eeij
4C
C.ARMY FÖR MATEMATIK, ASTRONOMI 0011 FYSIK UTGIVT AV
K. SVENSKA VETENSKAPSAKADEM lEN B.& 17. Nzo 12.
A CONTRIBUTION
TO THEUHEOHY OF SHIP WAVES
BY
EINAR IIOGNER
WITH 23 FIGURES IN THE TEXT
STOCKHOLM
ALMQVTST & WIKSELLS BOKTRYCKERIA.B.
BERLIN LONDON PARIS
R. FRIEDLANDER & SOHN WRELDON & WESLEY, LTD LIBRAJEI C. ELINCESIECE
11 CARLSTRASSE 28 ESSEX STREET, STRAND 11 RUE DR LILLE
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 threedimensional ship waves has been
Leveloped by Lord KELVIN', EKMAN2, HAVEL0CK3, HoFF4, and BEEI5. These authors have succeeded in giving a
matheaatical explanation of the system of ship waves and a
descripion of the outlines of its structure.
But the present theory is as yet rather incomplete.
Calulations 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. 450500.
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. 417430.}
L. Hopy, Hydrodynamische Untersuchungen, Diss., München 1909,
. 4791.
G. GENEN, Phil. Mag. Ser. 6, Vol. 36 (1918), _{pp. 4863.}
This signification has been introduced by Prof. JAMES ThoMsoN. . Lord KELVIN, Math. and Phys. Papers IV, p. 369.
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.
E. ROGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 3  and yaxes in the undisturbed water level and the zaxis
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
parhieles.
The forcive, that has a finite extension, may travel with uniform velocity U in the direction of the positive xaxis, 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?}
0z2Introducing 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\
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. Thifact 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 (6where 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+ß22
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 polarcoordinates in the aßplane ai introduced in (6) and (7) when avoiding the singular point in (7).
. 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, zsystem. Denoting quantities, attributed
o the point of pressure by a dash and, by F, its integral
Dressure, we find
Polarcoordinates in the aßplane
a=rcos8, ß=rsin8
nay be introduced. Thus we get
1LT f
2
id8,
dr47r
j
U2reos$g
We may conveniently write

FÒ
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 drJ cos
8 r_!_{4} cos28 where+

_{'472!J}
IPU r r aej(aY»V+2
U2a2_gYa2+ß2 (12) (13) dadß. (8)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 semicircles in the complex
rplane. 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 xJ cosJ
(I CO52i cos2 9 2Tiere ' signifies a displacement of the path of integration at the singular point out into the positive and" a displacemeni out into the negative imaginary rplane. 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 contributioTto 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
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) d927rtJe 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; 2r d
f
e8
i ds.J cos'3'J Vs8+a2
_! 2At first that part Z19
of the
integral d1 for whichz 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 halfpart of the symmetrical
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=
ô271j.
Thus we have
f
e8ifd&
i(dip
¿/_JJ
Vg2±a2<JZ()xRJSinip
and find for
>  +
i. <
_{f}
_{, =}
 fiogtg( 
9)
 log tg 1 for i(dip
2 ed2<J
and for ifdp
il
i/
e'2 < xRJ
sin ip = zR ¡log tg 9.o) + log tgFrom these expressions it is evìdent that, when x R is
great, J2 with increasing R decreases at least at the rate of
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 withespect to s we have
f
e8
ds e_8ds<
Vs2+a2fy
+JdslOo1+l±c)
S b a o iHere we intend, for the sake of brevity, to omit examinng 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
df
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 JÁHNIEEMDE, Funktionentafeln mitormein und Kurven. Leipzig und Berlin 1909, p. 21. 2
We SUPPOSe S <_{R}
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
whenincreases, 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
024it is true, proportional to
' 3'
and respetively, 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 (ecos}2+ sin
. dsis, by substituting
+ r for
, found to be symmetrical irelation to y = O, when having also
T
¡ ,. ,.sii
¡ _{e} CO83y81fl&4=jd5I
. . ds.j
j seos
+zx(cos32,s1n)
OROGNER, A CONTRIBUTION TO THE THEORY OP SHIP WAVES. 11
024 024
'he expressions
_{ô2}
and_{ooy}
are consequently symmetrical04
024nd and antisymmetrical 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)>oSubstituting
+ r for
in the second integral we getz
(p=4
I e0 sin dJ
cos &COSz9'z 2
Limiting the integral according to the restriction z () <O,
abstituting the value (11) for x() and introducing the new
ariable
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 (1and if we introduce polarcoordinates in the , yplane
 tg &
Fx
QURe
fvu2
+ le
(Lwhere
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/}
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,
derease as , and they vanish for sin
e =
O when z <0, evenwe 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, use14 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
(2that 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=±
vwhere 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 zp1ane, and by "nutside the boundar
ROGNER, A CONTRTEUTJOI TO THE THEORY OF SHIP WAVES. lo
b is quite the approximation (26) that is the cause of the inmite 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 theappliability 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
calcuation 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 statiotary phases as to every value of f9.
(28)
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
(3that 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
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 uplane 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 uplane along the imaginary axis from the two points
f ramification u = ± i outwards we make the analytic funeion 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}
18 ARKIV FÖR MATEMATIK, ASTRONOMI O. FYSIK. BD 17. tc:o 12
y ±
± 1) +(t4u2 + 1)2+ uv
The positive sign for Y is
to be chosen, when 'u, iisituated 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) +
+ (uu + 1).
(39The 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.
I. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 19 In the figures 27 the curve Fi(r, u) = O is shown by
lotted lines. As we have restricted ourselves only to consider
that halfpart 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 uplane, as shown by the curves fully drawn in the
gures 27, conveniently in such a manner, that the variaon of F alvariaong the path, at least in the neighbourhood of Le points u1, u, u, and of the upper limit u4 =  ctg (9 of
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 uraxis and we writfor 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 fromto thus becomes
o.
PxVu4 + i Reie(+1)+1(u3)fe_x11'dv=
2VQUPVu + 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.
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 24 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 contribuion is to be considered, and it vanishes of course when reloving 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
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) arzero 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 vplane 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
inegative, are marked with lines.
Fig. 8.
(4
At the boundary plane the hyperbola (42) degenerate
into the straight lines (see fig. 10).
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, situated in those parts of the complex vplane 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, except the arcs and , can be neglected, compared with
the condition that e
i when y j
situated atone of these arcs, gives a relation from which the order of
magnitude of R that is necessary to obtain the degree of
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)
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 velocv potential produced by a travelocvelling point of pressure:
in the region COD
5(R, (9,4= 6h/q2I8P
±
V+ 1
e {Iar(ta)COSF(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)
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, ocupied 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 uplan which are situated in the neighbourhood of certain pomiu1, 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+Yoddy01
(5where 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.
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} 8a=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()
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=iVF"(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=1_{17F"(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 C6"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) (61JIOGNER, 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
NCSI3)
+ ßj (65)=
ar U411R116'g2P
: + i N sin F(u3) +_{ß}.} 7vQ U'I8Rh/3The 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 beenntroduced 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 tgGr (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)
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 ismall 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) dicfwe
dw (69. HOGNETI, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 31
=
fev2+W)dw=
J
e ¿2+V)dw =1J
iv2eiW'dw. (70)The wplane has with regard to the character of the real
art of the exponent a similar appearance as the vplane hgures 810) and it is thus evident that we in the integrals
bove can use the arbitrary limits w, wn, and wm
coresponding 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 Ffunction
F(s)
=
_{o}f t
e dt re find() =
(S + 54  (S3 S4) +
7O viii Will .F1 + (S1S) + d (S + 84)] (72) + (71)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+ (6n5)! n=OThese 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 inegative and a2 positive (formula (45)). We write
W111 8111
I( loi)
=
f
e1(_IdIv28)dw=
._1f
eH8'+I)ds = WI1ii
8111 :J
sB e2ds (74 n. $11 WI SIL(iai)= f eu1I1t5'±tt$)dw
=
_1fe8+
ds =W111 8111
Sj
=..1ÇÇfs3e18'ds
(7where the new variable
$ = 1"j1?V
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.
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.
. 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
conidered in the series (73). For this value the 4th decimal is ikely to be correct.
For I a > 3.2 the semiconvergent 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 
21 + &
da \a
9jdc
9vhich 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
JAllÑKFEMDE, 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 .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(3i7).
The real and the imaginary parts of I (a) and 12(a) satisf
the differential equation
dI
4d81 116a48\d21
180a38\dI
28a2+
81
+ Jd2 +
81 da 811=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:
. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 37
K(r)=2fcos(w
+ w8)dw 2/1\
rsf 1 4 1i
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
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
2dv2
3K0.
This equation is also satisfied by the BESSEL functions of
the order ±
.'
After determining the arbitrary constants ofits solution we find:
for i < K(v) ei(W+1v')dw = 271: _{}
yvJJl(
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).
5. HOGNER, A CONTRIBUTION TO THE THEORY OF SHIP WAVES. 39 and for r>O
K(r) = f eW+)dw =
2vjVä + i
1'21 .r\
V3 + 1 / 21 , 'r jz VT3I +J_II
3/3
2\3YÏ
_{f} 2\3/
_{,j}
where'Y3+t
3si i1LJ(as)=
_{2} _{2iF(*)ì}_{2(+2)}
JV3_i
3sJ_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,
thereore 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
situabion 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 JAHNKEERDE, 1. C., p. 92.
2 Cf. the earlier authors.
s2
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=iand 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
considered.
At a certain distance inside the boundary plane, F" (Ua) being different from
zero and R being great,
oand 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)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)
+ sinURt VF"(u1)
(u + 1)G7(u2) sin F(u2) +j
+_{i/"}
U2According 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)42 ARKIV FÖJI MATEMATIK, ASTRONOMI O. FYSIR. BD 17. :o 12.
In reality, the formula (86), being the first
approximation of our formula (82) agrees with ERMAIc's final
formula1, 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 distribution of pressure, the intensity of which is propordistributional 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 amplitiides) 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. 419421. HAVELOCK here has overlooked the phase difference at the boundary planes between the transverse and the diverging 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 twodimensional waves from which he has constructed his
threedimensional waves.
Hor's formulae are not quite correct on account of some errors on p. 71. HOI'Fs calculations are, founded on some assumptions
introduced on pp. 5253 and 83 concerning the "ship", the physical
purport 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
interpretation 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.
E. ROGNER, A CONTRIBUTION To TEE THEORY OF SHIP wAvES. 43
1.
. ihere 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 computation 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 kcoordinates of the crests of the
different wave systems for diffedifferent 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 eaxis 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 phaseterms in (82).
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 6.944A5 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, areflisappearing 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 +
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
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
dif7veU (Jmetre)I8
'erent wave systems are indicated by dotted lines. Fig. 20
yFig. 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
\\\\\
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 systeniis 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
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 1921 it
is evident that thebounary planes are not sharply marked in the phenomenon, and
lus it
is easily accounted for the fact that values, differentom 19° 28', are obtained by experimental measurements of
Tryckt den i februari 1(23.
Uppsala 19. Almqvist & \V$kset? SoktrvckeriA.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.