The damping of an oscillating ellipsoid near a free surface

HYDROMECHANICS

o

AERODYNAMIC

o

STRUCTURAL MECHANICS

o

APPLIED MATHEMATICS

ARCH!EF

Lab.

y. Scheepsbouv

T!

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by J. N. Newman HYDROMECHANICS LABORATORY

RESEARCH AND DEVELOPMENT REPORT THE DAMPING OF AN OSCILLATING ELLIPSOID

THE DAMPING OF AN OSCILLATING ELLIPSOID NEAR A FREE SURFACE

by

J. N. Newman

Reprint of paper published in The Journal of Ship Research Vol. 5, No. 3, December 1961.

The Damping of an Oscillating

Ellipsoid Near a Free Surface

By J. N. Newman'

The six damping coefficients are derived for an ellipsoid with three unequal axes, which ¡s moving with constant horizontal velocity beneath a free surface, and oscillating in any one of six degrees of freedom. It is assumed that the ftow is irrotational and incompressible, and that the ellipsoid is either slender or deeply submerged, in order that the disturbance of the free surface be small. With these assumptions the six damp-ing coefficients ore derived and computations are presented for two particular

ellips-oids. Of special interest is the occurrence of negative damping at very high forward speeds.

IN the analysis of ship motions, considerable

im-portance has been attached to the damping provided by energy radiation in the form of outgoing surface waves. Although there are other damping mechanisms, notably

viscosity, these have been neglected in most analyses,

especially in the study of pitch and heave. For the case

of roll the importance of viscous damping depends

strongly on the shape of the hull, and one must therefore proceed with caution. Nevertheless for many applica-tions, such as surface ships without bilge keels or other

significant appendages, the neglect of viscosity seems

permissible, and the analysis can therefore be based upon

potential theory. A notable step has been made in this

direction by Ursell [I }2 who developed a theory for the

roll damping of two-dimensional cylindrical sections.

For fairly long, slender ships at low speeds a

two-dimen-sional or "strip" theory should be valid and Ursell's

results may be applied.

'Seaworthiness Branch, David Taylor Model Basin, Navy

Department, Washington, D. C.

a,, a, a, semi-lengths of principal axes of an ellipsoid

B1, = damping coefficients(j = 1, 2, 3, 4, 5, 6)

c = forward velocity

D, virtual-mass coefficients defined following equation

(11)

D nondimensional virtual-mass coefficients defined fol-lowing equation (31)

G = Green's function g = gravitational acceleration h = depth of submergence

1= '1

j =

index referring to direction of axis or motion

IT \ '/2

j,,(z) = spherical Bessel function, j,(z) J.+ i,', (z)

K = wave number, K = w2/g

u, velocity components of ellipsoid (j 1, 2,. . .6)

V(x,,.r,,z,) = gravitational potential of effipsoid

x,y,z Cartesian co-ordinates translating in space

Nomencla turc

In three dimensions there has been considerable work on pitch and heave damping, and Hishida [21 has studied

the sway damping of a submerged spheroid. Hishida assumes that roll and sway damping are similar, at least in their dependence on forward speed, and this

assump-tion seems to be justified by experiments with surface

ships. Nevertheless it is desirable to study the damping of pure rolling motion for a three-dimensional body with forward speed, and for such purposes an axisymmetric body such as a spheroid is unsuitable due to its circular sections. With the advent of model testing in oblique waves it is also desirable to analyze all six degrees of

freedom.

For these reasons the present paper considers the

damping of a submerged ellipsoid with three unequal axes, which is moving with constant forward velocity

and oscillating in surge, heave, sway, roll, yaw, or pitch.

In order to study this problem, the ellipsoid is

repre-sented by a distribution of singularities, and the damping

2_{Numbers in brackets designate References at end of paper.}

= Cartesian co-ordinates fixed in ellipsoid

=Cartesian co-ordinates translating in space

aj Green's integrals defi.ned by equation (6)(j 1,2, 3) = Kronecker delta function, &, = O if i ,

i

_i

- i

if

-

translational displacements of ellipsoid (j

-dummy co-ordinates corresponding to y

(j

-1, 1,

2, 3)

2, 3) O = root of equation (7) defining ellipsoidal co-ordinates;

polar co-ordinate

ej = rotational displacements of ellipsoid (j 1, 2, 3) A = ellipsoidal co-ordinate normal to confocal ellipsoids

dummy co-ordinates corresponding to z, (j = 1, 2, 3)

p - fluid density r wc/g

4' velocity potential

4,,- components of velocity potential defined by equa-tion (2)

coefficients are then found from energy radiation at

infinity.

With regard to the derivation of this theory, two items should be emphasized. The singularity system consists of a distribution of steady-state dipoles, representing the constant forward velocity of the ellipsoid, plus a distribu-tion of oscillating dipoles and quadripoles due to the un-steady motion. However the strength of the oscillating

singularity distribution is dependent on the forward

speed.

This is a consequence of the fact that the

boundary condition for the velocity potential due to the forward speed must be satisfied on the actual oscillatmg surface of the ellipsoid rather than its mean position with

respect to time. A similar result has been noted for the

ellipsoid by Eggers [3] and for the thin ship by Newman [4, 5]. In this respect the present theory differs from that of Hishida.

The second item to be emphasized is the use of energy

radiation at infinity to determine the damping

co-efficients. Physically the most direct method would be from pressure integration over the actual surface of the body. Mathematically, however, the two methods are

equivalent, and the present theory cannot be applied at the surface of the body without first correcting for the "image" of the free surface inside the body. This dif-ficulty is avoided by working with the velocity potential

at infinite distance from the ship, where to first order

these image corrections do not contribute. In this sense the present results are onry valid for a deeply submerged ellipsoid or else for a slender one. The limits of this

re-striction are not definitely known but related work has shown good agreement with experiments for depths of

submergence a few times the beam or depth of the body.

Numerical results have been obtained for various

speeds, frequencies and eccentricities of the ellipsoid and these are illustrated in Figs. 1-12. Of particular interest

is the fact that certain of the damping coefficients

be-come negative at very high speeds.

The derivation of the theory is divided into two sec-tions. In the first of these we obtain the singularity dis-tribution for oscillatory motion in an infinite fluid, and in the second section these results are applied to obtain the damping near a free surface. This is followed by a discussion of the results and by graphs showing the com-puted coefficients.

Motion in an Infinite Fluid

We consider an ellipsoid, defined by the equation

112

++=1

X2 12

where (x1, 12, 13) is a Cartesian co-ordinate system fixed with respect to the ellipsoid, and (a1, a2, at) are the

semi-lengths of the three principal axes. We may assume

without loss of generality that a1 a2 a3 O, whence

the xi-axis coincides with the major axis of the ellipsoid.

Adopting thinottion of St. Denis and Craven [6],

the motion of the ellipsoid is defined by the trnslational

velocity components (u1 u2,

u3) and the rotational

(1)

velocity components (u4, ub, u) relative to the (xi, 12, 13) axes. In an infinite fluid with the usual assumptions of incompressible, inviscid

flow, the velocity potential,

whose gradient is equal to the velocity vector, may be

written

a

It, 13) (x)

_{= E}

u,4,(x4) (2)

where in an infinite fluid the potentials may be written in the form4

Here V(x4) denotes the gravitational potential of the

ellipsoid of unit density,

V(x1) =

f1f[(xi Ei)2 I (12 E2)2 + (r3

-f11dE1ddE3 (5) with the volume integral taken over the interior of the

ellipsoid given by (1), and the constants a1 are given by the integrals

dx

a1 = a1aa3 I

Jo (a,2 + X)[(a12 + X)(a22 + X)(a32 + X)1h2

(6) In equation (4) and hereafter we adopt the cyclic

con-vention, i.e., a4 = a1, a = at, and similarly for aj and z1. Following Havelock [9] we deduce that since the gravi-tational potentials of a family of confocal ellipsoids are

proportional to their masses, the potential V(x1) may

be expressed in terms of a volume integral over a confocal ellipsoid

+

+

13

1

(7

a32 + O a22 + O a32 + O

and in the limit O ' at2, in terms of a surface integral

over the elliptic focal conic

A function of argument (z.) will be used to imply a function of the three variables (z1, z,, zs).

'Reference [7], chapter 7, section 6. Equations (3) and (4)

can also be verified from Lamb [8] by combining equations (2) through (5) of section 339 with the expressions for the ve1octy potential as derived in sections 114 and 115. The present

equa-tions differ from these references by a minus sign due to the

different definitions of the velocity potential.

av

-

-

(j =

1, 2, 3) (3)

2T(2 - a,) òx,

i

=

2

- a+22

2(a,+i' - a1+22) + (a,+it + aA+22)(aJ+1

f

av

ov

-2, 3) (4) 1, I X1i

- x12

i

(j =

\

ò;2

òx3+i1

Fig. t Surge damping coefficient for ellipsoid at/a1 1/7,

- 1/14,

b/a1 2/7, for various Froude numbers

c/(2gai) 'la

70 60

320 30

Fig. 2 Sway damping coefficient for ellipsoid at/ai = 1/7,

1/14, b/ar = 2/7, for various Froude numbers cl (2gat) 'la

r.0

4 6

Fig. 3 Heave damping coefficient for ellipsoid at/ai 1/7, 1/14, b/a1 - 2/7, for various Froude numbers

c/(2gai) 'I'

xi2

Fig. 4 Roll damping coefficient for ellipsoid at/ai 1/7,

= 1/14,

b/at = 2/7, for various Froude numbers

c/(2gai)'/a

Fig. 5 Pitch damping coefficient for ellipsoid at/ai = 1/7

ai/ai =

i/i4,

b/ar 2/7, for various Froude numbers c/(2ga1) I'

Fig. 6 Yaw damping coefficient for ellipsoid a1/a = 1/7,

ax/a 1/14, b/ai = 2/7, for various Froude numbers

c/(2ga1)'/a X22 a2 - a32

+

2 _{- a32} = 1, x3 = 0 (8) Thus 2a1a2a3 V(x1) =

[(a12 - a32) (a22 - a32) J','

fi

[iE2

E22

T" fi'

- ìdE1dE2

a12 - a32 -

2 - a32j

_\r,

b-O

(9)

Fig. 7 Surge damping coefficient for ellipsoid a2/a1 = 1/7, a3/a1 = 1/14, b/ai = 2/7, for various Froude numbers

c/(2gai) /

Fig. 8 Sway damping coefficient for ellipsoid a2/a1 = 1/7,

a3/a1 = 1/14, h/a1 = 2/7, for various Froude numbers c/(2ga1) ì' ¿ s 40 30 20 0

Fig. 10 Roll damping coefficient for ellipsoid e2/a1 = 1/7, aa/a1 = 1/14, h/a1 = 2/7, for various Froude numbers

c/ (2gai) V'

Fig. i i Pitchdamping coefficient for ellipsoid a2/a1 = i 7,

as/a1 = 1/14, h/a, = 2/7, for various Froude numbers

c/ (2gai) 1/,

60 50

Fig. 9 Heave damping coefficient for ellipsoid a2/a, = 1/7, Fig. 12 Yaw damping coefficient for ellipsoid a2/a, = 1/7,

= 1/14, h/a,

= 2/7, for various Froude numbers aa/a, = 1/14, h/a, = 2/7, for various Froude numbera

c/ (2ga,)'Is c/ (2gai)'/S

F

where the surface integral is over the surface boundedby neglecting squares of the displacements

, and 8,

it

the ellipse follows that

E'

₊

E22 ₌ y) = X,

+ (

+

81jX+2

-a1' - a,'

a,' - a,'

A detailed derivation of this equation is given in the Xf =

y, - (

+

O.,iy,+,

-Appendix. It will also be necessary to employ orthogonal ellips-Substituting (9) in (3) and (4) and noting from direct oidal co-ordinates (X, ,, which are defined as the three differentiation that roots of equation (7) when considered as a cubic in 8.

o

'i'

,, Details of this co-ordinate system are given by Lamb [8]

- (-)

= - x ()

(section 112). The co-ordinate X is normal to the surface

(iX1 \T/ _{uE, '' of each ellipsoid and is analogous to the radial co-ordinate}

and in a spherical system. The ellipsoid defined by (1) is the

o surface X = 0 and the normal component of the velocity

is proportional to O4/OX. The Cartesian co-ordinates

òXj+z \/

òxai V]

may be expressed in terms of (X, , y)as

it follows that = i-D,

fi

= [1

¿'s:

(1 ò

(1\

Ei+2 E,+i 0E

r)

E'

_1"

)

a,' - a,'

a,' - a,']

0E,

(1

(j=1,2,3)

(10)

E'

E2'

\'/

a1' - a,'

a,' - a,')

r

ò

f1\

o

/i\

òE+

r) - E,+

òE,+

(j= 1,2,3)

(11)

H=+-+--1

where a,

a,'

a,'

W = A I = A +I

D, -

( -

1 ₂

\

-- '" 'S - S-',

2 2

2v

' '

(2 -

a,) [(a, -

a, )(a, - a,)] '

'

_{OH O 2x O4}

a1a,a,

,,

Ox, Ox,

,.,

a, Ox,

- [(a,

- a,')(a,' - a,')]

' =

[

2 +

(a,i' +

,_i òx, o

= (14)

-

a+2)]

Fuhermore from (12) it follows that

a+,

-(j= 1, 2,3)

OH OH OH fOx, Oxj\

Equations (10) and (11) express the potentials , in

-

=

-

e

terms of certain dipole distributions over the elliptic focal

conic. The first of these expressions has been derived by

-- (, +

Havelock [9].

_a,'

The foregoing results express the velocity potential in terms of a Cartesian co-ordinate system (xe) fixed with

2e - + 2eewl

(--

03 - 0,' (15)

respect to the ellipsoid. Since we shall ultimately be a2

\a,'

a,

concerned with oscillatory motion near a free surface, it

is necessary to transform to a steady-state co-ordinate Thus the boundary condition on the ellipsoid may be system. To be explicit, let the unsteady motion of the written

ellipsoid consist of infinitestimal oscillatory translations o ,,

-

(, + 0,+,x,,

along the x,-axes and rotations

about the

_{,=j a,}

x,-axes, where the real part is to be taken in all complex

expressions. If (y) is the steady Cartesian co-ordinate i z1

+

(,

-

o, -)

(16)

system about which the (z,) system is oscillating, then

+

C 2 \, a,' a,

(a,2 + X)(a,' + Xa,' +

y)

Xj2 =

(a,'

_a+i')(a,'

_a1+,')

and thus

(j = 1, 2, 3)

(12)

Ox,1

x

OX

2a,'+X

If the ellipsoid translates with constant velocity c ) along the y,-axis and (y) is the perturbation velocity

potential, the boundary condition on the ellipsoid may be written [8] as

onH=0 (13)

The potentials given by equations (10) and (11)

satisfy the boundary conditions ò4, Ox, òX OX 2a,2

òif3

i+i

òXj+2 ox ox ox f+ii+2 I 2 2\ 1 i = 2 2 - a,+2

i

'j -

' '

2a1+i aa+2

Comparing these boundary conditions with (16) it is

apparent that to firstorder \ve may satisfy the problem

of the oscillatory ellipsoid in an infinite fluid with the

potential

3

= c + ic.'e"'

E

+

J+30,) +c02e'""

-

c0ie""42

(17) This result may also be derived from a physical argu-ment. In the (x,) co-ordinate system, fixed with respect

to the ellipsoid, the three translational velocity com-ponents are, to first order in , and 0,,

(i1e"

+ c, i2e" - c03e', i,3e" +

cO2e"")

where the contributions involving02and03 aredue to the

"angle-of-attack" components of the forwardvelocity c.

Substituting in equation (2), the total potential to first order is

=

(iwj& + e) ct,i + (iw2 - c3)e'

+ (i3 + 02)e"" + ioi" (014 + 02415 + 0)

and equation (17) follows directly.

Substituting equations (10) and (11) in (17), we obtain

-

_If

[

Ei2 E22

1''

-

a2 - a32 a22 - a32.j

JcD1

-- (t"

+ i,e'

,D, 0 1

k òE1 \r)

ji

J

()

3

+ iwe"t E o,D13

[Eii

_{Ò:+2 ()}

Ei+2

-i

+ c02e'D3

() - c83e'D2

-

()} d1d2

(18)

Equation (18) expresses the potential asa function ofthe

oscillatory (x1) co-ordinate system. We proceed now to

transform this expression to a function of the steady co-ordinate system (y). We have

(j =

1, 2, 3)

O)

I

r

[(Xi )2 + (x2 - Ei)2 + (X3 - E3)21'

i

- [(y -

)2 + (y2 - 2)2 + (

3)2]/'= G(y1, )

where

".4

'i, E, +

(,

+ Oj+'Ej+2

-and G denotes the Green's function,which in an infinite

fluid is simply equal to

the source function hr.

In

order to carry out the surface integration in (18) we expand the Green's function using Taylor's theorem:

3

[0G]

G(y4,

G(y1, E) + E ('i, -

E,)

+

j-1

J i-i

3

= G(y4, E) + e'' E a, + OJ+1E,+2 - O,+2EJ+i)

J-1

G(y1, E)

+.

Substituting in (18),andneglecting terms of second order

in and 0j, we find that

1

If

[i

E2 E22

-T 3 = 0L a12 - a32

2

-a32j

{cDi

--

G(y,E1) + cD1e'.

(,

+ Oj+IEJ+2

òE,

- 0i+2Ei+I) - G(y1,

] + ie"

[,D,

_[

-- G(y1, E)

_ò,

0E,

\

+ c62D3e - G(y,, E) - c03D2e" G(y1, E) } dE1dE2 (19) Equation (19) demonstrates the importance of a sys-tematic development in problems involving both transla-tion and oscillatransla-tions. It should be noted that there are two ways in which the forward velocity influences the oscillatory potentials in (19). The first of these is the in-fluence of the last two terms in the integrand, correspond-ing, as stated before, to the "angle-of-attack" velocities due to forward motion at an angle of pitch or yaw. The second influence, represented by the terms

+ 0,D,3

(E+ à

-

J+2 G(y,E)

o

;, 3

cD1e - E (, + 0,

Ej-i-2

+22) -

G(y1, E)

0E,

is due to the fact that the original steady-state

singu-larity distribution representing the translation is located on an oscillating surface. Thus these terms represent the change in location of the steady singularity

dis-tribution.

Motion Near a Free Surface

We consider now the case where the ellipsoid is

be-neath a free surface, and we make the usual assumption

that, to first order, the potential is given by the same

distribution of singularities as for the motion in an infinite fluid, but with the singularities satisfying the linearized free surface condition. It has been shown [11, 15] that it is necessary to add to this a singularity distribution which cancels the induced velocity on the body due to the free-surface portion of the original singularities. However this corrective distribution consists of

ele-ment.ary singularities which do not generate waves and which will not radiate energy at infinity. In other words, as Havelock has frequently pointed out,5 a higher degree of approximation is usually necessary in working with

the pressure on the body as opposed to studying the

asymptotic behavior at infinity. Thus we may neglect the corrective distribution provided the damping is derived on the basis of energy flux at infinity.

We proceed therefore to substitute for the Green's function in equation (19), the potential of a source be-neath the free surface. The steady-state term

¡'f

'I,

Jj.0 L

a12 - a32 a22 - as2

[CD1 - G(yE,Ei)] dE1dE2

in (19) has been studied by Havelock [9, 10] in connec-tion with the steady-state wave resistance and will be de-leted from the present analysis. The remaining terms in (19) are sinusoidal in time and it is necessary therefore to employ the expression for a submerged source which

translates with velocity c and pulsates in strength

si-nusoidally with frequency w/2w.

Let (x, y, z) be a Cartesian co-ordinate system moving in space in the x-direction with velocity e, with the plane z = O corresponding to the undisturbed level of the free surface and z-axis positive upwards. The potential of a

source of strength e" located at the point x = E, y = ii,

z =

may be represented [4] by the asymptotic

ex-pansion G(x, y, z; E, s,,

) = i

(S)1

,, , X,,,(u,,) sine o y 1Sifl2 u,,

exp{Xm(u4)[z

+ + i(x - E) cos u,,

+i(y n)sinu,,] ±

}+o(j) (20)

where

x = R cosO

y = R sin O X,,,(u) -2 N do

cos (u,, - O) > s41(u4

du,, g cos2 u (m = 1, 2)

i. = wc/g

cos u

=

Icos u

82(U) = 1

'Cf. reference 112], page 15.

[1 + 2r cos u ± (1 + 4r cos u)"]

and the (±) sign in (20) is determined by the sign of do

- cos (u,, - O)

du4

The second summation is over the N-roots of tbe equa-tion

Sin2 u,, ± (1 + 4r cos u,,)1'

ctn O =

sin u,, cos u,, satisfying the inequality

IOIT

Substitution of (20) in (19) gives the asymptotic ve-locity potential due to the oscillations of the ellipsoid,

valid at large distances from the ellipsoid. We shall re-strict the subsequent analysis to the case of a submerged ellipsoid with the a3-axis vertical, that is with a "beam-depth" ratio greater than or equal to one. The horizontal case or the more general problem, where the ratios of the three principal axes a1, a2, and a3 are arbitrary, involves only slight modifications of the following analysis, and

in fact the final results can be shown to be valid for

arbitrary values of a1, a2, and a3 without the restriction

a1 a2 a3.

Assuming, then, that the a3-axis is vertical, we set

E Ei, n = Ei, and = E3 - h, where h is the mean depth

of the centroid below the free surface. The appropriate asymptotic source function is therefore

G(x, y, z;E, n, ) 1/,

(8T\

=i)

Am(U,) sin2 O ç' . (1?

sin2 u,, - cos (u,, - O)

du,,

- h + iR cos

I

'

s,,(u,,) exp

(u3 - o) + E3 - iEi

cos u - iE2

i ±

and the oscillatory potential, from (19), may be written in the form A(u,,ido ) Sin2

=Re e"

{Sfl2

u I- cos (u,, O)

du,,

s(u,,) exp {X4(u) [z - h + iR cos (u,, - O)] E ft,P,(u,,) + 8,P3(u,,) J (21)

4J

,.-where

Ei2 2 'I,

P,(u)

=

fj;0

(i

a12 - a32 - a22 - a32)

o

(ND, - cXm(U) cos uDi) exp { Xm(U) [Es - iEi COS U

arid

P,3(u)

=

ff

(

-2 'I,

\

a2 - a32 a22 - a32) ò

[(wD+2 - cX,,,(u)Dj cosu)(EJ+i

-exp' X,,(u) - iÊ1 cos u

+ ic45J3(D2 - D1) - ic5,2(D3 - D1) P4(u) P6(u) - 1E2 sin ufldEidE2

= 1, 2, 3) (23) may be obtained by differentiation

A = Am cos u and B = Am Sjfl U

2 .

W0 = - f f

- c cos

O)RdzdO (26)

where the bar denotes the time average, p is the fluid

density, and

1 (ò

ò\

(25) z,,

= -

- C)

is the free-surface elevation. Expanding the second term of the integrand in a Taylor series about z O, integrating with respect to z, and neglecting terms of

third order in the potential,

'2,r

_r°

RdzdO W,3 = Jo

-

-

Rcosüdû

(27) g o òt òt

òjI=0

Substituting equation (21) for the potential, integrating

with respect to z, and neglecting cross terms of order

1/R, we obtain the expression

TV0 =

-f2,r

u,, 3

[P(u) + O,P+3(u)]

2 [sgn u,,]dO (28) P2(u) = X,,,(wD3 -P4(u) jXm(D4

-eX,,,D1 cos u)P(A,

cX,,,D1 cos u) B) ÒP(A, B)

P5(u) = Am(wD5 - CXmD1 cosu) ÒP(A, B)

icX3,,(D3 - D1)P(A, B)

P6(u) = Xm(Di - cX,,,D cosu)

rap.

sm u -

ap

cos u with

+ XrnC sin u (D - D1)P(A, B)

Ei2 E22

\/

a12 - a32

a22 - a3J

cos AE1 co.s BE,dE1df2 (24) which has been evaluated by Havelock [10]. Denoting m2 = a12 - a32 and n2 = a22 - a32 Havelock has shown

that

P(A, B) = 2' "ir"mnJ,i.{ (m2A 2 + nB2Y"

(m2A2 + n2B2Y'

vhere J,, is the Bessel function of the first kind, of order

n. Introducing the spherical Bessel function

'ir \"

j,(z) =

and denoting

q = [(ai2 a32)A2 1 (a2

-we have

P(A, B) = 2ir[(a12 - a32) (a22 - a32)]'/

[ji(q)]

q It is easily seen that the integrals P(u) are given by

P1(u) = iD1X,,, cos u(w - eX,,, cos u)P(A, B)

P2(u) = iX,(cD2 - CXmD1 cos u) sin u P(A, B)

Pj(u) = 2iri[(a12 a32)(a22 32) 1'" X,,, cos u(c..,Di -cX,,,D1 cos

P2(u) = 2iri[(a12 - a32)(a22 - a32) 1" X,,, sin u

(wD2 - cA ,,,D cos ¡) [j1(q)/q] Pa(u) = 2ir [(a12 - a32) (a22 - a32) J''

- cX,,D1 cos u)[j1(q)/q] = - 2m [(a12 - a32) (a22 - a32)]'/ Am2

sin u(wD4 - cX,,,D1 cos u)(a22 - a32) L(q)/q2] = 2mi[(a12 - a32)(a22 - a32)]''

A,,,2 cos u (wD5 - cXmDi cos u)(a12 - Q32) [j(q)/q21 - cXm(D3 - Di)[ji(q)/qfl P6(u) = - 2w[(a12 - a32)(a2 - a32fl'' X,,,2 cos u si'i u

(wA - cX,,,D1 cos u)(a12 - a22)[j?(q)/q2] cA,,, sin u(D2

-Substitution of these relations in (21) gives the oscilla-tory potential at large distances from the ellipsoid. This

potential may be employed to determine the energy

radiation due to the damping forces and moments. The analysis of energy flux for a pitching and heaving

surface ship has been carried out in [4] and the results

can be applied to the present problem without essential (hanges. Thus we shall only outline the remainder of

the analysis, details of which may be found in [4]. The average work done per unit time by the damping forces is

Carrying out the indicated differentiations we obtain Changing the variable of integration from 8 to u,, and the following expressions: taking into consideration the appropriate limits of in-The functions Pj(u)

from the integral P(A, B)

=

-o \

(

Ame 2Xmh

where

O

for i.'/

uo=

()

for r> 1/4

Let

denote that component of the force

or

moment in the jth direction which is in phase with the

velocity Uj, so that B5, is the damping coefficient. If only one degree of freedom is allowed,

W0

= B,,uj =

(j = 1, 2, 3)

= O_32B15 (j = 4, 5, 6) and thus we obtain the six damping coefficients

B11

_{E fO}

(1 + 4ros u'

TWm

P(u)

sm(u)du ( = 1, 2, 3, 4, 5, 6) (30)

or to adopt a more convenient notation,

32T 2

-o

B,

= - --

pa12a22a32 E

J

1 + 4r cos u)"

W m-1

[Q,(u)12 s,JU)dU (j =

1,2,3,4,5, 6)

(31)

CXmD3 cos u) cos u

[ii(q)]

cXmDjcos u)

[jj(q)]

CXmD1 cos u)

[q)]

q

(Diw CXmD1 cos u)(a22 a32)Xm sin w

[:)]

u

[]q2

cX,,,D1 cos u) (a12 a32)X,N Cos

c (D3 D1)

[i(q)i

[qj

(Dew cX,,,D1 cos u)(a12 a30)X,,, cos u

ri2(q)1

I

IS1flU

Lq J

- c(D2 - D1) [ii(q)1

_[q]

and O for

r'/4

uo =

{

_i

(\

for r>

1/4 co

\4r)

3

E [P,(u) + 9,P5+3(u)J

1+1 cos u 2 s1(u) 8,,(U)dU (29) Icos u] sz(u) =

1

g Xm(u)

- 2c2 cos2

+ 2r co u

(1 + 4i cos u)"]

(m = 1, 2)

a22

q = Xm[(at2 - a32) cos2 u + (a22 a32) sn2 u]"

D-21

_aj

(j= 1,2,3)

D+3

= 2(a+ 12

a22) + (a+ 2 +

a+22)(aj+ 1 aa+2)

(j =

1, 2, 3) It will be recalled that we have restricted the orienta-tion of the ellipsoid in that the mean posiorienta-tion of the

a1-axis is in the direction of forward motion, the mean position of the a3-axis is vertical, and a1

5

a2

5

a3.

However it may be verified, by derivation of the

func-tions P1(u) for each possible case, that equation (31)

holds for all values of a1, a2, and a3 without the

restric-tion a1 5 a2 S a3.

Of course it is necessary that the mean position of the a1-axis be horizontal and in the

direction of forward motion, and that the mean position of the a3-axis be vertical.

Discussion and Conclusions

The principal analytical results of this investigation

are contained in equation (31) wherein the six damping coefficients are expressed as integrals of rather compli-cated functions.

It is not surprising to note that the

form of these expressions is similar to the pitch and heave damping coefficients of a thin ship [4, 14].

In particular the pitch, heave, and surge damping co-efficients of the submerged ellipsoid become infinite at r = 1/4 in exactly the same manner as for the thin ship. It should be noted however that the damping coefficients of sway, roll, and yaw are not singular at this point and are bounded for all speeds and frequencies. Thus there is a fundamental difference between the three modes of oscillation in the vertical x-z plane, which have a

logarith-mic singularity at r = , and the three modes of

os-cillation perpendicular to this plane, which are

non-singular.

A surprising aspect of the results is that in spite of the complexities introduced in deriving the coefficients for

ellipsoids with three unequal axes, the final results, as

expressed by equation (31), are basically no more com-plicated than those of a spheroid. Nevertheless

exten-sive computations are necessary in order to study the form of the six coefficients B5»

For this pùrpose a

program has been prepared for the IBM 704-typé digital computer, based upon numerical methods which are out-lined in the Appendix. This program may be employed to find the damping of an ellipsoid of arbitrary

dimen-sions and depth of submergence, as a function of

for-ward speed and frequency. The results of such calcula-tionM are shown in Figs. 1-12 for two different ellipsoids. tegration, this reduces to

WD = Xme2m'

(1 + 4

cos u)" where Qi(u) = Q2(u) = = Q4(u) = Q(u) = Q6(u) = (D1W (D2w -(D3w

Both are submerged at a depth of '/ times the length.

The first of these two ellipsoids has a depth-length ratio (a3/ai) of

'/,

and a beam-length ratio

(az/a,) of

'/7, while in the second case these two ratios are inter-changed. Calculations have been made with a wider variation of the beam and depth ratios but the results

are not significantly different from the curves shown. Figs. I to 12 show, in addition to the three-dimensional coefficients computed from equation (31), the damping coefficients derived from a two-dimensional or "strip-theory" analysis. The derivation of the two-dimensional results is given in the Appendix. The two-dimensional results agree quite well with the thredimensional (zero-speed) results in the cases of heave, roll, and sway, and for high frequencies the agreement for surge, yaw, and pitch is also good. However at low or moderate

fre-quencies there are signìTfliánt discrepancies in the last three iiiodes. This is physically explainable by the fact

that for heave, roll, and sway the normal velocities at

different sections of the ellipsoid are all in phase with one another, whereas for the cases of surge, yaw, or pitch, the normal velocities at the bow and stern are 180 deg out of

phase, and will interfere with one another at low fre-quencies, where the body is small relative to a wave

length. At very high freauencies it is seen from the

curves, and shown analytically in the Appendix, that the two and three dimensional results are identical for all six coefficients, at zero speed.

It is epeciaIly interesting to note that for very high

forward speeds (i.e., Froudé numbers of about 1.0) the surge damping coefficient becomes negative, and for the

"thin" ellipsoid, Fig. 11, this is also true of the pitch

coefficient. Similar calculations with a sphere also show a negative heave danpin coefficient7 but no cases have

yet been found of negative damping in sway, roll, or yaw. The possibility of negative damping was antici-pated by Eggers [16] who found similar results for a

source and dipole combination. The concept of negative damping is not easy to accept, although the situation is

not unlike aeroelastic flutter.

In the case of surge a

physical argument can be devised,6 for it is well known

that the steady-state wave resistance of a submerged

body rises to a maximum with increasing velocity and thereafter falls off to zero at very high velocities. Thus

for velocities greater than that corresponding to the maximum, an increase in speed will give rise to a

de-crease in resistance, and vice-versa. From a pseudo-steady-state argument it follows that at these speeds,

the surge damping coefficient will be negative for suf-ficiently low frequencies.

The presence of negative damping implies a source of

energy other than the oscillating forces acting on the

body. At zero speed there is no other energy source, but when forward speed is involved there is a possible source of energy due to the forward velocity, just as for the ease of flutter. If the body is in a fixed position in space

and the fluid is flowing past it, this energy source is the

s 'p ny w

suggested by Marshall P. Tulin.

infinite kinetic energy of the stream.

If on the other

hand the body is moving in space and the fluid is at rest at infinity, there is a source of energy in the work done to overcome wave resistance.

In fact if the body is

moving in space with velocity c, then the total work done in overcoming both the damping and the wave resistance

is7

W0 + IV5 = Ir2

I.

[

Ip(--C--

fò4 ò

Jo

_{J-e' L}

\Òt òx

(

- c

coso)

pc cosO]RdzdO

as compared with equation (26) for W0 alone. Here p is the fluid pressure. Substituting the velocity potential

of the submerged ellipsoid we obtain, in place of (29),

') 2

¡t

_\ -ZA1,h

w

w

P'ç'

i

o

RLJ(l+4Tcosuy/

3

[P(u) + O,P41(u)]

(w - eX,,, cos u)s,,,(u)du

i- t

(32)

It is easily shown that

(w - c?,,, cos u)s,,(u) O and thus that

WD + ¡V O

Therefore the total energy flux at infinity is always

positive, and the negative damping may be interpreted

as coming from the work done to overcome the increase of wave resistance due to the oscillations.

Acknowl.dgm.nt

ThIs investigation was instigated by discussions of the Analytical Ship-Wave Relations Panel of The Society of Naval Architects and Marine Engineers. In addition to this group, the author is especially grateful to Dr. T. F. Ogilvie of the David Taylor Model Basin for innumerable

discussions and to Miss Patricia A. McCauley, Mrs.

Helen W. Henderson, and Mr. Thomas J. Langan, also

of the David Taylor Model Basin, for programming and computing the numerical results and for checking

portions of the analysis.

R.fer.nc.s

I F. Ursell, "On the Rolling Motion of Cylinders in the Surface of a Fluid," Quarterly Journal of Mechanics and Applied Mathematics, vol. 2, part 3, 1949, pp.

335-353.

2

T. Hishida, "Studies on the Wave-Making Re-

-sistance for the Rolling of ShipsPart 6," Journal of the

2osen Kiokai, vol. 87, 1955, pp. 67-78.

3 K. Eggers, "Uber die Darstellung von Körpern in Potentialströmung," Zeitschrift fzr angewandte Mathe-matik und Mechanik, band 38, heft 7/8, 1958.

4 J.

N. Newman, "The Damping and Wave

Re8istance of a Pitching and Heaving Ship," JOURNAL OF Smp RESEARCH, vol. 3, no. 1, 1959, pp. 1-19.

5

J. N. Newman, "A Linearized Theory for the

Motions of a Thin Ship in Regular Waves," JOURNAL OF

SHIP RESEARCH, vol. 5, no. 1, 1961.

6 M. St. Denis and J. P. Craven, "Recent Contribu-tions Under the Bureau of Ships Fundamental Hydro-mechanics Research ProgramPart 3," JOURNAL OF

SHIP RESEARCH, vol. 2, no. 3, 1958, pp. 1-22.

7

N. J. Kotschin, I. A. Kibel, and N. W. Rose,

"Theoretische Hydromechanik," band 1, Akademie-Verlag, Berlin, Germany, 1954.

8 H. Lamb, "Hydrodynamics," sixth edition, Dover Publications, Inc., New York, N. Y., 1945.

9

T. H. Havelock, "The Wave Resistance of a

Spheroid," Proceedings of The Royal Society, series A, vol. 131, 1931, pp. 275-285.

10 T. H. Havelock, "The Wave Resistance of an Ellipsoid," Proceedings of The Royal Society, series A, vol. 132, 1931, pp. 480-486.

li

T. H. Havelock, "The Moment of a Submerged Solid of Revolution Moving Horizontally," Quarterly

Journal of Mechanics and Applied Mathematics, vol. 5, part 2, 1952, pp. 129-136.

12 T. H. Havelock, "Wave Resistance Theory and Its Application to Ship Problems," Trans. SNAME,

vol. 59, 1951, pp. 13-24.

13 T. H. Havelock, "Waves Produced by the Rolling of a Ship," Philosophical Magazine, vol. 29, series 7, 1940, pp. 407-414.

14 T. H. Havelock, "The Effect of Speed of Advance upon the Damping of Heave and Pitch," Transactions of the Institution of Naixzl Architects, vol. 100, 1958, pp. 131-135.

15 H. L. Pond, "The Moment on a Rankine Ovoid Moving Under a Free Surface," JOURNAL OF SHIP

RESEARCH, vol. 2, no. 4, 1959, pp. 1-9.

16 K. Eggers, "Uber die Erfassung der

Widerstand-serhöhung im Seegang durch Energiebetrachtungen,"

Ingenieur-Archiv, band 29, heft 1, 1960, pp. 39-54.

17 Z. Kopal, "Numerical Analysis," Chapman and

Hall, London, England, 1955.

18

G. N. Watson, "A Treatise on the Theory of

Bessel Functions," second edition, Cambridge

Uni-versity Press, Cambridge, England, 1959.

19 A. Erdélyi, editor, "Higher Transcendental

Func-tions," vol. 2, McGraw-Hill Book Company, Inc.,

New York, N. Y., 1953.

APPENDIX Derivation of Equation (9)

The volume of an ellipsoid is the product of the semi-.

lengths of the three principal axes times 4ir/3. Thus the ratio of the mass of the ellipsoid (1) to the confocal ellipsoid (7) is

and the gravitational potential is, for any value of 8, a1a2a3

V(x)

-[(a12 + 8)(a22 + O)(a32 + O)] dEidEdE2

33

JJJ [(xi

E)2 + (X2 - E22 + (X3 - E3)2]h/ ( where now the volume integration is over the interior of the ellipsoid defined by (7). For O = O this reduces to

(5). Now let O - -a32. Then the ellipsoid defined by

(7) degenerates to the elliptic focal conic or the plane

area defined by equation (8). As O ' - ai2 the integral

in (33) tends to zero while the ratio of the masses tends to infinity.

To find the limiting value of (33) let O =

-ai2 +

Setting = O in the integrand and carrying out the integration between the limits

f

Ei2 Ei2 \i/2

E (

i

-\

a12 - a32 a22 a22

<E3<E(1

E22

a12 - a32 a22 - ai2

Ei2

we obtain in the limit as E - O

ff(

2a1aa3

V(x1)

-[(ai2 - a32) (a22 - a32) }/

i

12 ci

i

a12 - a32 a22

[(xi - Ei)2 + (X2 - Ei)2 + (Xi

where the integration is over the surface

E22

a12 - a32

+

a22 - a32 1, Ei = O

Numerical Analysis of Results

To facilitate computations based upon equation (31) we nondimensionalize the physical parameters by

substituting w2a, g

B = ai/ai

C = ai/ai

D = 2h/ai

and we change the variable of integration to K

+ 2r cos u ± (1 + 4r cos u)1'

2r2 cos u

With these substitutions equation (31) transforms to

32ir 3B2C2f'

(Kr 1)Kr

-B, =

pa13&2 w

-

[(Kr

l) - K2]''1

Ei2

l'

-

a32) dE1dE2 (34)

-a1a3a3

where ji(q) Q1(K) =

_{(Kr - 1)2}

(D1 - rKD1) q (D2 - rKD1)

(Kri)2

q Q1(K) = w(D3 - rKD1)ji(q)

q1

= cai(B2 - C2) [(Kr - l) -

K2]' (D4 - rKD1) q2 Q6(K) = w42il(1 - C2)K(D1 q2

- c(D3 - D1)

q2

[(Kr - 1) - K2]'

= wai(i - B2)K

_{(Kr - 1)2}

(D6 - rKD1)

- c(D2 - D1)

q2

[(Kr - 1) - .KI" ji(q)

(Kri)2

q and q =

[K2(I - B2) + (Kr - 1)4(B2 - C2)]''

The prime after the integral sign in equation (35) de-notes that only the intervals where

(Kr - 1) - K2

O

are to be included in the integration. The four zeros of

this function are

2r - i

(1 - 4r)''

K1.2.

_2r2

2r + i

(1 + 4r)"

K3.4

-2r2

but the first two are complex for r> , while for r = O

K1 and K4 are infinite. Thus the integral in (35) may

be decomposed in the following manner: Q2(K) = w Q4(K) Q6(K)

[(Kr 1)

-'F(K)dK = f rKi rK,

i'\

(I

_\J

+ I

+ I

)F(K)dK

JK, JK4/ if

O<r<3,

(ft: +

f:).K)dK

if

r

where F(K) denotes the integrand of (35). Since this function ha a square-root singularity (either infinity or

zero) at each of the finite limits of integration, some care is required and it is necessary to consider both the finite integral

rK.

j F(K)dK

J K,

and semi-infinite integrals of the form

j.K

j F(K)dK or F(K)dK

JK,

The finite integral is, by a linear change of the variable of integration, of the form

f'

1(x) ,,

=

f f(cos 9)dO

_,(1 x)

o

where f(x) is regular in the interval of integration. In-tegrals of this form are readily evaluated from the Gauss-Tchebysheff quadrature formula

AT

j-"

12j-1 \

f(cos O)dO

= - E

sm (

i)

Jo

n1_i

2n

(

i

+ E

(36)

where E,, is an error term which goes to zero with

in-creasing n.

The semi-infinite integrals are treated by changing the variable of integration to

z = ( jK - K )"

(n = 1, 3, 4)

Then, for exampi",

L. F(K)dK =

2f F(K4

+ x')xdx (37)

and in this form the integrand is a regular function of x.

In order to evaluate this integral we subdivide into an

infinite number of finite integrals of length A:

2 F(K4 + x2)xdx = 2

E,

F(K4 + x2)xdx

O n- A,

(38) and for each of these finite integrals Gauss-Legendre quadratures [17] may be employed. In programming

these integrals for the digital computer, a 16-point

quadrature formula was used, and the degree of accuracy

was then controlled by varying the parameter A in

(38). The infinite series in (38) was terminated when the contribution from the last term did not affect the final

answer to one part in 2, or better than seven significant

figures.

The parameters n of (36) and A of (38), which control

the accuracy of the numerical integrations, were

esti-mated from the approximate behavior of the integrand,

but a "safety factor" was placed in the program arbi-trarily to increase the accuracy. By making trial cal-culations the proper value of the safety factor was

de-termined, such that increasing beyond this point did not

significantly affect the final computed damping

co-efficient&

A

F(K)dK

if r=O

a,2a32

B11 = ST2Wp

a1

fo

cos 0)2 cosO sin2 OdO (43)

Thu "Strip Theory" Analysis

If the ellipsoid is long and slender, and the forward velocity is zero, the flow near any transverse section

(except near the ends) will be approximately

two-dimensional and the damping coefficients can be ob-tained from slender body or "strip" theory. This tech-nique employs the solution of the analogous

two-dimen-sional problem of an oscillating submerged elliptic

cylinder.

The problem of a rolling or swaying elliptic cylinder has been solved by Havelock [13]. If the vertical and

horizontal axes of the ellipse are of length 2a and 2b, re-spectively, and the depth of submergence is h, Havelock finds that for horizontal oscillations of amplitude d, the wave amplitude at infinity is

A 2TKad

(a + b)

_{a b}

'el1(K(a - b2)"t)

(39)

while for roll of amplitude O, about the centroid of the

ellipse,

A = TKO (a + b)2eI2(K(a2 - b2)"8)

(40)

where

K = w2/g

Following an analysis similar to Havelock, it is readily shown that for heave oscillations of amplitude d,

I

-i-A = 2TKbd

(a

ehIi(K(al - b2)")

(41)

- bJ

Finally, in order to analyse surge by the strip theory

we must consider an elliptic cylinder which dilates with constant eccentricity. If the area of the ellipse is Tab + .6 cos t so that ô denotes the amplitude of the change of area, then

A = Köe'"Io(K(a2 - b2)"I)

(42)

The average rate of energy flux per unit time, in two dimensions, is

1 A2

pg

-2

and this must equal the work done by the damping force at each section.

To obtain three-dimensional damping coefficients we substitute in the foregoing equations the local values of a and b and integrate over the length of the ellipsoid. If

z1 = a1 sin O,

-

<0<

it follows from the equation (1) of the ellipsoid that a = a3 cos O

b = a2 cos O

Substituting in (39-42) and integrating we obtain the

following six damping coefficients, as derived by the strip theory:

t10(K(a32 - a22Y/

a22B22 =

= 8T2pala22aa2e2K 11(K(a32 - a22)h/2 COS o

0)2cos10d0 (44)

B« = 2T2()pai(a2 + a3)4e2

{12(K(a12 - a22)'' cos 0) 2 _{odo (45)}

o

a32Ba = a22 Ba = 8ir2wpa13a32a32 (a2 +

_\a3a2)

a3\1

e2

'2

{Ii(K(a3 - a22)/ cos_°)t2sin2 O cos8 OdO (46) o

These expressions may be given in closed form in terms of modified Bessel and Struve functions. It can be shown that the foregoing strip-theory equations reduce to

B11 = 21r2wpa22a32

{i

+ 12 +T [4K2(a3 _{- a22)}

i]

(11L -

10L1)}

(47) a22B22 = a32Bn = 2T2wpa1a22a32 (a2 + a3' 2K

'\a3a2J

{i

- 312 +

[i

_{- 4K2(a32 -} a22)] (11L0 - I0L)} (48) T2

e2"

c,pa'(a2 + a3)4 K2(a32 - a22) { [105 + 44K2(a32 - a22)lIo - [20K2(a32 - a22) + 315112

-

[iuc1a3

- a22) - 40

105

+

_{K2(a32 -} a22)] (11L0 -10L1)} (49) 2 72 _2fa2-f-a3 = a2 Ba = - wpaj a2 a3 ( 8

\a3a2

e2'-'

K2(a32 - a22)

{

[15 + 4K2(a32 - a22)]Io

+ [4K2(a32 _a22) - 45112

+

[iiiaa

- a22) - 24 +

K2(a22 a32)]

(11L - 10L1) (50)

where the argument of all Bessel and Struve functions is 2.K(a32 - a22)",

and the Struve function of imaginary argument is

de-fined by

B =

(x/2)l'++l

F(m +

) r(m + n + )

The equivalence of equations (47-50) with the preceding

integral expressions may beverified by expanding in

powers of

K(a32 al2)V and making use of the expansions8

I(x)L.o(x) - In(x)L1(x) =

--.x 2 (x/2)2

-,,to m!(m + 1)!(2m+ 3) and9

I"/2

o [I,(xcos O) ]2 COS2"1 O sin2' O do

cos2

o

,,.om![(m + n)!12(m + 2n)!

Ê

(4xcos0)l'2"(2m+2n)!do

i

(4X)22" (2m + 2n)! 1'( + 4)

2 mO m![(m + n)]2 I'(m + 2n + w + )

It is interesting to note that in the limit of zero speed

and a1 -

(or a long, slender ellipsoid), the

three-dimensional coefficients given by equation (31) tend to the foregoing results. To show this we set r = O in (31)

and take the limit as a2/ai and az/ai approach zero. From (6) it follows that

2a9 2a2

a1 = O

a2 -

a3 =

a2+a3

al+a3

and thus the entrained mass coefficients D, tend to the limits

D1= 4

D2=

D6=al+

=D_al+al

2a2 2a9

D4 + a3)2 4a1a3

Therefore in the limit of large a1, and zero speed,

B2

= -

32T pa12a22aslK3e_2Khf Q,2(u)du (51)

W o

where

Qj = 4wcosu q

Q2 = 4w

(a2

+ al'\

_[ii(q)1

sin u

a2)LqJ

= (al

+ a

[ii(q)1

al q

'The first equality follows from reference 1191, section 7.14,

equation (5) after substituting z = ix; the second equality is obtained by expanding I,(E) in an infinite series and integrating term-by-term.

The first equality follows from the Neumann series for the iquare of a Bessel function (cf. Watson [18],section 2.61).

and

= wK + a9)4(a2 - a2)1 [jt(q)l

4a2a3

q:j

Q = 4wKa11 (a2 + a1 rj1(q)1 a

)cosu[_-]

Q, = 4wKa2 (a2 + a3\ Í--9)1 sin u

\

al J q2 J

q = K[(a12 - a32) cos2 u + (a22 - a32) sin2 u]'1'

Let

x = K(a12 - a32)'h2 cos u

so that in the limit of Ka1 -' ,xwillvaryfrom -

to Since the only contribution, to first order in (1/Kai), is from the vicinity of ti.= T/2, we may set

sin u = 1 Z Ka1

dx

Ka1

and, taking advantage of the symmetry, (51) tends to

the limit

B,

= pK2a1al2a32e2

f

[J(x)]dx

(52) w Jo where - _{WZ j1(q)}

QI-

_2Ka1

-q

a2C2= a1CJ1

=

4w(al + a3)

ji(q)

= 4a2a3 (al + a3)(a2

az)2

a3Qs = a2Q, = 4wai(a2 + a3)x

q

and

q = [z2 + K2(a22 - alt)1/'2

The integrals in (52) are all of the form

[j,.([x2 + K2(al2 - a32)]'s)]2_x"'dx

[z2 + K2(a2 - al2)]'

where n = 1 or 2 and m = O or 1. These integrals are special cases of the discontinuous integral of Sonin&° and it follows that

r(m+4)

2[K(a2'

-fo

/2

_{Izn_m+,/,(2K(a32} -COSM+h/'t o do

IO Ibid., section 13.47, equation (7). cos u = du = L(x)

= E

«-o

=

fo

Substitutmg Sonine's first integral" for the foregoing where we have performed the integration with respect to integrand we obtain O by using the integral expression for the product of two

modified Bessel functions.12

1 .12 ./2 Using this reduction for the integrals in equation (52), do j d,,& the strip equations (47-50) follow directly. Thus we

= [K2(a,2 2)

have proved that the strip theory is analytically valid for '2-2,,, (2K(a32 - a22)' cos O sin 4') 4' cos2m4' zero speed in the limit where the ength tends to infinity.

-

1 _{2 '7"(a32 - a22)'/ sin 4') .5jfl2}

2+1

cos2" 4'

2

f.12

-

T[K2(a32 - a22)J1+' 'n-m

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