putting y=a, that to this order the circular

shape holds.

In the vicinity of the origin, moie generally, on expanding

ç (.r/2.v3/48 + ...) (1 +y2/S...)

--

1

u2!2 +.c7+y4/384.v4/384+...

= 2C x/r(1 + ternis in r2),

(idI cosO_{(1+termns in}

dr

lien ce

-.r.d0 = 4(1

₍ + terms in

r)

=

(rm+r6/12).. to this order.

jIy(iro(ItJna mica i Inertia Caffieients. 177

When r a, fus is the kinetic energy integral, the valu&

being a2(1 + a712). This applies to the case of a boundary at distance ii- from the centre of the circle, and generalization for any distance of boundary c (large in relation to a) gives ira2(l +a2ir2/12c2).

In a very similar wanner w=C .cothz/2 * represents a

small circle midway between two plane boundaries at distance ±n-, moving pau'ailel to the boundaries, tim entriined area being ira(1 + a'm-2/3c). In each ease the plane through the

centre perpendicular to the direction of motion may be

regarded as a free surface, the entrained area for the

semi-circle being, of course, half the above.

(3) Lamina between two Plane Bounilaries.

Applying Lamb's solution t, cosh w = ¿cosliz, which

applies to the motion of a lamina of width 2 .cos' (1/s)

midway between two plane boundaries distance

ir apart,

and moving in a direction pari1Iel to the boundaries. At the surface of the lamina,

x=0, y<cos

l/u),

r=0,

tfld

4, = coslr'(cosU),

-1,

so that (14,

1s=4 cosh1(cosy)dy

= 2wlogj

2ir log sec b,

if b=cos(l/) is the half width of the lamina.

This gives. on expansion,

-

ils 27r(b2/2 + b4/12 +...) =wb2(1+b2/t5...).

This applies to the case when the width of the lamina is moderately small in relation to tim distance between the

boundaries, and may he generalized for bbundaries at a

o,', distance e from the centre of the lamina as irb (i +

* Lamb, 'Hydrodynamics,' p. 08 (5th cd.). t Loe. cit. p. 508.

IThil. Mag. S. 7.Vol. 9. No. 55. Jan. 1930.

178 Dr. J. L. Taylor on some

The factor may be compared with that for a circle of

radius b in the corresponding case considered in (2) above. (1') Circle completely surroi,,adetl by Riqi:? Bou na7ary.

The first order correction io the inertia coefficient for

a circle cuelo-eJ by a rigid hiouiidarv, corresponding to the three-di inensional problem of an infinite ci renTar cylinder

enclosed by a fixed cylinder of any cross-section, can be

re:idilv obtained. providNi a solution is known for the fluid motion due to the translation of a cylinder whose section is the inverse of that of the fixed cylinder, with respect to an internal point, namely the Cent re of the circle.

Taking this point as the origin, and superposing a uniform stream on the known solution, inversion gives a doublet at the origin, which may iherciore be supposed to be the centre of a small circle, moving inside th0 fixed boundary.

In ternis of z', which is equal to 1/; we have, when ¿ is

large, a solution of forni

w

z'+a/:' +

and, accordingly, when z is small,

'e

_{l/z +az +...}

c= cose/r+arcoso+

=-_cosO/r2+acosO+...,

neglecting terms of higher degree in r.

Putting r=b, this corresponds to the

circle of radius b moving in the direction

(1/b2- a).

=

_4&.

(le

case of a small

Ox vitli velocity

(.=b)

= (l/ba2b2).

Correcting this for unit velocity, since tue kinetic energy varies.as the square of flic velocity, the eiìtraiñed area is

(i/1i2a2b2)

(1/b2a2

= 7rl'(l+2ab2)

to this order in b.

Thus, it is only necessary to know the coefficient

_{a in the}

original

solution in order to

determine

tue first order

IJqilrod,yna,ìiical Inertia Coefficient s. 179

correction to the inertia.

if tuo boundary is a circle of

unit radius which invertsinto itself, a= 1, giving 7Tb'(l + 2b2)

or more generally when the radius of the outer circle is e, 2b2/c2) in agreement, for the case of(li/c) small, with the exact solution, which in this case may readily be shown

fc2+b2 to be 7Tb

\Vhen the outer boundary is a squsre the solution of

Part I. (4) may be applied. a=1, giving irb7(l +2b2/cl) in

this case also, tlì

.si(le

of the square being K=1854.

Generalizing the solution for a square of side 2e give

li2(1-r K2b2/2c2) or rb2(1+172b2/c2), slightly less, as would be anticipated, than for.a circle of radius e.

PAI1T IV.Mo'rloNs IN THREE DDIENSIONs. (1) Infinite Circular Cylinder.

If a cylinder of radius a be supposed to execute flexural vibrations of sinai! amphit nie, the velocity being gi en by b. coskz, z being measured along the axis, the ahiprolriate

olnt ion for the ¡notion of the external fluid is

-C. K1(kr) .cosO.coskz

with a suitable tinie-Ictor, K1 being Bess,l's function of order

unity of

the second kind (and

" of imaginary

argulneilt "), which is selected so as to make the ¡notion

vanish at infinity. The ¡notion is pat1le1 to the plane :e,

and the constant C is deteriiiined by the boundary condition

=b.coskz.cosO

(r=a)

dr

a2(

Ki(kaï

.b2cos2kz.

\ka.K11(lea),

If die fluid vere constrained byiL series of planes

perpen-'lic ular to O, so that the motion took place in two dimensions Oiìlv, the corresponding expression would be . b2. a2. coskz,

N2

=

O.k. Ki'(A-a)cos9.coskz,

C=b/(k. i''(ka)},

¿lì

.a.dû

180 _{I)r. J. L. Taylor on some}

so that the expression in brackets, which is always less than

unity, indicates tite ratio in which the eñergy is reduced, as compared with the two-dimensional casa. _{Its value R}

in tenus of a/X or ke/2, X being tite

'ave-lengih of the vibration, is given below

aix 1/4w 1/2w 3/4w 17w

R 0781 05S8 0464 03S0

If, instead of seeking a solution of Laplace's equation, we

had uod the equation (V2+k12)=O, appropriate to a

compressible fluid, 277/k1 bein the length of titecon1presion wave of the same frequency as tite vibration, the _{same form}

of solution would have applied, provided _{k> k1, (k2 k12)4} being substituted for k. _{This lias the effect of increasing}

slightly the kinetic energy.

(2) Ellipsoid.

A solution can be obtained which fulfils the boundary condition for what is praetieaily_{a two-node flexural vibration} of a prolate ellipsoid of revolution.

in terms of the usual coordinates

=

y = k(1 2_.1)4cosw;

=

appropriate to a prolate ellipsoid, the foci _{of tite titeridian}

being the points (±k, O, O ),

= O. P3). Q'() . cose

is a known solution of Laplace's equation

* Laiiib, _{p 130.}

Q'(

(1 -n. 1/2

(2._ 1)4.

- i

-(

(15_3),

(152-3) log

-4

J n_

- II!Idro11!i1amzcul Inertia Coe/fidents. 181

The velocity potential can be ¡nade to satisfy the boundary condition

=(a2_x2)

+2.xy

(=:)

by adjusting suitably the values of C ilnd a. Since

(a2_.c2)+2..r,f

- k222) + 2X32(2 1) (2

this gives a2 k2(2-02) 5

C=k.a23/2(o2_I).1t

_(=),

2T=

-dn

=

-j d.

:*4).../:1__,h2»2__1)

.dw

-

i k '

-

(/L1 4). k. o) -1

o

(::

k(1-2).(2i)do

7Tk.(-1)

.C2.Q3()

j

1 2

_Í'523\2d

J,L1

2 J 32rk3

-Q3'())

-

.a . S

(=O)

The assumed boundary condition (1) above corresponds to au approximate type of ilexural vibration in which the amplitude is proportional to (a2 x2), the positions of the

nodes" being given by

182 _{Dr. J. L. Tayloron some}

For a slender ellipsoid, _{is very little different from unity,}

so that the nodes are approximately

_{± v'l/5 of the major}

semi-axis (A-) from the centre of tite eilipsotd. _{This agrees}

fairly closely with tite actual positions for a beati having

the same mass distribution as an eliipoid of uniform density.

The effect of the departure of the actual amplitude curve f'rotn the parabolic shape_{is prob:blv unimportant, hut as the} pritttary object of the investigation is to compare tite inertia

of the water with that of the ellipsoid itself, this may be allowed for by calculating the kinetic energy of the ellipsoid

for a hypothetical vibration

_{in which the amplitude is}

proportional to (a2v)

_{as the basis of comparison.}

The area of the circular section of the ellipsoid being

1) or

-the kinetic energy, assuming unity density, and velocity

proportional to (a) is given

by

2T1

r(o2-_l)Ço

_{(a2a2)2. (k22.v2) .dr}

-

k70(O2_1)(7_14,2+204).

The ratio

T/T, -

_-

2(9 2) 2

142±

o+

i 4/(2_1)_(I5_ 3)log0

- 9O + 3G +

4/(2_

_{1) + 3(l 5o2_11)lo4}

This tends to the value unity as _{tends to unity, i.e., as} the ellipsoid becomes indefinitely sittall

_{in diameter, the}

length remaining constant. r110 huid _{motion is then} practi-cally two-dimensional. _{The value for various}

ratios of tite

axes c/b is given below, the s'dueof

_{o being 1V1_c/b;_}

Tito rotatory inertia of _{the ellipsoid has been neglected,}

since in any practical _{application of the results the} effect

of this factor would

_{be separately estimated,}

if it were necessary to take it into account. _{ft can readily be shown}

c/b _{0045 010}

0141 ₀₂₀₁

Il3drodynczni icet Inertia Coef/Icients. 183 that for ait ellipsoid of uniform density tite rotatory inertia is given by

ii(o21)

/

2

T

' _1,

from which the corrected ratio TIC'!'1+ II) may be obtained.

Tue results given above, both for the ellipsoid and for the iufnite cylinder, show that, even when the (halneter is ino(lcriLtely silla ¡ in relation to

tite wave-length of the

vibration, the three-dimensional character of tile fluid motion cannot he ignoied.

In the case of tile cylinder,

the distribution of tite pressure due to the fluid inertia is, however, identical a itli that s'lien tite _{¡notion is confined}

to two dimensions, being proportional to

Cos . cos kz.

A similar resu1 t. itolds i it ti me casc of ti e clii psoid _{as regards}

tile attiount of tile normal pressure on tile Snrface, but a

sntitll correction has to be applied to tite C01u11)O1)eilt ill tile

direction of motion towards the ends of tile ellipsoid _on

account of tile inclination of the surface to tite axis.

¡n applying the results obttined to bodies of other forms-for i nsta1ce, the hull of a lmip,rcasonai le _{accu r:mey should} be obtained by assummmimmg a distribution of tite "added mass'' of vater accordiin to th _{emit pineal formula given at t ho}

end of Part I. above, and correcting the total

_{aitioutit in} accôrdance with tile ratio found for an ellipsoid of similar

proportions. This applies to ril)rations ill a vertical plane, but for, horizontal ¡notion the correction factor will be iuiuch

nearer urnity, simice the finid ¡notion, as indicated in tile

exam pies of Part II., is more local in the t\vo-dinleimsional case.

The author is iimdebteml to Professor j. l'rOu(lmali for some valualmle suggest ions in tile preparation of titis_paper.

jluthoì"s jtote.\\llile this paper lias been ut the

_press,

Prof. H. M. Lewis has published (Aitier. Soc. of Nay. Arch., Nov. 1929) a øiution corresponding to Part I., example (3).

He also treats the cae of an ellipsoid (Part IV. (2)). but

neglects the term in fl the boundary conditions, and

arrives at different results in consequence.]

Ii