A contribution to the Theory of Ship Waves

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C.

ARMY FÖR MATEMATIK, ASTRONOMI 0011 FYSIK UTGIVT AV

K. SVENSKA VETENSKAPSAKADEM lEN B.& 17. Nzo 12.

A CONTRIBUTION

TO THE

UHEOHY OF SHIP WAVES

BY

EINAR IIOGNER

WITH 23 FIGURES IN THE TEXT

STOCKHOLM

ALMQVTST & WIKSELLS BOKTRYCKERI-A.-B.

BERLIN LONDON PARIS

R. FRIEDLANDER & SOHN WRELDON & WESLEY, LTD LIBRAJEI C. ELINCESIECE

11 CARLSTRASSE 28 ESSEX STREET, STRAND 11 RUE DR LILLE

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L

4RKIV FOR MATEMATIK, ASTRONOMI OC11 FYSIK.

BAND 17. N:o 12.

A Coiitributioii to the Theory of Ship Waves.

By

EINAR HOGNER.

With 23 Figures in the text.

Communicated March 8th by G. GEA NQVIST and O. W. OSEEN.

The theory of the three-dimensional ship waves has been

Leveloped by Lord KELVIN', EKMAN2, HAVEL0CK3, HoFF4, and BEEI5. These authors have succeeded in giving a

mathe-aatical explanation of the system of ship waves and a

descrip-ion of the outlines of its structure.

But the present theory is as yet rather incomplete.

Cal-ulations of the waves produced by a ship of a given shape

ave nOt yet been successful, and so, for the ship, the authors

ave substituted a pressure of a given distribution on the

rater surface, called a »forcive» (GREEN has calculated the

rave motion produced by a submerged sphere.) Further the ieory is valid only for waves at distances from the ship that re considered great as compared with the wave lengths, but

Lord KELVIN, Math. and Phys. Papers IV, Art. 38 (1905).

Popular Lectures and Addresses III, pp. 450-500.

V. W. EKMAN, Arkiv för matematik, astronomi och fysik, Bd 3, N:o 2, Stockholm 1906.

Arkiv för matematik, astronomi och fysik, Bd 3, N:o 11, bockholm 1907.

T. H. HAVELOCK, Proc. Roy. Soc. Ser. A, Vol.81(1908),pp. 417---430.

L. Hopy, Hydrodynamische Untersuchungen, Diss., München 1909,

. 47-91.

G. GENEN, Phil. Mag. Ser. 6, Vol. 36 (1918), pp. 48-63.

This signification has been introduced by Prof. JAMES ThoMsoN. . Lord KELVIN, Math. and Phys. Papers IV, p. 369.

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2 ARKIV POli MATEMATIx, ASTli0NOMI O. FYSIX. BD 17. x:o 1

even at such distances the theory fails in the proximity o

the two » boundary lines »' or more exactly: » boundary planes At these vertical planes, radiating from the ship and forming a angle of 19 28 with the vertical midwake plane the formulae o the theory give infinite values of the amplitudes of the waves

Different causes are given by the different authors to explaii this fact. The real cause, i. e. an approximation introduced

not valid in the proximity of and at the boundary planes

has been pointed out by HAVELOCK, who in fact shows tha

the wave amplitudes have finite values at the exact boundar

planes. Further the theory does not give any waves outside th

boundary planes, on account of which fact the water surfac according to the theory would be discontinuous along thes

planes.

Thus, it is evident that, even at great distances from th

ship, the most prominent feature of the wave system, th

high waves along the boundary planes, have hitherto evade

the mathematical treatment.

The aim of this paper is to investigate the ship waves

produced by a forcive, travelling with constant velocity ove

the water surface, also in the proximity of and outside th

boundary planes. We will, in conformity with the earlie

authors, restrict our investigation to waves at great distance from the forcive. Only the case of water of infinite dept

will for the present be considered.

It is to me a duty and pleasure here to acknowledge m great obligation to my highly esteemed teacher Professor D:

C. W. OsEEN of the Upsala University. He has pointed out th

it must be possible to solve the problem without any mathematic

complications and he has suggested this investigation. For a

the valuable advice he has given as well as for the great intere

he has taken in my work, I here beg to express my heartfelt thank

Up to that point from which our investigation will star we shall develop the theory of the earlier authors in a fori

given by Prof. OSEEN fl some of his lectures.

A system of orthogonal coordinates x, y, z will be introducei

fixed in relation to the motionless mass of water and with ti

Cf. the authors quoted.

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E. ROGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 3 - and y-axes in the undisturbed water level and the z-axis

vertical, positive upwards.

The water is supposed to have no compressibility, no viscosity,

md no surface tension, and the motion of the water further

will be supposed to be irrotational. Thus a velocity potential

can be found, such that

Oq âg

u= v=---- w=

(1)

Ox Oy Oz

where u, y, w are the velocity components of the water

par-hieles.

The forcive, that has a finite extension, may travel with uniform velocity U in the direction of the positive x-axis, md it may be stationary in a system of coordinates , y, z,

braveffing with the forcive. The whole phenomenon then is

stationary in this system of coordinates and is a function nly of , y, z, where

=x Ut

[f the origins of the two systems coincide for the time t= O. The velocity potential must satisfy the equation of LAPLACE

0292

+ +

ò

0?

0z2

Introducing the usual approximations in the hydrodynamical

equations in the form of EULER and supposing the equation )f the water surface to be

ve have the well known surface condition

+

0 "\OzJ=o

und the surface elevation outside the forcive, where is

sup-)osed to be zero

U(Oçv\

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4 AKIV FÖR MATEMATIR, ASTROENOMI O. FYSIR. BD 17. N0 12

Here are

= the density of the water,

p. = the pressure on the water surface,

g = the acceleration of gravitation.

For the validity of these approximate formulae certaii

conditions, as well known, must be satisfied. Thus the velo

city components u, y, and w of the water particles must b small as compared with the velocity U of the forcive. Fron

this follows that the amplitudes of the surface wavesmust b

small relatively to their lengths. Therefore our theory wil

be valid only when the tangent plane of the deformed wate surface is nearly horizontal.

The water must be at rest at an infinite

depth. Thi

fact may be expressed by

(5

According to F0URIBR's theorem the forcive p,y) sub stituted for the ship may be expressed by thefollowing formula

p, j)

=

dadß jfPo, Yo) eila o)+(Yyo)] ddy0 (6

where Q is the region occupied by the forcive.

At first calculating the velocity potentials correspondin

to each element of the surface pressure and afterwards sum

bing up these velocity potentials, we find that

+

u r r iadßJfp

ae(o)+ß(YYo)]+&

o,Yo) d0dy0 (

42eJJ

-

U2a2__gVa2+ß2

2

satisfies (2), (3), and (5) if the surface pressure is given by (E

(and if derivations under the signs of integral are allowed).'

In reality, derivations under the signs of integral are allowed whe z < 0. This is immediately seen if polar-coordinates in the aß-plane ai introduced in (6) and (7) when avoiding the singular point in (7).

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. ROGNER, A COENTRLBUTION TO THE THEORY OP SHIP WAVES.

It is now convenient at first to consider the special case vhen the forcive is a point of pressure, coinciding with the

)rigin of the , y, z-system. Denoting quantities, attributed

o the point of pressure by a dash and, by F, its integral

Dressure, we find

Polar-coordinates in the aß-plane

a=rcos8, ß=rsin8

nay be introduced. Thus we get

1LT f

2

id8,

dr

47r

j

U2reos$-g

We may conveniently write

-

42QU

(9)

vhere, if we put

=Rcos), y=Rsin)

(10)

x($)

= cos8 + isin8 = Rcos(3' E))

(11)

we have ¡ cl9' 1' 9(1X+z)

(D=I

2 dr

J cos

8

r-_!4 cos28 where

+

-

'472!J

IPU r r aej(aY»V+2

U2a2_gYa2+ß2 (12) (13) dadß. (8)

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6 ARKIV 1'ÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 17. rc:o 12.

The integration with respect to r at first may be carried

out. The singular point

-Ic

r=

cos2 ¿.

may be passed round along small semi-circles in the complex

r-plane. The value of the integral is indeterminate, but we

conventionally fix 2

r d9' (' er(i)

dr +

f d&

r"

9(+z) d (14)

JcosJ

r x

J cosJ

-(I CO52i cos2 9 2

Tiere ' signifies a displacement of the path of integration at the singular point out into the positive and" a displacemeni out into the negative imaginary r-plane. Thus one value o

the indeterminate integral is selected, this value, as seen below, corresponding to the case of motionless water at great distances

in front of the ship. Also the expression (14) evidently satis

fies the differential equation (2), and the boundary conditions

(3) and (5).

Now we change the paths of integration in the following

manner For z > O we follow the positive imaginary axis tc

r = IA (A -, + ). From this point to r = A we follow the

circle quadrant with the radius A, and from r = A to r =

the real axis. in this case the singular point gives a contri

bution to the second integral in (14). For <O we follow

the negative imaginary axis to r = - IA, the circle quadrani from this point to r = A and at last the real axis froni . = A

to r =

. In this case the singular point gives a contributioT

to the first integral in (14). lt is easily seen that the contri

butions from the circle quadrants and from the real axis fron:

= A to r =

vanish for A = x.

Using CAUCHY'S residue calculus and putting

cos29'

= a

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5. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 7

r d

D= i

i

-. de

J c0s'&J s+za

_!

o n; an; 2 2 x(iX+z) d P %(IX+Z) d9

27rtJe C0823'

J

e COS&

cos2J

(15)

Let us estimate an upper limit of the value of the integral

an;

}

d3'

z/=j

i .

J COSJ s+ia

n; J) 2 We have I d&

e8

ds=d1=

J cos28J Ys2+a2

z (J 2 n; 2

r d

f

e8

i ds.

J cos'3'J Vs8+a2

_! 2

At first that part Z19

of the

integral d1 for which

z e > O may be examined e is to be chosen small and

different from zero. We suppose X(19) = O, consequently

cos(0e) ==o,

+ (n +

sin0=(l)cose,

and cos = (_ 1)'z sin C.

In the following we always purpose to confine ourselves

to consider only that half-part of the symmetrical

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8 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 17. :o 1

we thus have 9= and (90 ± j4=Rsinip

Putting

(& + )=RsinJ=e, we find sin=j or approxi

mately ô1=

ô271-j.

Thus we have

f

e8

ifd&

i

(dip

¿/_JJ

Vg2±a2<JZ()xRJSinip

and find for

> - +

i. <

f

, =

- fiogtg

( -

9)

- log tg 1 for i

(dip

2 e

d2<-J

and for i

fdp

il

i/

e

'2 < xRJ

sin ip = zR ¡log tg 9.o) + log tg

-From these expressions it is evìdent that, when x R is

great, J2 with increasing R decreases at least at the rate of

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HOGNER, A CONTRIBUTION TO THE THEORY OP SHIP WAVES. 9

gR

Now we state that quantities, decreasing more rapidly

han , are to be neglected. Consequently ¿1 is to be neglected.

For that part /13 of z1, for which O < <e an upper

imit may be esteemed as follows. For the integral with

espect to s we have

f

e8

ds e_8

-ds<

Vs2+a2

fy

+JdslOo-1+l±c)

S b a o i

Here we intend, for the sake of brevity, to omit examin-ng the case when cos c O at the same time as O <. () < e. his happens when O < e < ô' or 7V ô' < E)< v (ô' is a small ositive angle). Tri reality, these regions are of little interest

o our main subject. Consequently we may here suppose

OS ==O and we now choose e so small that a2 may be

eg1ected compared with 1.

G1 being the constant f ds + log 2, we thus have

f1/2

0ds< 01loga

,rid

13

-

f

d

f

e8

ds<

loo-

I

=

f

(e1

\

COSt

j/2

+a2 bcos9j COS2i

(J

-OXt

,o +

J (

xR sin - 8)\

dS' 2

=

01log

p cos9'

j

cos2 'JO C

=

f

ds=0,2194. See JÁHNIE-EMDE, Funktionentafeln mit

ormein und Kurven. Leipzig und Berlin 1909, p. 21. 2

We SUPPOSe S <R

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10 ARKIV FÖR MATEMAPIK, ASTROROMI 0. FYSIK. D 17. N0 1 Introducing the approximations cos cos an

we obtain

(C - log z log e + i + 2 log cos

Reo

Thus .4

decreases at least at the rate of

when

increases, and also this part of the integral 4 is to be ne

lected.

Thus we have shown that the integral ¿1 and, cons

quently, also 4 by our degree of approximation are to b

neglected when z > O. In the same way we can show tha 4 is to be neglected also in the case of z < O.

Those parts of the velocity potential, of the surfac

elevation, and of the velocity components of the water pa

tides that correspond to the term 4 of the function 0 ar

04 024 04

024

it is true, proportional to

' 3'

and respe

tively, and, in reality, these expressions should have bee:

examined instead of 4. But this examination would hay

brought our investigation out into mathematical consideration

of too great a length and has therefore been omitted. We will, however, also from a physical point of vie prove the validity of the approximation in question.

The integral

r2

r

e(1

í=IdI

j

j s cos .9 + ix (ecos2

+ sin

. ds

is, by substituting

+ r for

, found to be symmetrical i

relation to y = O, when having also

T

¡ ,. ,.

sii

¡ e CO83y81fl&

4=jd5I

. . ds.

j

j seos

+zx(cos3-2,s1n)

O

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ROGNER, A CONTRIBUTION TO THE THEORY OP SHIP WAVES. 11

024 024

'he expressions

ô2

and

ooy

are consequently symmetrical

04

024

nd and anti-symmetrical in relation to = y = O.

In the following we shall find that the wave motion

orresponding to 7i 4 is to be neglected at great distances i front of the forcive. If we suppose that the total wave

iotion at great distances in front of the forcive is to be

eglected, we in this region must omit also the contribution

ue to 4.

With regard to the symmetrical character of z! e wave motion due to 4 consequently must be omitted also t great distances in the rear of the forcive.

These considerations are valid also in the wake of the

)rcive. The assumption that the wave motion vamishes at reat distances in front of the forcive can be accepted in all

ases physically possible, because the forcive then has a finite

itensity.

Omitting 4 we obtain from (15)

2 2 1 xiX()+zJ d19 r x[iX(8+z]

0 = - 2i

e

-

J e COS 2 X(3)>o

Substituting

+ r for

in the second integral we get

z

(p=4

I e0 sin d

J

cos &COSz9

'z 2

Limiting the integral according to the restriction z () <O,

abstituting the value (11) for x() and introducing the new

ariable

(13)

12 ARXIV FÖR MATEMATIK, ASTON0MI o. PYSIK. BD 17. :o I we may conveniently write

E

im

f

e (U2+ 1) +ytt) du.

-Here is to be noticed that the positive value of the squa

root always is to be chosen because

-The integral in (16) and its derivatives with respect

, y, and z are uniformly convergent for z < O. Because

only have to deal with negative values of z, we thus get ti

velocity potential from (9)

s

y

= -

UReI

Yu2 + i e( 1)+iz(VU)1 du (1

-and if we introduce polar-coordinates in the , y-plane

- tg &

Fx

QURe

fvu2

+ le

(L

where

F(u)=(cose + usin E)Vu2 + 1. (1

¿)q ô

Observing that b = we get the velocity componeni

of the water particles from (1). The terms in ii and y whic

we obtain by derivation relatively to and y, occurrin in the upper limit, are found to be

and

ic

J//(2

i

7QUy \y/

(14)

HOGNER. A CONTRIBUTION TO THE THEORY OF 8H11' WAVES. 13

According to (10) these terms are proportional to

and COSe

Rsin2e Rsin3O

For sin e 0, these expressions, when R increases,

de-rease as , and they vanish for sin

e =

O when z <0, even

we take z as small as we please. Consequently these

arms by our degree of approximation are to be neglected.

Further observing that the surface elevation , accordiig D (1) and (4), is given by

U

= -

(u)

re thus get the following expressions for the velocity

compo-.ents of the water particles and for the surface elevation:

etgS

ii (R, e,z)

=

.imf

(u2 + 1) e 1)+F(u) du (20)

ctg e Px2 (R, ,z)

= .

Im

f

u(u2 + 1) du (21) - etg e (R, e,z)

-

eJ

(u2 + 1)/2 e 2+1)+ixEF(u) du (22) - z - ctg e

(R,e)

Imlirn

f

(u2 + 1) du. (23)

For an approximate evaluation of the ixitegrals, occurring i (18) and (20)(23), most of the earlier authors, availing

aemselves of the assumption that R is great as compared

ith

=

-g'-- (and apparently also as compared with z1)i, use

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14 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 1. x:o 1

the »principle of stationary phases». Developing F(u) fror

those points u» that satisfy the relation

i

F'(u=

a)

[u»cos1 + (+ 1) sin ] =0

(2

that gives

(2

they write, putting u = u» + y

F(u) = F(u») + jF"(ua) (2

with which approximation the integrations have beeii carrie out in a manner well known.

Only when u1 and u2 are real, i. e. for tg I

stationary phases are found in the integrands, and only i'

those regions, where this happens, a considerable wave motio:

is possible. These regions CO C' and A OA' are bounded b

the vertical » boundary planes »

tge=±

v-where k is an integer (fig. 1).

or e=±19°28'+k»v

Fig. 1.

I

At first considering points in the water mass, situate

inside' the boundary planes, we find, with RAvELoCK, tha

We by "inside the boundary plane» denote that a point is situate

between the boundary plane and the z-p1ane, and by "nutside the boundar

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ROGNER, A CONTRTEUTJOI TO THE THEORY OF SHIP WAVES. lo

b is quite the approximation (26) that is the cause of the in-mite values of the wave amplitudes at the boundary planes hat the earlier theory gives. In fact

1

F"(a) -

+ [cos El + ua(3 + 2) in f9] (27)

anishes at the boundary planes, where u and u2 coincide in

ne point u:

j

ctgl\

i

¿o

Jtge=---

vi

±1/ri

nd I F"(a) is consequently small in the proximity of these

)lanes.

Even if R is

great, the conditions for the

appli-ability of the principle of stationary phases, consequently, ure not satisfied in the inside proximity of and at the

bonn-[ary planes if the approximation (26) has been introduced. Using the approximation

F(u) F(ua) + F" (ua) (29)

vhich is valid only at the exact boundary planes, HAVELOCK

und GREEN calculate the wave sections along these very planes.

o more, however, is this approximation applicable to

calcu-ation of the wave motion in the inside proximity of the

boun-[ary planes. For this purpose we must use the approximation

F(u) = F(Ua) + F" (aa) + F" (us). (30)

Because

F" (Ua) 3 (sin (9 U, COS 1)

(u, ± 1)6/2

s different from zero in the neighbourhood of and at the oundary planes, we, with this approximation, have fulfilled he conditions for the applicability of the principle of statio-tary phases as to every value of f9.

(28)

(17)

For the idea of developing F(u) from the point u8 I m indepted t

Prof. OSEEN.

16 ARKIV FÖR MATEMATIK, ASTRONOMI O. rsIK. n 17. ic:o F

The earlier theory gives no surface elevation outside ti]

boundary planes. In reality, it may seem as if the princip] of stationary phases would give this result, because F'(u) i this case does not vanish for any real value of u (see formu]

(25)). However, as F'(u) vanishes at the boundary planes fc

u = F'(u) must be small in the outside proximity these planes for all real values of u near u0. Though 1?

great, these values of u give contributions to the integrals ths

are to be considered for points R, (9,z in the outside proximity C

the boundary planes, and the wave motion thus is extende

also outside these planes.

In order to get a simple approximation for F(u) in thj

case we (with regard to the relation F" (u0) = 0) for points i the outside proximity of the boundary planes, may define real value u3 in the neigbourhood of u0 by the relation

F"(u3) = O or

u +

+ ctg(9=0

(3

that only has one real solution, and use the approximation F(u) = F(u5) ± vF'(u3) + -F"(u3). (3

For points R, (9,z at a certain distance outside the boundar

planes, for which F' (u3)I has a value not very small, it i

evident that the integrals in (18) and (20)(23) and thus als

the wave motion vanish.

The application, however, of the principle of stationar phases, when we have to deal with the approximate expre

sion (30) for F(u), whose derivative vanishes for two res

values of ua -- y of which only one coincides with one roc

of F'(u) = 0, would present some complications. We shal

therefore, in our investigation use complex integration for th

evaluation of the integrals in question, which method als gives a clearer idea of the degree of approximation that i

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S. HOGNER, A CONTRIBUTION TO TEE THEORY O SHIP WAVES. 17

For this purpose we write

U U,. + Uj

1'(u) = F,. (Ur, u) -f i F (u,.. ui).

Then the exponent, occurring in the integrands in (18) nd (2O)(3), becomes

z(u2 + 1) + ixRF(u) = z {z(u,. u2 + 1)

- RF(u,., u» +

± j {2Zri + RFr(r, u))

(34)

We are intent upon displacing the path of integration

)ut into the complex u-plane in such a way that the real part )f the exponent, if possible, may be negative and numerically

reat, when the distance R from the point forcive is great.'

&t such points of the path where this happens, the integrands )ecomes small, and the integrals along such parts of the path,

uith regard to the uniform convergence of the integrals in 18) and (20)(23), may be neglected, if R is taken great

nough.

The principal term - xRF(u,, u) in the real part of the

xponent becomes zero at the curve

F (u,., u) = Im (cos E) + u sin E)) Vu2 + i O. (35)

For real values of u, the positive sign of the square root ias been found to be essential. By making two cuts in the

omplex u-plane along the imaginary axis from the two points

f ramification u = ± i outwards we make the analytic fune-ion Vu2 + 1 uniform, as easily seen in such a way that its eal part never changes its sign, if u never transgresses these

uts. If we put

Vu2 + i = X (u,., u) + i Y(u,., u) (36)

e find, taking X and Y real and X> O:

Here and below: great as compared with

X 9

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18 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 17. tc:o 12

y ±

± 1) +

(t4-u2 + 1)2+ uv

The positive sign for Y is

to be chosen, when 'u, ii

situated in the first or in the third quadrant, but the nega tive sign, when u is situated in the second or in the fourtl

quadrant.

Introducing (36) in (35), we find the curve

Fi(r, uj) = Ycos E) + (ujX + u,- Y)sin E) = 0 (38 which (cf. formula (37)) consists of the two branches

and

+ 1)2 + uu = u + u,-ctg E) +

+ (u-u + 1).

(39

The curve (39) consists of two branches, one or both o

which end at the points u = ± i and have the asymptote

ctg E)

2

Singular points of the curve (38) are obtained from th

relation

F''ua) = O

which (cf. formule (24) and (25)) gives the two double point u1 and u2, which at the boundary planes coincide in u0 (sed

above formula (28)). When the point R, E), z is situated outsid

the boundary planes, in which case u1 and u2 have imaginar

values, we introduce the point u3 (see formula (32)) that at th

boundary planes also coincides with u.

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I. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 19 In the figures 2-7 the curve Fi(r, u) = O is shown by

lotted lines. As we have restricted ourselves only to consider

that half-part of the symmetrical phenomenon, for which

;in (9> 0, it is easily found that the term -i.REi(u, u) Of

the real part of the exponent (formula (34)) is negative in the egions marked with lines in the figures.

Fig. 2. Fig. 5.

Fig. 3. Fig. 6.

Fig. 4. Fig. 7.

The path of integration may be placed out into the

omplex u-plane, as shown by the curves fully drawn in the

gures 2-7, conveniently in such a manner, that the varia-on of F alvaria-ong the path, at least in the neighbourhood of Le points u1, u, u, and of the upper limit u4 = - ctg (9 of

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20 ARRIV FÖ MATEMATIR, ASTRONOMI O. FYSIK. BD 17. :o 1

the integrals, may be as great as possible.' On account of known quality of the analytic functions, this happens alon the curves Fr = constant. It is evident that, when z < 0, th

path can be laid so that xz(u u + 1) and thus also th

whole real part of the exponent will be negative at ever

point of the path situated within the regions marked with line

From what has been said above it is clear that, if th distance R from the forcive is great enough, the integral

vanish along all parts of the path, except the small arcs , and in the neighbourhood of the points UL, u, u2 and u4, at which arcs F is small.

At first the contributions to the integrals in (18) an

(20)(23) from the arc

may be examined. We may iii tegrate along the perpendicular to the ur-axis and we writ

for this purpose:

u = u4 + iv

and, observing that F' (u4) = F' (- ctg ) = 1, approximatel

F(u) = F(u4) + iv.

Because those parts of the path of integration where is not small do not give considerable contributions to th integrals when R is great, we may extend the integratioi from y = ± to y = 0. Supposing R great also as compare with Iz we may e. g. consider the factor

1/u2 + i e('+

in (18) constant at the arc

The contribution from

to thus becomes

o.

PxVu4 + i Reie(+1)+1(u3)fe_x11'dv=

2VQU

PVu + i

sin xRF(u4)

This contribution is by our degree of approximation t be neglected with regard to the factor R in the denominato]

Cf. L. BRILLOUIN, Annales de l'École Normale Supérieure, 3:e Séri Tome 33, 1016, p. 21.

(22)

î. HOOENER, A CONTRJEUTION TO THE THEORY OF SHIF WAVES. 21

E'or the same reason the contributions from the upper limit o the expressions (20)(23) also are to be neglected.

The figures 2-4 corresponding to the state of things in

he region D OB (fig. 1) thus show that in this region (and course also in D' OB') no wave motion that is to be

con-;idered at great distances from the forcive, takes place.' uch a wave motion is only found in the region COD (and

rj'OD), obtained by the contributions to the integrals from

he arcs y and 2'2 in fig. 5, and further in the region BO C

and B' O C') close outside the boundary plane O C (and O C'),

his wave motion being obtained by the contributions to the

utegrals from the arc in fig. 7. Only near the boundary

)lane, where the passage between the branches of the curve = O is narrow and I F1 I thus is small at , this contribu-ion is to be considered, and it vanishes of course when re-loving from the boundary plane.

Considering Y2 + i constant at the arcs y, 72'

Lnd developing F(u) in TAYLoR's series from the points

t,, u, and u

and using the approximations (30) and (33)

ve, from formula (18), get the following expressions for : in the region COD

(R, @,Z)=_7V1'TRe

Vu+ 1.

CZ(u+1) +iZRF(Ua) f (ua)± '"''a)]dv (40)

and in the region BO C

(R, + 1.

Ree (

+I)+j(v,) f e'

ti""

+1F8)]

dv. (41)

This effect has been reached by the choice of the path of

(23)

22 ARKIV FÖR MATEMATIK, ASTR0N0H O. FYSK. BD 17. :o

These expressions are also valid at the boundary plan

OC and in

its neighbourhood, where F"(ua) and F'(u) ar

zero or numerically small, because the terms

F" in the e

ponents also are considered.

Putting

V = V. + îV

we find that the real part of the exponent in the integran

in (40) becomes zero at the real axis

vi = O

and at the hyperbola

F .F"(da) ]

,

,.'

(ua) ;,

,=1.

[F (Ua) F (fla) -i ,,, i r ..) ,,, 1F (na) i F (Ua)

Now it is easily found from (25), (27) and (31) tha

we. in the region COD, have F"(1) < 0, F"(u2) > O au

F" (aa) > 0. The complex v-plane then has the appearanc

shown in the figures 8 (for a = 1) and 9 (for a = 2), wher those regions in

which the real part of the exponent

i

negative, are marked with lines.

Fig. 8.

(4

At the boundary plane the hyperbola (42) degenerate

into the straight lines (see fig. 10).

(24)

HOGNER, A CONTRIBUTION TO THE THEORY OP SHIP WAVES. 23

The real part of the exponent in the integrand in (41)

becomes zero at the real axis

Vj = O

and at the hyperbola

The derivatives F'(u3) and F"(u3) always are positive

in the region B O C. The real part of the exponent is ne:

gative in the regions marked with lines in fig. 11.

21'(u3) 6F'(u3) -

- 1.

F (ui)

F (u3)

vi

Fig. 11.

(44)

At the boundary plane also this hyperbola degenerates into the straight lines (43).

If R is great enough, we, without considerable change in the values of the integrals in (40) and (41), may extend

the integrations between arbitrary limits v, y11, and vin, si-tuated in those parts of the complex v-plane where the real

parts of the exponents in the integrands are negative. We

can, if agreeable, choose vz= +

, v31=x,

and vm=icc.

The condition that R must be so great that the values

of the integrals in (40) and (41) along the whole paths, ex-cept the arcs and , can be neglected, compared with

the condition that e

i when y j

situated at

one of these arcs, gives a relation from which the order of

magnitude of R that is necessary to obtain the degree of

(25)

24 ARRIV FÖR MATEMATIK, ASTRONOMI 0. FYSIK. BD 1. N:o 12

The above considerations are valid also near and at thE

midwake plane though all the derivatives F(u0) vanish al

this plane. It is easily found that we in the neighbourhood ol

i

F,

the midwake plane F (u3)e 2sin&

48

and that every derivative F'(u2) con tains at least one factor sin more than F(u0). If we mak the dimensions of the arc 3 proportional to wher

<p < 1, our approximations thus are valid also for sin e *0 In the integrals in (40) and (41) the new variable

w = y

6

and the indications

62/3 (x R)1'F" (ea)

=

-2 [F"(ua)j21 6 (z R)2'F" (u3)

[E" (u3)]'/

may be introduced. Then we get in the region COD

(R, e,z)=

(b 1, 2, 3) (4ö) j20 6h/8x8 f V'u + i e(u 1)+iF(u1)fe133+w3)dw + t$ '

1F (ui)

-+3

Vu. +1 e(1L2+1)+1

)fet(+) dw}

(46)

1/ F"(u2)

(26)

HOGNER, A CONTRIBUTION TO THE THEORY OF SHIF WAVES. 25

ï(B, 0,4=

+ 6h/8x2/3 ± i

Re9(u

1) 3)JeiC+tJ3)dw. (47) UR" YF"(u3)

Introducing the indications

Ii(a)=Iir(G) + iI1()

L(a) i2r() + iI2(a) =

fe1+udu

dw

here 'ir, Iii, '2r, 121, and K are real, substituting the value 13) for x, and treating the expressions (20)(23) in the same

aanner as we get the following expressions for the veloc-v potential produced by a traveloc-velling point of pressure:

in the region COD

5(R, (9,4= 6h/q2I8P

±

V+ 1

e {Iar(ta)COS

F(a)

-7VQ U2uIRhI8a i

- Iai(a) Sin F(ua) } (49)

and in the region BO C

(R, 0,4=

61/8 2/, +

K(r) cos F(u3).

-

U'R'

VF"(u3)

ad similar expressions for u, , w, and .

(48)

(27)

26 ARKIV TÖR 3IATEMATIR, ASTRONOMI O. FYSIK. BD 17. :o 1

In order to calculate the water motion and the surfac

elevation produced by a forcive of finite extension we sta] from the fortnulae (18) and (20)(23). Introducing the orth gonal coordinates , y again, we for have the expressio (17).

Substituting -

for ,

y - y for y, dq for

, an Po' y0) d0 dy0 for F and integrating over the region Q, o

cupied by the forcive, we find the total velocity potential.

YYo

(, y, z) = X

-'Jfp

yO) dod Yo f Vu8 + 1.

0(u i) +ix[() + (yy0)ul 1/u!+ du. (5: From the treatment of the integral in (17) we find th

also to the value of the integral with respect to u in (5

only those parts of the path of integration in the u-plan which are situated in the neighbourhood of certain pomi

u1, u2, and u3, give considerable contributions. If and

as here assumed, are great as compared with and Yo respec ively (i. e. the coordinates for a point within the region c the forcive), these points u1, u, and u3 are identical wit

those introduced above (formulae (25) and (32)).

Introducing the indication

G(u) = G,.(u) + i G(u) =

=

Yo) Ci0+Yo

ddy01

(5

where P is the integral pressure of the forcive

P = f f p, Yo) ddy0

1)

and observing that V2 f i e_

o+yoi varies slowly

compared with e1z(yu) and again introducing polar e

ordinates for , y, we may write approximately Compare EKMAN'S function F(t, y), 1. e. 1907, p. 5.

(28)

EL HOGNER, A CONTRIBUTION TO THE THEORY OP SHIP WAVES. 27

in the region COD

'

+1) + ixRF(ua)

J

e[F

+F"(ua)J

dv (53)

and in the region B O C

1(R, e, z)

=

_7/;xy_+

i Re G(u).

e"" +1) +i,GRF(u,) J e'' [rFua)+ F"(u)]dv. (54)

Treating these formulae in a similar manner as we have

5reated (40) and (41) above, we obtain

in the regiou COD

6'1'gP

'ç Vu + i

p(R,1,z)=

U2t/8R 8

a=i VF"(ta) U+ 1){[

('ia)lar () - G (us) Ij ()] cos F(u(,)

.qR

I

- [Gr(a)Iai(fJq) + Gj(ua)Iar(aa)J sin F(i0) (55)

and in the region BO C

ßh13q213P

Vu ± i

p(R, e, z) =

-7VQ L2t8RI8jfFl

K(U){Gr(s)cos

E()

(29)

28 ARKIV FÖR MATE1ATIK, A5TRO1OMI o. FY5IK. BD 17. i:o 12

From the formulae (20)(23) we, in the same way, ge

similar expressions for u, y, w, and . Thus we may write

in the region COD (fig. 1)

çv(R, E), z) =

=

69 P

VUa + 1

e'' Ma

{f F(ta) +

aa} (57 7rQU2l/2Rl/8a=i

VF"(a)

u(R, E),z)= fi1/agl218F

± U

+ i

e' Ma sin

F(Ua) +aa} (58

=

U4'R",1

VF"(u0)

v(R, E), z) =

'Ua(u + 1) (u+i)

Ma sin {y2 F(ua) + a} (59

ve U R

a=117F"(a) w(R, E), z) =

2

I9R

ßhI3yl'I8 (u + 1)8/2

eTh .21/ta COS F(a) + a} (60

n U42IR1I,=1 V.F"(a)

(R, e)=

6/8g23P

u + i

Ma jn { F(Ua) +a«} - 'vQ U811'R'1' 8 a=1 /

TF iva)

'"f and in the region BO C

6"g"P

1'tu+ i

8(u+l)

. icos

F(u3) +ß} (6 7vQ U22I8R8/8 VF" (u3) u(R, E), z) =

6"y"P

+ i

e''

.

.vsin{.F(us)

+ßj 71Q U41R11 1/F"(u3) (61

(30)

JIOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 29 (R, O,z)=

6h/sgih/8P u3( + 1) t+i)

Nsin fqR a 1U2F(u3)±ß} U4h/1 R111

/ E"

(H3) 6'1g121P

(u + 1I

NCS

I3)

+ ßj (65)

=

ar U411R11

6'g2P

: + i N sin F(u3) +ß}. 7vQ U'I8Rh/3

The pressure in the water mass, if the pressure on the

vater surface outside the forcive is zero, is with our degree

)f approximation given by

p(R, O,z)=(U.ugz).

(67) The new indications that in these formulae have been

ntroduced have the following signification:

Ifa = [Gr(Ua)Iar(aa)

-- Gi(a)Iaa)J

V +

= K(i-).

Gr(s)V1 +

[Gr(ua)Iai(aa) + Gj(ua)Iar(a) a Gr(ua)Iar(aa) Gj(a)Ic,j(Oa) [G(u3) 2 [ Gr(u) Gr(ua)Iai(aa} + Gi(a)Iar(iia) = arc tg

Gr (fla) 'ar (aa) - G (Ha) 'ai (aa)

G1 (Ha)

= arc tg

Gr(3)

The functions E(ub), G(Hb), Ia(aa), and K(v) are given by

he formulae (19), (25), (32), (52), (45), (71)(73), (77)(79),

nd (80).

(64)

(66)

(31)

30 ARKIV FÖR IATEMATIK, A5TROuOMI O. FYSIK. BD 17. :o 12

In the formulae (57)(66) the positive signs of the squar roots always are to be chosen. The square roots in Ma am Y are to be chosen positive if

.

{}

<.

The formulae (57)(6fl, also valid in the proximity o

and at the boundary planes, thus give the velocity potential

the velocity components of the water particles, the surfac elevation and the pressure in the water inside and outsid

the boundary planes, produced by an arbitrary forcive of finit

extension and thus they give the solution of our problem provided that the integrals Ia(s) and K('t) are known. Oi

account of the restrictions, introduced above, these formulad are valid only for distances R from the forcive that are grea

compared as well with the wave lengths along any plan

J U2

E) = coust. these wave lengths being proportional to

as with the extension of the forcive and with the depth

Further they are valid only for

+/

<v (

being i

small positive angle), except the region swept over by th4

forcive. The waves in the region v E)

-f and in th

wake of the forcive can be calculated after a transformatioj of coordinates and a convenient division of the forcive. Th function, 0(u). depending on the distribution of pressure withii

the forcive, can be calculated for a given forcive according t any exact or approximate method. It is not necessary tha

the forcive is symmetrical in relation to any direction.

Izi

The treatment of the integrals 11(a), I (a), and K(v) no

remains to be performed.

Developing the factor ei# in series we find for I (a

and I(o)

i

will WI!!

.11(a) =

fet()

dw = f et(W2+ w) dic

fwe

dw (69

(32)

. HOGNETI, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 31

=

fev2+W)dw

=

J

e ¿2+V)dw =

1J

iv2eiW'dw. (70)

The w-plane has with regard to the character of the real

art of the exponent a similar appearance as the v-plane hgures 8-10) and it is thus evident that we in the integrals

bove can use the arbitrary limits w, wn, and wm

cor-esponding to Vi, vn, and Vm.

Performing the integrals appearing in the terms of the

enes in (89) and (70) along the paths shown in

fig. 12 Iwil, Iwrr I, and

Iwiiil=

)

Fig. 12.

nd introducing the F-function

F(s)

=

o

f t

e dt re find

() =

(S + 54 - (S3 S4) +

7O viii Will .F1 + (S1-S) + -d-- (S + 84)] (72) + (71)

(33)

AEKIF FÖR 31ATE1ATIK, ASTROENOMI O. FYSIK. BD 17. :o 1 where

s -

''

l)'F(4n + (6nT t=0

(-1)'F(4n

+ 1I).

(6n+2)! + S3= (6n+3)! (-1)'r'(4n+ (6n--5)! n=O

These series are convergent for all values of a. Sem

convergent developments, convenient for the calculation whe i al is great, are easily found for 11(a) if a < O and for I2(

if a> O.

This is quite what is needed because a1 always i

negative and a2 positive (formula (45)). We write

W111 8111

I( loi)

=

f

e1(_IdIv28)dw

=

._1f

eH8'+I)ds = WI1

ii

8111 -:

J

sB e2ds (74 n. $11 WI SI

L(iai)= f eu1I1t5'±tt$)dw

=

_1fe8+

ds =

W111 8111

Sj

-=..1ÇÇfs3e18'ds

(7

where the new variable

$ = 1"j1?V

(34)

c. HOGNER, A CONTRIBUTION TO THE THEOBY OF SHIP WAVES. 33

md the indication

the square roots always being positive) have been introduced.

Performing the integrations in every term of the series n (74) and (75) along the paths shown in the figures (13) and

14) (Isil, IsnI, Is'ml, and Isiul =

vhere

Fig. 18.

Re(-is')O

md observing that the integrals along the arcs between

Lfld 5m vanish, we obtain:

_1

{T1 - T2 - i(T, + T2))

I1(_IciI)_Y9II

_1

{T1T2 + i(T1± T2)) (78) 17(1 'I - V2111 (76) (77) T1 = ( flF(6fl + ) / i 12,... (4n)! (T (79)

(1)flF(6fl+3)(1fi3

(4n+2)!

kIi

The integral Ii(a) has been calculated by means of the

onvergent development ((71) and (73)) from ci = 3.2 to

= + 3.2. The values of the real and of the imaginary parts

f

the integral are given in table I,

and are graphically

)resented in fig. 15.

(35)

34 ARKIV ÖR MATBMAPIK, ASTRONOMI o. FYSIK. BD 17. :o Table I. Fig. 15. .1. -:...

I

--_-'

a Iir(6) Iii(G) a Ii(i)

9 +0.6781. -0.7153 .2 + 0.779 34 -1.329 98. -3.0 + 0.6956 -0.7412 .4 + 0.80035 -1.29957. .8 +0.7132. -0.769 4 .6 + 0.840 20. -1.24481 .6 + 0.7313. -0.800 .8 + 0.90096 -1.16102. .4 + 0.7493. -0.8857. + 1.0 + 0.981 68 -1.04146 .2 + 0.766 8. -0.8743 .2 + 1.076 41 -0.877 08. -2.o + 0.78806 -0.91695 .4 + 1.171 53. -0.656 89. .8 + 0.797 87. -0.96350 .6 + 1.24240 -0.37148 .6 + 0.808 76. -1.013 65. .8 +1.25050 -0.01935. .4 +0.81681. -1.09459 +2.0 + 1.144 15. +0.38067 .2 + 0.819 26. -1.l2101. .2 +0.8692. +0.7726. -1.0 + 0.81720 -1.174 96. .4 + 0.898 2. + 1.0494 .8 + 0.810 89 -1.22595 .6 -0.2179. + 1.0694. .6 + 0.799 94. -1.271 06. .8 -0.7937 + 0.686 1. .4 + 0.787 90 -1.807 10. +3.0 -1.0156 -0.0418. 9 +0.777 77. -1.830 90. .2 -0.612 2. -0.7729. 0.o + 0.773 54 -133947.

(36)

. HOØNE}1, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 35

The corresponding values of I0(a will be obtained from bhe following relations:

I2r(Y = Iir(i);

12(o = Ii(a).

For I I= 3.2 up to the 65th power of a has been

con-idered in the series (73). For this value the 4th decimal is ikely to be correct.

For I a > 3.2 the semi-convergent developments (77)(79)

nay conveniently be used, and it is then only necessary to

onsider a few terms. The coefficients in the series (79) have the following values:

Table II.

All the above numerical calculations have been performed

vith a calculating machine Trinks Brunsviga».

The integrals 11(a) and 12(c) satisfy the differential equation

d21 /1 .4u2\dI .2a -

2-1 + &

da \a

9jdc

9

vhich is also satisfied by the BESSEL functions of the order

11

.

The integrals in question thus can be expressed by

neans of these functions. After determining the arbitrary

cosi-;tants of the solution of the differential equation, we find

JAllÑKF-EMDE, I. C, J). 167. P(6n + l) r(6n + 3l) (4n)! (.4n + 2)! o 0.9873

i

6.767578 0.347 717 101V7 2 1.914 822. io Jf7 5.205 921. . 10 V 3 1.765133.

ioV

7.1727 . ioV7 4 3.897 377 . io°Jí7 1.888 128. 10hiV 1.118388. . 7.558615. . iO"Vr 6 5.618 177. . 4.554 775. io'V 7 3.999 86 3.782 37 . lOY 8 3.881 87. iO2411 4.14058 .

(37)

36 ARRIV 1'ÖE MATEMATI, ASTRONOMI O. FYSIK. BD 17. :o 12.

-

11(a) = f ei(t)dw =

12 a3 '

=ae{(l

+iV3)J(Ç)

+

(l_iV)J_)1

12(a) = f dw =

=

+ (1 +

From these formulae the BESSEL functions t1(s) and J(s

can easily be expressed by I (3

1")

or 12(3

i7).

The real and the imaginary parts of I (a) and 12(a) satisf

the differential equation

dI

4d81 116a4

8\d21

180a3

8\dI

28a2

+

81

+ Jd2 +

81 da 81

1=0.

We now turn to the integral K() and write

K(v) = f e>dw =

2j/

f cos (wmw) dw

-where Ji' w has been substituted for w and

12\2/

m=(--)

v.

This integral has been treated by AIRY and DE MORGAN.

The latter has given a convergent series for the integral which

with our indications, is read:

(38)

. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 37

K(r)=2fcos(w

+ w8)dw 2

/1\

rsf 1 4 1

i

741

+3 3 3

239+

j 2

/2\

n 2 5 2

_F)cos+234+3

3 237k

(80)

By means of this series AIRY has calculated the integral

Ki(rn)=fcos(w8_mw)dw

from m=-5.6 to m= +5.6.

We here reproduce AIRY's table1:

Table III.

AIRY has calculated the values from m = - 4.0 to m = + 4.0 also by L method of mechanical quadrature. The values thus obtained, differing only ittle from those obtained by calculation by the series are not here reproduced.

in K1(m) n K1(m) in K1(m) m K1(m) .6 + 0.000 u .6 + 0.08277 .4 + 0.84040 .4 -0.78021 .4 +0.000is .4 +0.04442 .6 +0.91481 .6 -0.76516 .2 + 0.00028 .2 + 0.05959 .8 + 0.97012 .8 -0.66044 -6.0 + 0.000 41. -2.0 + 0.07908 + 1.0 + 1.00041 + 4.0 -0;474 19 .8 + 0.000 68 .8 + Ojos 77 .2 + 0.997 86 .2 -0.226 45 .6 + 0.00093 .6 + 0.13462 .4 + 0.95607 .4 + 0.05193 .4 + 0.00188 .4 + 0.17254 .6 + 0.87048 .6 + 0.32258 .2 + 0.00204 .2 + 0.21839 .8 + 0.73930 .8 + 0.544 75 -4.0 + 0.00297 -1.0 + 0.27283 + 2.0 + 0.564 90 + 5.0 + 0.68182 .8 + 0.00429 .8 + 0.88622 .2 + 0.85866 .2 + 0.70818 .6 + 0.00621 .6 + 0.40889 .4 + 0.11722 .4 + 0.61515 .4 + 0.008 78 .4 + 0.48856 .6 -0.12815 .6 + 0.41460 .2 + 0.01239 .2 + 0.575 07 .8 -0.862 37 -3.0 + 0.01730 0.o + 0.66527 + 3.0 -0.563 23 .8 + 0.028 98 .2 + 0.755 37 .2 -0.708 76

(39)

38 ABKIV FÖB MATEMAPIK, ASPR0N01I O. FY5IK. BI) 17. ic:o 12,

These values are 'graphically presented in fig. 16.

For great values of the variable STOKES has given a semi

convergent series for the integral in question', but with regard

to the fact that the wave motion in the region BO C is to

be neglected, except near the boundary plane, the series (80) is sufficient for our problem.

4k1..

im

Fig. 16.

K(r) satisfies the differential equation

d2K

i

2

dv2

3K-0.

This equation is also satisfied by the BESSEL functions of

the order ±

.'

After determining the arbitrary constants of

its solution we find:

for i < K(v) ei(W+1v')dw = 271:

y-vJJl(

2 8) +

,_J 2

31 1 3]/3 fi

1 C. G. STOKES, Math, and Phys. Papers, Vol. H, p. 320.

2 CI. STOKES, 1. e., p. 334, formnla (11).

(40)

5. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 39 and for r>O

K(r) = f eW+)dw =

2v

j-Vä + i

1

'21 .r\

V3 + 1 / 21 , 'r jz VT3I +

--J_I---I

3/3

2

\3YÏ

f 2

\3/

,j

where'

Y3+t

3si i1L

J(as)=

2 2iF(*)ì

2(+2)

J

V3_i

3s

J_4(iS) = -

- 2

Now we are able to interpret our final formulae

57)(67). These formulae show that the wave motion with

ncreasing depth below the water surface decreases as

Dur assumption that z is small as compared with R,

there-ore does not considerably restrict the validity of our formulae.

We will now give a description of the shape of the ship waves, given by the formulae (61) and (66) for the surface

levation .

The resultant wave system inside the boundary planes

3an be considered as constituted by the superposition of two flifferent wave systems the >'transverse» one, corresponding

bo the term a = 1, and the »diverging» one, corresponding

bo the term a

2. Outside the boundary planes, however,

the resultant wave system is simple. The mutual

situa-bion and the direction at the boundary planes of the crests

)f the different wave systems are from the expressions

or a, a.,, and ß (formulae (68)) easily found in general to

e dependent on the aceleration of gravitation, on the velocity )f the forcive, on the distribution of pressure within the forcive,

md the direction also on the distance from the forcive. The JAHNKE-ERDE, 1. C., p. 92.

2 Cf. the earlier authors.

s2

(41)

40 AK1V FÖR MATEMATLK, ASTROENOMI O. FYSIR. B» 17. N:o 12

resultant wave system inside the boundary planes without

discontinuity joins the system outside the boundary planes.

In order to

give a concrete example we will considex the waves produced by a forcive, symmetrical in relation to

= y = 0. This is the case treated by ERMAN (1. o. 1907)

and it

is according to formula (52) characterized by G = 0,

which according to (68) gives

Ma = Gr(a) . lar (aa) i +

N = Gr(8) . K(v) l (aa) Ci = arc tg Iar(aa)

ß =0.

(81)

The formulae for the surface elevation then become:

in the region GOD

(R O) 6'1g'1P U + i . Ma

in1yF

fgR (Ua) J cea}

-r U'R'

a=i

and in the region BOG

(R O)

6'8g'P

± Go .K(i) sin2 F(u5). (83)

e U31I.R"VF" (u3)

These formulae are of course valid also in the proximity

of and at the boundary plane, in the special case here

con-sidered.

At a certain distance inside the boundary plane, F" (Ua) being different from

zero and R being great,

o

and a

according to (45) take great values. In this case we have approximately according to (77)(79) and (45)

(82)

(ii)F()

= (1 V21a11 1Ji VF"(u1) i)

6 g R V - F" ()'

(F" (ui) <0) (84)

(42)

E. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 41

t t

(1 + i) r' (1) (us)

V2lI

(F" (u) > 0). (85)

With these approximations the expression (82) takes the following form, not valid in the neighbourhood of and at the

oundary plane:

(R, E) 2igiP J(u, + 1) G(u1)

. F(u1)

-+ sin

URt VF"(u1)

(u + 1)G7(u2) sin F(u2) +

j

+

i/"

U2

According to (71)(73) and (80) we have

Ii(0)

('?) r(k)

'2(0)

=

( + r(Ä)

The surface elevation at the exact boundary plane,

calcu-[ated from (82), thus becomes

and calculated from (83)

6iF(1)gP (u + 1)G7(u0) gR sin F(u9). (88) (R, e) 3i U3Ri sin F(u0)

-+ (R, 9) 6iF(*)gP (u + 1)G,.(u0)

3n U8Ri

VF"(u0) (86)

+ F0

+ (87)

(43)

42 ARKIV FÖJI MATEMATIK, ASTRONOMI O. FYSIR. BD 17. :o 12.

In reality, the formula (86), being the first

approxima-tion of our formula (82) agrees with ERMAIc's final

for-mula1, and in case of the forcive being a pressure point,.

characterized by G = 1, it also agrees with the formulae of

HAVELOCK somewhat corrected. Comparing formula (86) with1

H0PF's final formulae (33) 1. e., p. 72 and (47) 1. c., p. 84, we find, if some errors in these formulae are corrected, that

our formula (86) does not agree with Hopr's formula (33),

but, in fact, applied to a circular forcive of uniform distribu-tion of pressure, the intensity of which is propordistribu-tional to the

square of the velocity of the forcive, it becomes identical

with Hopr's formula (47) Lord KELVIN'S formulae, referring

to a special forcive of infinite extent (to which case our

calculations are not applicable) differ (though only concerning the wave amplitudes) from our formula (86).

The formulae (87) and (88) valid at the exact boundary plane, evidently are identical with each other, although the waves inside this plane according to (87) are composed by superposition of the 'transverse and the diverging waves that

EisíAx, 1. e. 1907, p. 14 formula (19). Observe also on the same

page the explanation of the increasing wave resistance (and wave ampli-tiides) with increasing velocity in case of the extension of the forcive not

being small as compared with the wave lengths.

2 HAVELOCK, 1. e.. pp. 419-421. HAVELOCK here has overlooked the phase difference at the boundary planes between the transverse and the di-verging waves. The same error is found in GREENS final formulae (52)

and (53), i. e., p. 60. The pattern given in a figure by Lord KELVIN shows no phase difference, which nevertheless appears in his formulae. This pattern

thus does not show the isophasal curves but the envelope curves of the crests of the two-dimensional waves from which he has constructed his

three-dimensional waves.

Hor's formulae are not quite correct on account of some errors on p. 71. HOI'Fs calculations are, founded on some assumptions

intro-duced on pp. 52-53 and 83 concerning the "ship", the physical

pur-port of which he does not explain. In reality they purport, in the first

case, resulting in the formula (33), that the number of annular waves

emitted per length unit of the path covered by the "ship" is independent

of its velocity, and in the second case, resulting in the formula (47), that

the intensity of the forcive, which, when the initial surface elevation is zero,

always must be the cause of the initial velocity potential, is uniformly distributed within the circular area of the "ship" and proportional to the

square of its velocity. Professor Hor has kindly endorsed this

interpreta-tion of his formulae. The dissimilarity between HoPs's formulae (33) and (47) indicates that the character of the wave system is essentially dependent

on the character of the ship even though its extension is infinitesimal as

compared with the wave lengths.

(44)

E. ROGNER, A CONTRIBUTION To TEE THEORY OF SHIP wAvES. 43

1.

. i

here show a phase difference of of a wave length instead of

(f a wave length in the earlier theory. These formulae in reality agree with HAVELOCK'S formula for the waves at the exact boundary plane.'

The table IV presents the result of a numerical com-putation according to the fornnilae (82) and (83) for one

crest of the transverse wave system, the corresponding crest

hf the diverging wave system, and the corresponding crest

outside the boundary plane, in ease the waves are produced by

a pressure point (thus Gr = 1) travelling with the velocity U = 3.13 ms per sec. ( i

metre').

The pattern of the wave crests is in this special case given by the equations

+ a1 = + 2kir

n:

F(u2) + a2 = + 2kn

where k is an integer. The k-coordinates of the crests of the

diffe-rent wave systems for diffediffe-rent values of are denoted by ,, ,

and , the corresponding quantities according to the earlier

theory (thus calculated from (86)) by (g,) and (Eu). The quantities

?,

, and A, are the projections on the e-axis of the wave lengths,

measured along the planes - = coust.

A,, A,, and A, are

the amplitudes expressed by the unit

7vQ U' (i metre)'1 The

indications (A,) and (A,) denote the corresponding quantities

according to the earlier theory.

a1 and a

are the phase

terms in (82).

(45)

44 ARKIV FÖR 5ATEMATIK, ASTEONOMI O. FYSIK. BD 17. ic:o 12. y metres 1 I1 0.355 0.360 0.865 0.870 1003.81 1005.17 1009.84 1014.44 1018.08

The mutual situation of the wave crests of the different

wave systems is shown by fig. 17. The lines fully drawn

show the crests calculated from our formulae (82) and (83),

whereas the dotted lines show the crests according to the

earlier theory (formula (86)). The crests of the diverging

waves pass through = y = O, and have there the tangent plane y = O. In the case, here considered, the phase term

ß being a constant. the resultant wave crests at the boundary

planes form an angle of 540 44' with the midwake plane (cf.

Hopr, 1. e., p. 73).

The figure 18 shows with lines fully drawn the amplitudes

of corresponding waves of the three simple wave systems

according to (82) and (83) for the case in question and with dotted lines the amplitudes according to the earlier theory.

The waves are here illustrated also in the proximity of

y

-metres metres

-

( ' 11 metres f . 2 metres metres rn 0.000 921.27 0.00 921.27 0.00 6.288 0.10e 025.06 361.83 925.00 361.88 6.815 0.200 941.17 681.38 941.17 681.38 6.419 0.300 972.03 922.81 972.04 922.80 6.629 0.885 989.79 981.26 989.81 981.25 6.751 0.840 992.91 988.27 992.94 988.25 6.772 0.845 996.25 994.85 996.so 994.82 6.795 0.650 999.87 1000.94 999.oe 1000.85 6.820 0.842 1001.43 1003.28 1001.60 1003.07 6.841 1 1002.67 1004.95 1002.95 1004.66 6.840 metres 6.840 6.850 6.881 6.918 6.944

(46)

A5 0.400 0.815 0.109 0,027 0.005

md outside the boundary plane. The infinite amplitudes of

he diverging waves at the line

= y = O, obtained from the formulae in case the forcive has an infinite intensity, are

flisappearing if the forcive has a finite intensity and if we

Lpproach to y = O along a wave crest1, but it does not

lisappear if we approach e. g. along a plane = const. + O. Our

ormulae, however, by our degree of approximation are not raM near this line.8

Cf. HAVELOCK, 1. e., p. 421.

2 Because u2 = and F"(R2) = O at the midwake plane, formulae (82)

nid (86) in case the forcive has an infinite intensity give infinite values of he amplitudes of the diverging waves at this plane. For a forcive of finite

ntensity, on the contrary (G(u2) for u2 -+ decreasing as as is easily leen by performing the integra.tions for an element, of the forcive), the

bmplitudes in question become zero.

8. ROGNER, A CONTRIBUTION TO TRE TREOSSY OT 5H11' WAVES. 45

A., (A ) (A2) «L ir 0.088 + 4 )85 7.52 0.086 7.54 0.2500 1V +0.25007V )97 1.032 0.096 1.085 0.2500 7V + 0.2500 1!: 132 0.380 0.132 0.380 0.2516 7V + 0.2513 1V 187 0.829 0.188 0.332 0.2558 7V + 0.2325 7V ?07 0.384 0.208 0.836 0.2595 7V + 0.25557V 3S 0.348 0.241 0.351 -0.2655 7V + 0.2605 7V 0.879 0.314 0.400 0.2845 7V + 0.27957V 135 0.899 0.400 0.467 0.3025 7!: + 0.29957V 100 0.400 7V s +

(47)

46 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 17. ì:o 12, Fig. 17.

s,'

,s,'st Ji Is,'2W Q', as, a- tgi;-&, Fig. 18.

The superposition of the transverse and the divergiug

waves has been graphically carried out in the neighbourhood

(48)

E. ROGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 47

plane in fig. 19. This figure shows the waves, produced by

the pressure point, above considered, at a distance of about 1 000 metres behind the forcive. The equidistance between

01n2I8.

the level curves is 31 The crests of the

dif-7veU (Jmetre)I8

'erent wave systems are indicated by dotted lines. Fig. 20

y-Fig. 19.

shows a photo of a relief model of these waves. Fig. 21

shows the vertical section aa along a resultant wave crest

see fig. 19).

The resultant wave system, as is seen from the figures

19 and 20, consists of hills appearing around the intersections

etween the crests of the transverse and the diverging wave ystems and hollows between them. The hills and hollows

ppear in rows, diverging from the ship, these rows being

XÈ%tL

'N

\\\\\

(49)

48 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 1'. :o 12.

separated by streaks where the wave motion is considerably

reduced by interference between the two wave systems. When

Fig. 20.

the sea is

calm, this structure of the resultant wave systeni

is easily seen by the alternating bright and dark streaks which.

Fig. 21.

diverging from the ship, are seen to cover the wave systeir

(see fig. 22). The photos reproduced in the figures 22 and 2

(50)

IIOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 49

fig. 19 and the actual ship waves, though in fig. 23 near

e ship.

Fig. 22.

Fig. 23.

From the figures 19-21 it

is evident that the

boun-ary planes are not sharply marked in the phenomenon, and

lus it

is easily accounted for the fact that values, different

om 19° 28', are obtained by experimental measurements of

(51)

Tryckt den i februari 1(23.

Uppsala 19. Almqvist & \V$kset? Soktrvckeri-A.-S.

AKIV FÖR MATEMATIK, ASTKONOEMI O. FYSIK. Bl) 17. :o F

the angle between the boundary planes and the midwake plan One value 19° /4 found by Lord KELVIN and Mr PuEvIs'

remarkable, seeing that the plane

= - tg 19°

1/4

=

0.341

passes nearly through the highest points of the outmost wave

these points being situated at some distance inside the bou dary plane (see figures 19 and 21). Lord KELvIN does n

mention whether he by this measurement has considered ti highest points of the outmost waves as marking the boundai

planes. There is, however, all probability that he has and

the matter stands thus this measurement shows a beautifi

agreement with the theory expounded in this paper.

Obraz

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