On the linearized theory of wave resistance for a pressure distribution moving at constant speed of advance on the surface of deep or shallow water

9

-

SKIpSMODELL:TANKEN

NO.RGES TEKNISKE HOGSKOLE TRONDHEIM

ON THE LINEARIZED THEORY OF WAVE RESISTANCE FOR A PRESSURE DISTRIBUTION MOVING .AT CONSTANT SPEED OF ADVANCE ON THE SURFACE OF DEEP OR SHALLOW WATER

by J. K. LUNDE SKIPSMODELLTANKENS MEDDELELSE NR. 8 JUNE 1951 N T H - TRYKK Trondheim 1951. -StudiecentrIum _{T. N. 0}

VOW Sche-epsbouw en _{NavigDa.fielft} Md. Scheepsbouw.

SYNOPSIS

, This paper is concerned with the linear theory of wave resistance

experienced by any distribution of surface pressure moving at a constant speed of advance on the surface of deep or shallow water. The theory developed is of some interest because of the obvious connection between a pressure distribution a Lid a small fast motor boat, but the analysis does not however include spray, the generation of which is independent of gravity. Whilst surface waves are not the immediate object they have

been analysed theoretically in one or two rather partictilar and

restric-ted cases.

The exact problem is by its nature a hard one because the solction depends upon finding a potential function which satisfies, among others, a certain non-linear boundary condition at the free surface whose shape is not given, but which must be determined as part of the solution.

The linearized problem of the deep water case is an-old one and

dates back to Kelvin who examined the waves produced by a concentrated

pressure point by regarding the displacement

of a point in the free

sur-face as due to a series of pressure impulses applied along the course of the pressure point and behind it. In this analysis Kelvin made use of his theory of stationary phase.

In a series of papers, published in the Proc. Roy. Soc.London, Have-lock has examined the wave .resistance experienced by different types -of

pressure distributions whilst some of his papers also deal with the wave motion. Use is made in these papers of an artifice, originally due to Ray-leigh, which consists in assuming that the fluid has a slight amount of

friction proportional to the relative velocity. The integrals concerned give then a solution which corresponds to the surface waves trailing aft of the disturbance. This hypothetical law of friction is not professed to be

altogether a natural one, but it represents in a rough way the effect of small dissipative forces and it. has furthermore the mathematical conven-ience that it does not interfere with the irrotational cl-...iracter of the motion. The frictional coefficient is however, only retained in the intermediate ana-lysis to a degree sufficient to attain its chief purp...)&e, and is made zero in

the final result, thus indicating that it is not important that the hypothetical law of friction should be an absolutely natural one.

Hogner examined in some papers the waves produced by a concen-trated pressure point moving on the surface of deep water -by-making use of the method of steepest descents. The same author obtained, in a later

paper, an expression of the wave resistance experienced by a pressure

distribution by direct integration of the horizontal component of the pres-sure which acts on the distribution.

In a recent paper Peters re-examined the waves produced by a

con-centrated pressure point moving at constant speed of advance on the sur-face of deep water.

Mentioned should also be made of the papers by Lamb, Terazawa, Green, Weinblum, Inui, and Andersson. (See the list of reference at the

end of the paper. )

It is then found that the problems are, to a. certain extent,

indetermi-nate in so far as we get waves in front of the _{pressure distribution as}

well as to the rear. On this solution _{we then find it necessary to}

su-perpose a freely moving double pattern in such a way as to cancel out approximately the wave motion at a distance in advance of the

distri-bution leaving waves only to the rear, the combined result being the practical solution. It is, however. indicated in two appendices at the end of the paper, that we may obtain the velocity potential for steady motion by considering .a pressure distribution started from _{rest and} made to move With constant speed of advance. This method gives

fur-thermore the practical solution at once with waves only to the rear of

the pressure distribution.

In the course of the analysis it is, because of certain reasons .

suggested that we may not be able to determine what influence

_a

rea-sonable small variation of the wave profile along a ship will have on the .wave resistance even-if the wave profile could be measured with great accuracy.

The paper does, not deal with the theory of_{wave resistance for} displacement ships'. This theory has been examined in a number of

papers by Havelock whilst Wigley has compared measured and

calcu-lated resistance for different models. The theory of wave resistance for displacement ships has -also recently been examined by Lunde ina paper to be published in the Transactions of the Society of Naval Archi-tects and Marine Engineers.

1. Velocity .Potential in the Case of Deep Water.

Let the x- and y-axis of a right hand, orthogonal, and fixed re-ference frame be in thefree surface of a moving body of inviscid and incompressible water with the z-axis directed vertically upwards. Suppose that the water, which fills the space z < 0, is moving in the

negative direction of the x-axis with a constant velocity c.

Let ps be

the intensity of the pressure distribution applied on the free surface within the area nand let the surface pressure be zero outside cl .

The motion which is 'assumed to be irrotational is given by the velocity potential , where

(1.10) = cx +.0 (x, y, z)

In this equation cx is the velocity potential of the Uniform stream and (x, y, z) is the velocity potential of the disturbance due to the pres-sure distribution, thus the components' u, v, -w of the perturbation are given by - ckx,

in the x-, y-, z-direction respectively.

Physical considerations suggest, that the presence of a- distri-bution of pressure over the limited area on the surface of the moving water will produce negligible effect on the velocity potential cx at

suffi-cient great distance upstream. The pressure equation may accordingly be written

P , _{21 /P 2}

-77 + [ c + Ox) .4) + g(z + + + gzi

y co

where is the wave elevation. Assuming that the perturbation velo

-cities, caused by the presence of the pressure distribution, are so small

that their squares may be neglected in the comparison with the other terms in the pressure 'equation we get

, P

+ c (/) g( + ) kr,.+ gz)

x 03 :

On the free surface we have 13'0 apart from within the region II where the surface pressure Ft given by Ps.: Thus the free surface

con-dition becomes inside

(1.11)

+c Ox+g

= 0

being the condition to be satisfied by the perturbation pressure at the free surface, and

(1. 12) r c

g

everywhere on the surface outside

-equation of the boundary surface, we have dridt =, Ft + uFx + vF3; + wFz 0. In our case the free surface is given by z = (x, y), thus

F(x, y, z, t) = z - y), and we have

(c +)

, +day - (i) = 0

Just as above, we neglect small quantities of -second order, that is

the none-linear

termXtX4-Ycl)

t

Y' thus

(1.13) ctiz = -w

This is the usual approximate expression for the condition that a fluid particle in the free surface moves with a normal velocity equal

the normal velocity of the free surface. From the last equation- we

have that (1:11) may be transformed to

inside n , and (1

li) to

(1. 15) + K. = 0

for points on the free surface outside n , and where Ko = gic2 . Both

(1.14) and (1.15) express the condition to be satisfied at the free surface

but to the approximations accepted these conditions may be regarded as

fulfilled at the average free surface, that is for z = 0.

As physical considerations also suggest, that we have no waves at

a large distance in front of the distribution, we must have

(1.16) for z = 0 and x -4 CI3 and furthermore

( 1 . 1 7 ) ci) 4

+; 0 for

z < 0

XX yy zz

which is the equation of continuity of an incompressible fluid in irrota-tional motion.

We note that by neglecting small quantities of second order in the different conditions the problem has been linearized. It should, however,

be realised that arbitrarily small causes may produce finiteeffects in

hydrodynamics, but this can hardly be accepted as universal. if this was so, since every physical experiment is actually affected by countless minute influences, we should have no chance of ever to predict correct-ly the result of any specific experiment. Since the results of many

ex-periments can, however, be predicted with practical certainty it is prob-ably not an universal principle, but is preaurnable restricted to certain

5

-We also note that it is not the complete velocity potential of (1. 10) which enters into the different conditions given above, bit. rather the velocity potential '1) (x, y, z) causing the perturbation

Velocity.

It will readily be seen, that an integral function which satis-fies the condition that the surface pressure is ps inside n and zero

outside on the free surface of the liquid is

(1. 18) ps(x, y) = Lim 1 ff ps (h, k)dhdk--fr d 9 J eKz+iK z -0 4 'n'2 C2 -Tr

where

(.75 = (x - h)cose + (y - k)sine

Thus from an inspection of (1.14) and (1. 15)-it seems as if the

velocity potential, due to the perturbation pressure, will take the form

of

(1. 19) 46,.= f p (h, k)dhdk fn. -d eKz+iK-4)F( 6 ,K) dK

-7T

..

-where F( 9 , K) is a function of 0 and K and which we now determine

by substituting (1. 18) and (1. 19) in (1. 14). Thus

F( 8 ,K) -

i K

47T2 PC (K..--,K0 sec2e ) cos 0 and Kz+iKi5 (1.20) Re i f s

k)dhdk f sec e de1e

KdK 4 Tr2P c ç1 71 0 K -Kosec 28

with c = (x-h)cos 0 + (y -k)sine , Ko = g/c 2e and *here Re stands

for the real part. It Will readily be seen that (1.20) satisfies both (1. 15)

and (1. 17).

We now consider the condition expressed by (1. 16) and accord-ingly write (1. 20) inthe form

(1. 21) ct = - ₂ 1

tt-77/2

p s (h , k)dhdk .1 sec 0 da jrajsin KL cos° Ycos KFsinO)eK_--zKdK

Tr pc CI 0 0 K - Ko seC 2. .9

2 2 2

(1.22) c ₌ Im where L < 0 axis from 0 to of the integral

For L >

of the -contour quadrant when 1 ff psdhdk 77-2/2 X 2 2 tZ-ila(t - k ) . 2 t - _-Kot co

s dhdk cos Fr dr Im

f

c°e

i

i-

s -iLs

0 Kn :

s-7

- VT-7Fr2 and by substituting t2 =r2+,s2 andr = k noting

that t

k we get

a)

1.

co sFX d

X. hn fC°ett-ila(t2-X2)

t2- X.2_ K t

tdt

where L x-h and F= y-k and where Im stands for the imaginary part.

The same result would have been obtained by substituting X = K sin9

14- t in (1. 21), noting that t .

The integral in t is now transformed by contour integration. For

L <0 we use the contour indicated on fig. 1 consisting of the positive half of the real axis, a quadrant whose radius- tends to infinity and the positive half of the imaginary axis. As the integrand has a simple pole on the

positive part of the real axis at t

_1--K0/2 +1)2 +(K0/2) we cut out the

part of the path within a small sircle of radius 7 about the pole, and then make 7 tend to zero thus taldng the principal value in the usual sense.

It will readily be seen that we get no contribution from the quadrant when

its radius is made infinite large and L< 0, thus

fa) etZ-1L(t2-. A.2)

tdt=

t 2- X2 _{- Kot}

K0/2+X2 +(ç/2) 23 cos L [ 2(Km/2) 2+K. .1 OF(K0/2)

)ik _{(IC0/2 )2}

2+ k2 1 _{Kelm cos rnz - (m +2) sin mz}2

mdrn

(m 2+ 2 + K 2rn2

K0/2 + +(K0/2)23

noting that the result from the integration along the real is real and accordingly does not contribute to the value

0 e. a contour wich is the image in the real axis

show- n.on fig. 1. Again we get no.contribution from the

its radius is made infinite large, thus

tdt =

eZ EK0/2+4k2+(K _{-2)2 tos( LE 2 iCS3Z2)2 +K,Jx.-+(icrizr}

+ ((0/2) 2 [K /2+ +(K0/2) 2]

2

2!

co -Lm

+k )

2r2,

2

e. _{LKom cos m}z- m + _{A ) sin mz]}

mdm

where L > 0, noting as above that the result of the _{integration along the}

real axis from 0 to X is real.

Putting m

_{pi sin 8 , X= pi cos 0 in the integral term in the result}

above, and X = Ko sine _{sec 0 in the other term,, we substitute the result} in (1.22) which we now express as the sum of two parts, thus,= 501, + 01** ,

where (1. 23)

_{L >0=}

, e 03 7T/2cos(Fpi co s9)

ff.

sdhdk J - . 2 2 2

v

Pc

_dP1 o _p

+ Ko sin 0

X EK0 pisin 9 cos(zpisin 0 )

sin(zpisin 0 )) sin 0 d 8

(1. 24)

(1. 25).

Lpi

dp₁₀ /2cos (Fp cos_{2 2 2}

_{9) X}

pi + Ko sin a

K pi sin ecos(zpisin 0)-p12sin(zp1sin 0) ] sin 8 d 9

7Tp C 1

_{gpsdhdk f}

2 7T pc 0 2 2 2 2 2 k

) +K

_tn cow% ' L > . J2 , sec 28

(1. 27) ec

L>0

2

K° SI psdhdk 2e K0z sec 0cos (K L s'ece )cos(KoF sine sec2 )secs 0 4:1

77P c

According to (1.12) the wave profile outside n is given by

= - c 95 (x, y, o) and from the" velocity potential above we get when

g x

expressing as the sum of two parts or

e + e"

that

-7T pg

-Lpi 2 71/2 cos(Fpicose )sin2t9

e -pi dp de 0 2 2 gb P1 + Ko sine (1.28) y >0 7 2

KO _{psdhdk I sin(K Lsec e )c os (KoF sin e sec2,60)sec49 de}

2

IT p g

(1.

29)'rr

C° Lpi 2 1r/2 cos(Fpicos )Sin

2 8 9 1C0

p-dhdk 5 e

'pi dpi f

d °id< 0 _77-2p

g fl

0 o pi2+ Ko sin 02 2 (1.30) = L < 0 K2

If psdhdk

fr/s2in(K L sec 0) cos(KoF sine secse sect+ d

p g 0

We notice that (1.27) and (1.29) give a non-oscillating local dis-turbance which owing to the exponential factor dies out quickly with

in-creasing distance from the pressure distribution. The two expressions (1.28) and (1.30) on the other hand represent the dominant part of the

part of the disturbance. Thus the condition expressed by (1. 16) is not satisfied and it becomes necessary to superpose a freely moving wave ;pattern in such a way as to cancel out approximately the free surface

disturbance at a large distance in front and double the waves at a long

distance to the rear. The term to be added to (1. 20) must furthermore satisfy (1.17) and also (1.15), by itself at every point on the free surface. A suitable term is accordingly (1;26) with limitation on the value of L. The velocity potential may therefore be written

ersin EK(x-h)cos601 cos EK(y-k)sint9JKz

(1. 31) ci)

-

Psdhdk f7r/:ece d e KdK

ir2pc ft 0 0 K-Ko seca 0

7T/2K

z see°

2 r

+ KO ill psdhdk_ f 0 cos iKo(x-h)sec 0] cos [Ko(y-k)sinesec9]sec3ode

7rP c

which satisfies all conditions given in the foregoing as well as the con-dition that ckz- 0 when z In Appendix I is shown a different method by which the same velocity potential may be obtained.

The wave profile in front of the pressure distribution is now given by (1. 27) which, as explained above, dies quickly out with increasing dis-tance from the distribution. To the rear the profile is given by (1.29) which also dies quickly out with increasing distance, together with twice the value of (1. 30). We have accordingly for large distances from the

pressure distribution

and (1.32)

71/2

Ko2 f f ps dhdk I sin Kosec 0 -h cos

o (x-h)< o 7TP g -7r/2

+ (y-k)sin 91 sec40 d e

It thus appears that the wave pattern a long way away from the pressure distribution is built up of plane component waves advancingin all direction including within w /2 on either side of the axis of x and that

the amplitude factor depends upon the direction of these component waves. It is sometimes possible to express (1. 32) along the axis of x

in terms of tabulated functions, but this is not the ease for points away

from this axis, and the best one can do then is to find the asymptotic

2 a

expansion for large values of (x +y )2. We may, however, show that (1. 32) exhibit the usual features wellknown.for the waves behind a body =moving on the surface of a fluid. To do this we consider the particular

case of a concentrated pressure point of magnitude P situated at the

_ 2 7r/2

(1

. )ZPK°J

_(-IxI, r _C&

IT)-'17(-1 _{Trp g 0} sin(KolxIsece )cos(Koy sine se sec 40:19

aft of the pressure point and-where.. as mentioned before, the local dis-turbance, which dies quickly out With increasing distance from

the.pres-sure point has been neglected.

As thE Bessel function of the second kind of order zero is given by

Where (-1.)rn( .2m)

,z,

(4 v2-12 )(4 v2-32 ) m =o (2z)2rn 2 ! !8z) 2 4, v2 - 3 2 )(4v2 52) (4 v2 4! (8z)4 co 1.)n4 ( 21, 2r/11-1) 4 v2,12

('2

-.12 )(4 212-32 )(4v2 + . 2 (zz)M+1 1.! . m _!.(8i)2 4P g Kol 71/2 dt = - .1 cos (x sece)sec ed

the wave profile along the axis of k aft of the disturbance May accordingly be written

P _{a3 yo(Kol}

1

pgic, Blx12

which for large of Ixl approximates to

(1.34) _{er lx1-;0)Peig} -1)m [ 3(1,2m+1)+(31 2m+.1)] co ( )

sin(K lx1 -f) E

m. 0M+1

(2 Kol xl ) 4 - cos(K lx1- IT) F., m [ 3(1, 2m)+(3,2m CO rn (2 Ko X)2M Ko2 E 3y (K 4 p

1)- y (K, lx!

decreases as x-2 _{The two first terms in (1. 34) may be written}

*ic _{PK! 2 z}

The local disturbance which dies out quickly with increasing distance from the distribution may, -for the-concentrated

_pressure,

be found from (1. 27) or -(1. 29) and is given by

CO

(1.35) mrx, Kop xi pi 17/2cos(yp1cosel )sin28 _{d 0}

77.2pg 0 pi cipi _{1)12 +Ko2sin26} 03 [ e plTr2PgK dp o.0 me -13;IP1 p4 1 77/2

I

cos (yp, cos d8

Ct 11 rr/2 f 'cos(ypi cose ) d el + Kosin 6

because (x, y) = _{(-x. y). Assuming that Ix' >0, we have from the}

last expression 7T _{3 wx} 31E(3C, y) = P [ Tr2p g Ko 2(x2+y2) 3/2 + 2(x2+y) 5./2

f e

c° -131 0

We have, however., that

i` - 4

f

_{-cos ( .ypicaS} FlPi 0 2 2 2 d e (1. 36) CI (x, 0) - P 2 wp g Ko 7r/2 cos(ypi cos e) 2

p2i +Ko sin2 0

We now make use of the following result

de]

3 _

d [log pci +

d

co 77/2

pi 2 _{dpi I cos(ypicos 6)de}

I 9

and (x, y) decreases accordingly at _{least as fast as const./ I}

in fact the local disturbance has its largest _{valueon the axis of x and}

is here given by

( -Ix!, 0)=

- --a

_{P g rIK x}

3cos (K

x -

r5

(Kol - )

e x1P1

1 co exp i

Hv (z) = Y1, (z) + (-1-z) -

f

r ( v+i )r (1)

where H v (Z) is Struve's function of-order 7), and where the relation

requires - 77/2<18< 7T/2- 7T/2-113< argz<v/2 +13.Putting v=0, z =Kol x land

u=ic lp, we find that (1. 36) may be written

(1.37) (x, 0)-,p. 2 p g Ko d 3 Cli716( 1°gi -u (1+ ) Z2' Ilio(KoIx 1- Yo(Kol xl) ]

Using the asymptotic expansions for Ho- (z) and yo (z) we find that /10(z) - Yo (z)= 2/( 'az) for z > 15.9 and (1. 37) may therefore

be written for large values of x

(1.38) x, P K 2 6

27Tp g (Kci x (K0,)4

We now return to (1. 33) and want to find what component makes

up the transverse and diverge systems. We therefore substitute sec 0 =

coshr and xiy = tan a

, thus

"9)

( IXI Y) >0

7TPpKg02 fw Korw( T

dT

where r2= x2+ y2 , w(z) =i sin a. (cot acoshT - sinh Zr ).

The first few terms in the asymptotic expansion of this expression could be found by applying the method of steepest descents. .The contour of integration in (1. 39) can be deformed into an equivalent set of paths which pass trough the zeros of w' ( ) in the complex plan, that is, through the saddle points, the curves chosen being such that the real part of w ( r) is concentrated in as short a stretch of the paths as possible. An

approx-imate value of the integral can then be determined from a consideration of the integrand in the neighbourhood of the saddle points and by making use of Watson's lemma. We have then, from the physical point of view, evaded the interference effects which would occur with any other type of contour. The saddle points are given by w' ( 7-) = .0 or '

(1.40) tan 6 = (cot a. + Jcot' a.

1

13

-The two roots of tan Bare only real if cota 247 that is if

-19°28' 19028'. Thus the main part of the-wave pattern is

contained within the two straight lines, the so called lines of cusps,

radiating from the pressure point to the rear, each making the angle

19°28' with the line of motion. As 8 is- the direction angle of the plane component waves and as tan 0 has two roots, the smaller root

corresponds to the transverse waves and the larger to the divergent

waves at each point. For cot a = 2A--2 along the lines of cusps where the direction angle of the transverse waves coincides with that of the divergent waves we get tan 0 = 4272 or = .± 35°16'. Thus the

transverse waves are made up of component plane waves symmetri-cal about the axis of motion and whose direction angle in the plane bounded by the negative half of the axis of x and the positive half of the axis of y varies between 0°-and 35°16', whilst the direction angle of the divergent waves in the same plane varies between 35°16 and

900 The expression given by (1.40) represents really the envelopes of the straight crests (or troughs) of the plane component waves which

makes up the wave pattern, and along these envelopes the component

crests; or troughs as the case may be, combine together to give prom-inent features of the pattern. Owing to a difference in face, however

between that part of the envelopes formed by the transverse component waves and that part formed by the divergent component waves, the en-velopes are not quite the lines along which the wave pattern is an absO-lute maximum, in fact the absoabsO-lute maximum of the transverse waves

is situated slightly in front of the corresponding transverse part of the

envelopes.

The family of straight lines corresponding to the crests of the plane component waves of the travelling wave system may be represen-ted by

2 77 2

(1.41) x cos 8 - y sin 9 -N 1-7(o-cos = 0

where 6 is the direction angle, Kó. g/c , N

1,2,3,

. and Ko the

wave length. Finding the envelopes in the usual mariner by

differenti-ating in this case with respect to 0 and equdifferenti-ating to zero we obtain

x = (N Tr/2'K0) (5 cos 64-cos 30 ),y = (N 77/2 Ko) (sine + sin 30). By substituting it will be found that these expressions satisfy (1.40).

Differentiating with respect to 6 the expression of x and y we find that there are singular points or cusps on the envelopes for ta.n8= 4 2/2,

that is at the point where the transverse and divergent wave system meet. From the value of xl y" - y' x" when 0 varies between 0 and

97/2 we find that those parts of the envelopes which are due to the di-vergent waves are convex to the axis of x, and those parts which are due to the transverse Waves are convex-to the axis of y. At the

pres-sure point we have dy/dx= 0. The envelopes are indicated on fig. 2

3.nd they have a remarkable similarity to the general wave pattern formed at some distance behind any body moving on the surface of a fluid. We

due to the crests of the component waves represented by (1.41) by simple graphical methods. Representing the component crests with

straight lines for a number of values of 9 in the range from - 77 / 2 to IT /2, we would find the fan _{wave pattern represented by the} envelopes emerging from such a clia.aram. At,points at some distance outside the region covered by these envelopes -the component crests. and troughs tend to cancel each other out on the average.

If we want to find the first few terms of the asymptotic expan-sion of (1.39) by making use of .the steepest descent we would have

to consider separately the three cases cot 2a - 8 0 as the paths

are not the same for all values of , thus a single asymptotic

ex-pansion is not valid for all values of a . In fact the boundaries of the

sector of half angle 1928'0 are lines along which the-Stokels

phenom-enon occurs, that is, the asymptotic expansions change their character

completely upon crossing these lines. _{The first term in the asymptotic} expansion could also be found by making-use of the principle of station-ary phase. In this method we expand the power of the exponential func-tion in (1.39) about its stafunc-tionary value for large values of Kor and retain only two terms in the expansion, thus assuming that the terms

after the second are negligible. It will be found that the terms which we have to retain are not the same in the sector between ± 19028' and along these lines, this is due to the coincidence of the two roots in

(1.40) along these lines.

2.

General Wave Resistance Theory.

It may seem as the most direct way by which to find an expression for the wave resistance would be to regard it as what it is in fact, namely, the combined backward resultant of the water -pressure acting on the

pres-sure distribution. Assuming that the upward drawn normal at a point

(x, y, o) on the pressure distribution makes the angle a , /3

, y with the

axis of x, y, z respectively, and that the angles a, are approximately

equal to 77/2 whilst y is very small, we have that the wave resistance R experienced by the pressure distribution is given by

R =

ff p (x, y)sin

y clx dy

As, however, y is assumed to be a small angle and as the surface

ele-vation is given by t we have sin y . Substituting this expression

in the formula for the Wave resistance given above and making use of

13) we now have

10)

R

ff ps(x, y) (

z)z

dx dy

- 15

which holds whether the pressure distribution is moving on the surface of deep or shallow water.

It will be found, , however, that (2. 10) does not always leads to the simplest method for purpose of calculations and a different method will

accordingly be _given now.

Let A and B be two infinitely large vertical planes fixed in space

at right angle to the direction of motion of the pressure distribution, A

in advance and B to the rear. _{Consider the rate of increase of the}

ener-gy of the water in the region between the pressure distribution and these

two planes, and consider also the forces operating on the water at the boundaries of this portion of water. The water possesses kinetic energy

due to its motion and potential energy arising from alterations in the

sur-face elevation. _{As the pressure distribution moves forward with a} velo-city of advance c the length of the wave train-is Increased ahead of the

plane B at a rate equal to c. Due to this, total energy, kinetic and

poten-tial, will be flowing into the region _{across B at a rate E(B). Similar let} E(A) be the rate of kinetic and potential energy flowing out of the region across the plane A. Let the' water pressure at any point on B be p, and let u be the component of fluid veloditi inwards at right angle to this plane. The water to the left of B is doing work on the Water to the right at a rate pu per unit area at each point on the plane. Summing up for the whole plane we call W(B) the rate at which work is being done across this plane

on the water in question. _{This term is of course connected with the well} known theory that the energy in a plane wave is transmitted at a speed equal to the group velocity. Similarly we let - W(A) be the rate of work across the plane A on the fluid between A and B.. Hence, equating the

to-tal rate of work upon this portion of water to the rate of increase of its

total energy we get

Rc.+ W(B) - W(A) = E(B) - E(A)

where R is, as mentioned before, -the- wave resistance experienced by the

pressure distribution at the speed of advance c.

If we now take the plane A further and further in advance of the

pres-sure distribution the quantities E(A) and W(A) approximates to

zero since

the distribution is advancing into still water. If we also take B further and

further to the rear, the disturbance due to the pressure distribution approx-imates to a free wave pattern. Thus we have in the limit

(2.11) R = 1 (E(B) - W(8) ]

where E(B) and W(B) are calculated from the free wave pattern to which the disturbance approximates at a very long distance to the rear, thus

showing that the wave resistance depends on the supply of energy to the periodic part of the disturbance only and not on the local part of disturbance.

An other way to look upon this problem will apparently be to assume

that the rear of the wave system moves forward with a less velocity than the front. _{The whole procession Will then be. lengthened at}

_{a certain rate}

and the energy to this process is supplied by the .pressure

distribution.-We now proceed to calculate the quantities in (2. 11) in terms of the velocity potential ct) due to the perturbation velocity. We draw a third plane B1 parallel B and a-distance c behind it. The rate of

in-crease in kinetic energy between A and B will be due to the transfere of the kinetic energy between the two planes B and

B1 into the region

in question. The kinetic energy between the two planes B and B1 is given by

(2.12) - yb ds =

(C70)(

) dV

where On represents differentiation along a normal, drawn into the wa-ter between B and B1 and where V = ?--x +1-3-T° + in the

usual notations. The first integral is taken over the boundaries and

the second throughout the fluid volume between the two planes B and

B1

We consider shallow water of uniform depth d, because the par-ticular case of a deep water may simply be obtained by taking d, infi-nitely large. Hence integrating over the boundaries, which consist of

the two planes B, B/, the free surface and the bed given by x = xo

x

_{xl' z}

0 and z = -d, respectively we get for the integral to the

left in (2.12)

CO

(a. 1 3 ) p C z)

dY + 5 dY I

[ ( cfix)2+ cto

as clearly the contribution from the three planes y = ± co and z =- d,

which also have to be considered is zero. This is the expression for

the kinetic energy which in unit time is flowing into the region between the planes A and B. Substituting do z from (1. 15) we find that (2. 13)

may be written

(2.14) c

f

dY , (( Oxf + z Ko

I (00xx)

k -OD

The rate of increase of potential energy between A and B will be equal to the transfer of potential energy between B and B1 into the region in question and is accordingly given by

- ,

(2. 15)

gpc

f zdz f

=

pgc 5

2dy + co i_co x)2] z 0

-co -co

where we have made use of (1.12)

From (2. 14) a.nd,(2. 15) we get that total rate of flow of energy, kinetic and potential, across the plane B into the region in question is

given by (2.16) p c ((

dy f

[()2

17 -P ₍ X)2 C64) (2. 18) _{R K} _co 0 co

z +

1 ( (4)

- etq5] z 0

dyi E(B)

,K0 X mix

The pressure in the fluid is given by p. const. - g p z -cp .

Thus the average rate of work done across the plane B may be wriften (2.17) fa) dy

1

pudz .+ c P 5133dY f ( Ox)2 dz W(B)

-co -co

Substituting (2.16) and (2.17) in (2.11) we get an expression for

the wave resistance for shallow water, namely co

P°

f dyf [

) - dZ

Z -d, 20C

- -ct

and where we in the case of deep water simply substitute infinity for d, .

The disturbance produced by the pressure distribution will be of a

complicated character but the further we go to the rear the more it

ap-proximates to a freely moving Wave pattern, consisting of the periodic part, and following on with the same speed as the distribution.

The simplest form,of free waves on the surface consists of simple

harmonic waves extending over the whole surface. Taking a stationary origin with a wave propagation along the axis of xl, the surface elevation

for simple sine pattern of amplitude a may be represented by (

a sin K(xi

- ct )

where K = K tanh kd, and where the velocity potential is given

by

ac

cosh K (Y _{d,) cos K (}

= sixth Kd,

If a simple plane wave of this type is advancing in a direction making an

angle 8 with Ox we have

a sin K [(x -ct) cos 9+ y sine

cosh K (y + d,)

ac

cos K [ (x1-ct)cose + y sine] cose

sinh Kd,

where K is the positive real root of K - Ko se-c2e tanh Kd, o .

Generalising by supposing that we have simple plane waves ad-vancing in all direction including within 17 /2 on either side of the axis

of

x1 and superposing these waves we may assume that the surface

elevation is given by

v

A sin K(x cos 0 + y sin 6' ) d

-7T/2

refering now to an origin moving at a speed c. For a final

generali-sation we assume that we have a number of both plane sine and cosine waves with different focus point and that their amplitude factors depend on the direction of each component wave. The wave pattern may now be written 7r/2 n = f

no r

f ( ) sin K [ (x-x )cos .-717 2 n vi2 +

I

f' (0 )cos --77-/2r: or K E (x-xr)co56 + (y-yr)

or after the sine and cosine have been expanded

Tr/2 77/2

= 5 f(e) sin K(x cos /-1-y sine )d 0+

I

P ( Ocos K(x cost) +y sine) d

-77/2

that is

7T/2

(2. 19) = I (Qsin A cos B+Q2cos A sin B+Q3cos A cos B+Q4sin A sinB)de

1

where A K x cos& , B = K y sin 0 and where the Q's are functions of 0

in general and the form of the pressure distribution and its speed of ad-vance. The corresponding velocity potential now becomes

7r/2

(2. 20) = c I (Qicos A cos P.-Q2sin A sin B-Q3 sin A cos B +Q4 cos A sin B )

X

X

coshK (z+d,)

cosadO

sinh K di

where the relation between K and e is given by

(2. 21) K - Ko sec2 0 tanh K 0

sin 0] d

We assume that c 2< _{gd, so that (2. 21) has a pair of real roots}

for each value of e within the range of integration. It is, however

only the positive real root which has any interest here.

In order to evaluate the different terms in (2. 18) we make use of a theorem derived from Fourier double integral theorem (see Ap-pendix II)

5 du .1C° F( a )cos u (y - a )d a

0 _-CO

If we let

1° (F1 cos yu + F2 sin yu) du (2.22)

iG(y)= r

(G1 cos yu + G2sin yu) du where

F1

_{F2' G1, G2 are}

functions of u, then

(2. 23) fa) F(y)G(y)dy = 7T fc° (F, + F2G2) du

-co 0 I

assuming that the integrals are convergent. In order to make use of (2. 23) we have, however, to change from an integration in 0 to one in u where u _{K1 sin}e , thus differentiating with respect to 8 and

using (2. 21) we have

>Ksin yu]

ci) - Kc fl(Q,cos A - Q3 sin A) cos yu + (-Q2 sin A + Q4 cosA)X

o 0

Making use of (2. 23) we get that

77 c2

Ko

-

19-(coth K d,-K d, coseh2K d,) cosh K (z+d,)

cos4 d

0

(1+sin20 -Kod sech 2K d,) sinh K d,

/77/2(C1 2sin2 A + Q22cos 2A + Q; cos 2A + Q42 sin 2A

0 1

d0 cos SO (coth K d,- K d, cosech 2K d,) du

Ko (l+sin 20 - Ko d' sech2K d,) and (2. 20) may accordingly be written

+2(Q1C134°2(34

where we after having made use of (2. 23)' have repla.ced the variable u

by 9. The other terms in (2.18) may be evaluated in a similar Manner

and we get after some reduction that the wave resistance may be

ex-pressed as

(2. 24) R vp c24 +Qi +Q3- -K24- ) (cothK d,-K d, cosech2 K d,) cos39 d

"

where K is given in terms of 6 by (2. 21) and where c 2< gd, . The last

condition had to be imposed in order that (2. 21) should have a pair of real

roots for each value of 9 within the range of integration. This means that the graph of ta.nh K di on the base of K d, Must for each value of 0 be intersected by the straight line KAK sec2 0 ) which goes trough the origin. ' As the largest angle the tangent to the graph of tanh Kd, makes with, the

axis of K dI is 77 /4 we must have

-d . C S 02 5

d(K d,) Ko se,c29 Kodi

where as before

'

jac 2.

thus if c

for all Values of in question. But can hair`e-in..Order to -satisfy the above

gd,/c2, thus the smallest value of 60

-cos

1gdifc

.

If therefore c >gd, we have to substitute for the

_o

lower limit of the integral in (Z. 24) where 00 c o 1 4 gd, / c =

-1

cos vgKo

If we now consider the particular case of deep Water we take d, infinitely large and (2. 19), (2. 20), (2. 21), (2- 24) reduce to

( 2. 25)

2

(coth K cosech2K dt) cosh K(z+c11)

cos' A sin A lecos70 do

(1 4- sin 20-Kod sech2K d )sinh2K cl,

2< gd, _{this condition is satisfied} if c2- > gd, the largest value cos 2 0

condition is given by cos2

is in this case given by 90 =

(P sin A cos B +

0 I 1 . 1 cos A sin B +P cos A1cos131+P4sin Aisizthdd

(2. 26)1) c fr/2(P1 rcosA.Ico-sii1-P2sinAlsinB1-P3sinAlcosB +P4cosA1sinB1) X

0 _{K zsec2}

cos 9d9

where A1 = K x sec

1

, B Ko y sine sec

2 , lci-Ko sec 29 = and Ye

2 VIA 9 9

(2. 27)

_R.

₄ S₀ (P- +P₂ P₃

+ P- ) co:0 d e

₄

It should be remarked that both (2.24) for shallow water and (2. 27) for deep water do not only apply to a pressure distribution but

are in fact quite general. They may accordingly be used for any type of body moving in a fluid at constant speed of advance and for which a velocity potential may be found.

3.

The Wave Resistance in the Case of a Deep Water.

We now make use of (2. 25) and (2. 27) which applies for deep

water in order to find an expression for the wave resistance of the

pres-sure distribution. It will be noticed that (1. 32), which is an expression of the periodic part and therefore also of the wave profile at a long dis-tance to the rear of the disturbance, may be written in the form of (2. 25) if we put

,.p 2Ic2 a 2

I r L-1 .'"- ---"

p'cos

A2cos B2 sec49 dhdk, P2 = ---CL. J J

2K-

/3-S sin A2sin B2sec4e dhdk

7rp g 11 _{7rp g n}

21(02 s _21c.2

P3. Lc p _{sin A2cos B2 sec4 0 dhdk, Fs4---=-'4J ps'a cos A2sin B2sec40 dhdk}

np g 0 _{7Tp g n}

where A2 = Koh sec 0 , B2 = Kok sin 0 sec20. The wave resistance given by (2. 27) becomes now

2 77/2

R _77p C r fp 2+ p 2+ p2

40

1 2 3

r4

) cos 38 d e

which may also be written

21 -2 Tr/2 R ---13-1 (P2 + Q2 ) sec se d 0 2 cos (

sin ( Ko (xcos e + y sine ) sec211 dx dy P

Q )

where we have replaced the coordinates h, k by the current coordinates

x, y.

Both (3.10) and (3. 11) may be written in a number of differentways. We may for example substitute cosh u for sec , thus

with (3.13) I )7....sks

J)

or substituting a tan 2 2 63 -(3. 14)- r 2 .-(3.15) S T fn ps cos Ko (x + ya )4--F-TsT211) dy sin

If we assume that the pressure distribution is continuous and zero at its outer boundaries we may integrate (3.11) with respect to x, thus

K x cosh u + y sinh u cosh u dx dy cos

sin

We may compare (3.10) and (3.11) with the wave resistance of a distribution of doublets on a surface within the liquid. Suppose we neg-lect the depth of this surface at every point and assume that the axis of the doublets are parallel t o the axis of x. The wave resistance of such distribution may be shown to be given by

°

f

+ J2 )coshtidu 2 e 711) C 7T/2 R = 16 wp K04 r (F,2 Q 2) sec561 d 0 I I

with

We may accordingly say that the wave resistance in the two cases would be the same with the connection between the doublet density and the pressure distribution given by 4 Trp g M= cps.-

-Let us now compare in similar manner (3.16) and (3.17) with a

distribution of sources ,over a plane whose depth below the free surface we neglect at every point. The wave resistance of such a distribution may be shown to be given by

v./ 2

= 1 6 7rP K02 (F. 2 _{Q22 )} sec 30d 0

2

with

cos

x cos!) + y sint9 .) sec ) dx dy Q1) fl

sin (

P2)

Q2.)"

' and the wave resistance in these two latter cases would be the same with the connection between the source density

, and the pressure

distribu-tion given by cpxs = 4 7rp

g o.

The generalised expressions such as (3. 10) and (3. 11) may be put in a different form when pressure distributions 'symmetrical about the origin

are considered. Le t the 'pressure system be given by p_{2 2} = ps (r) where

r = x + y2 . Substituting x = r cos a

, y = r sin a, and

ps = ps (r) in .

(3.11) we now get

CO

P

. 5

ps (r) r

5271. cc's

Kr sec 26, cos ( a-9

Q ° sin

Reducing the range of the integration with respect to a from

0, 2

77. to 0, IT we find the 0-function reduces to zero and the

P-function to

CO

P (

z (Kr sec 2° ) dr

₀ 2 7r osec20 )

where

(3. 18)

F( y

)

f

ps (r)r

(r y ) dr

and where Z0 ( y ) is the Bessel function of the first kind of zero order.

The expression of the wave resistance (3.10) may now be written

ffo-

_ri cos_sin

23

-K (x cos 0 + y' sin 0 ) sec 29_o dx dy

2 7T/2

77 K0 s 2

(3. 19)

R = 4

sec[F(Ko sec

ta

p C

where the F-function is given by (3. 18).

We consider as a partidular case the pressure system given by

(3. 20)

p (r)

Bf/(f 2+ r2 ) 6 /2

It will of 'coutse be hoted that this .piessure distribution has no boundaries,

but by taking f small it approximates to a pressure concentrated over a Cir-cular area. It will for example at a radius 2f from its centre has fallen to

about 1/11 of its maximum value and at a radius 3f to about 1/32.

From the integral

.71/2.

An. (.:o: )

f

cos

where n is an integer.

notation, given by

A_3 ( a.). AS An_1 ( a )

A-1 ( a ) by successive

From the integral order, namely

we have

aM. fv-11,

X(af)

2/1r ( 1) v-p.

where Kv (x) is the modified Bessel function and where the result is valied

for -1 < Re

( V ) < 2 Re ( p. ) + 3/2, we get F(Ko sec2e ) =

-K f sec2 sesi

Be 0 . Substituting this t'esult in (3. 19) we have

(3.21) R, 477 2B.2 -2 Kof Oec20

sec 50 d 0

pC2 "

This integral may be expressed in terms of Bessel functions which are

tabulated. We use the notation

2n+1 _e

- a secede

The integral considered is therefore, in this

= -A'n ( ), we may obtain A_3 .( a ) from

differentiation.

expression of the modified Bessel function of zero

2

] d 0

( a /2)

2

cosh t

- 25

- , (c o s ht +1 ) 7/2 , a: sec 20

2 )(0( _{)= J e} 2

dt = 5 sece e

d9 =

A-1 ( )

0 0

-Thus remembering that k1 (a ) = -K0( ) - ( a ) and Ko ( a.) = -K1 ( a)

we have by successive differentiation a

1 - 1

A_3 ( )

-e?

Lr iN0ea/ 4"-FC) and (3. 21) may accordingly be written

2 2, 77.K B

-K f

2 K f 2 1 (3. 22) R _{° [ ko(Kof) +} + _(K0f)1 ) pc

If we on the other hand consider a little sphere.of radius b_ moving at a depth f below the surface of the fluid, and we substitute for the sphere a horizontal doublet of moment M, where M=1 b c and where c is the con-stant speed of advance it will be found that its wave resistance is given by

(3.23). R= ip g Ko2 be e -Ko [K. (K0f ) + }i

+(K f )

2 Kof 1 o

Thus if we put B gp 1)3 in (3. 22) we get the interesting result that the wave resistance of a pressure distribution given by (3. 20) and moving

on the surface of deep water is the same as the wave resistance of a little

sphere of radius b moving at a depth f below the 'surface of the water.

It.

should be noted that B is not a function of the speed of advance in this com-parison as is the case of the moment of the doublet.

Before considering a second particular case we ricite that the velocity potential of a horizontal doublet moving alopf the axis of x in an infinite stream is given by ch = (x2 + y2 + z2 ) -31 where M is the moment of

the doublet.

It may be shown that the velocity potential of a prolate spheroid

moving at a speed c along its major axis in an infinite stream is given

by

ea ( (aer - 1.211' (x - h)

(3. 24) = Bc r dh

[(x-h)2+ y2 + 21 312

where B= 1/ I( 4e/ (1-e2) - 2 log [ (1+e)/(1-e)] . In this expression a. b

are the polar and _equatorial radii and e the eccentricity of the median sec-tion given by e= 472--T2-3/a. We note that the Velocity potential given by (3. 24) may be considered to be due to a doublet distribution of moment Bc [ (ae)2- h 2] per unit length along the major axis from -ae to ae. Thus

froth a comparison of the firit particulai; case considered we should im-agine that a pressure distribution given by

ae CO :x) P+iQ = Z g pfB (ae)2-h2i

f f

...CO -CO -ae ae [ (ae) 2 -h 21 dh (3.25) ps= g p fl3 -ae [(x-h)2 +y 2+f 2l

would give the same wave resistance as a prolate spheroid moving at a depth f below the free surface of the fluid.

Substituting (3. 25) in (3.11) we have

which by putting x-h = xl, r2 x12

y2tan

a. = y x / reduces as far as

integrations with respect to- x and y are concerned to a double integral considered above, thu's

2

-K f sec° ae

i Koh sec 6

P+iQ 47rg p e ° [(ae) 2 -h 2 e _dh

-ae

It will readily be seen that Q = 0 and that the P-function may be written

2

-K f sec 0

_{o ,}

P. = 8 ir gp B(aere _{( 1 t2 cos(Koae t sec 0) dt}

Making use of the integral

2(z/Z)v ,t

J (1-t 2) v .cos zt dt

r( v+

which is valid for Re( v )> we get

, .

1a

-I taej/ 1 -K,f sec'2_2O p j 3 ` B e 47312 ,(Koae sec 0) Ko sec 9

and the wave resistance given by (3. 10) may now be written

/22

(3.26) 128 7r )sg pB2 e

0

Kof. _{[j3/2(Koae sec 0 ) ]2sec ea}

-2 sece

This .is, however, exactly the same expression we get for the wave resistance of a prolate spheroid moving at a depth f below the free surface.

Thus we have the result that the pressure distribution given by (3. 25) and

2

e

i K0sec2(x cos 6+ y sin 0

[(x-h)2 +:y2 f2]9/2

27

-moving on the surface of deep water gives raise to the same wave

resistance as a prolate spheroid moving at a depth f along its major.

axis at the same constant speed of advance.

By making e zero we now verify that (3. 26) reduces to (3. 21)

the expression of the wave resistance of a sphere. For the Bessel

function of the first kind we substitute

113/2 ( a ) =ETT ( siana _{- cos} CL)

in (3. 26) and express sin a. /a. and cos a. in infinite series. Expanding

also the different terms in the denominator of B in infinite series we

find that e3 in the factor in front of the integral expression cancel out and that (3. 26) reduces to (3. 21).

If we, on the other hand write down the expression for the

dis-turbance produced by these two pressure distributions given by (3. 20) and (3. 25) respectively it will be found. however, that they do nof a-gree with the disturbance produced on the surface by the bodies which

gives raise to the same wave resistance. This is due to the

non-oscil-lation local disturbance which are different whilst the periodic part of

the disturbance is the same for the pressure distribution and the cor-responding submerged body. As the wave resistance, however, depends

upon the supply of energy needed to maintain the periodic part of the disturbance, and not upon the local disturbance which as we have seen, dies quickly out with increasing distance from the body or pressure

dis-tribution, it was to be expected that a. pressure distribution and a

sub-merged body which produces the same periodic part of the disturbanCe also would have the same wave resistance. This result appears to have

an important bearing on the relation between the wave profile along a

ship and the corresponding wave resistance, because it.indicates that

it will be very difficult indeed to determine accurately what influence a reasonably small variation of the wave profile, measured along the ship, will have on the wave resistance when deep water is considered."- It is

of course released that the bodies considered above do not break the surface of the fluid as in the case of a ship, but there does not seem-any'

reasons why this should invalide the 'conclusion drawn. It has been pointed out in the course of the discussion of papers dealing with wave

resistance that more emphasize should be laid on the wave profile along the ship and that it should be carefully analysed.- It appears, however,

from what has been pointed out in the foregoing, that the value of such an analysis may be largely exaggerated leading probably only to very general conclusion. To this must be added the practical aspect of the

measurements required for such an analysis, it being not possible to

measure any wave profile with even nearly the accuracy by which the resistance may be measured'.

-The wave resistance expression given by-(3. 10) may also be .ob-tained from (2. 10). From (1. 31) or from (1. 23) to (1.25) we get

(3. 27) co 7T/ 2 3 3 ) (3. 28) I (c,z) - 1 psdhdk eLP P1 P c o s ( Fp c os0 1 fc singe d. 0' .t<o 772p,c2 n p12 Ko sin 92 " 2- 77/2 3i* 2

(3. 29) 1 (1)z f fp dhdkf cos(K01.. sece)cos(K Fsin asec20)sec5ad

f.(S) TrpK02c

These 'expressions have now to be substituted in (Z.10). The two functions given by.(3. 27)-and (3. 2,8) lead then to-an integral expression which may be

written formally when noting that the first require x-h >0 and the second

x-h < 0

where:Sand T are the limits of the pressure distribution in the x-direction.

Consider the first term 'only. Changing the Order of integration and then interchanging x and h we find that this term may be written

. .T

- p.s.dx, .pse -h).

Thus the two terms in the above integral expression cancel each other out, and accordingly (3.27) and (3. 28) do not contribute anything to the wave

resistance. As these two expressions correspond to the local disturbance the result obtained is in complete agreement with the development leading to (2.-10 and (2. 27). ,

-We consider .now, (3. 29) which when substituted-in (2. 10) leads to an expression which may be written formally

..f.,

psdh.. cos(

..

(x-h)]. . =

-where we have first of all changed the order of integration and then inter-changed,x and h. Hence..this integral function may be written

T

+...

f ps dx psdh Z

04fpsdhdkf

e'1

3 d

f cos(Fpitose

2 1 p c PI Pi, 2 2 2 sin e d e 2 _{p +Ko sine} h) . dh +

pse

x .

cos [..(x-h)]

/'

...,dh

. . f.x pscih. cos [ . -h)]. .

where h, k has been replaced by the current coordinates x, y. We notice

that these two last expressions are in agreement with (3.10) and (3. 11)

obtained in a different manner.

4. Velocity P.btenti_al in the Case of Shallow-

Water.'

The conditions given in section 1 and which the velocity potential had to satisfy hold good also in this case and need accordingly not be re-peated, but in addition we also have the condition-that the normal velo-'city at the bed must be zero, or cAz= 0 for z where d, is the uniform

depth of water.

"

The pressure ps acting on the surface within. the region may be ex-pressed by the integral function (1.18). The condition to be satisfied at the bed, and also the conditions given by (1.14) and (1.15) suggest that the velocity potential due to the perturbation velocities will take the form of

F( 0, K)

-- -

29-with no limitation on x-h. The complete wave resistance expression may'now be written

7T/2

K2

R=°- 55ns(x, y)dxdy ff ps(h, k) dhdk I cos [IC (x-h)sece ]X wpc 2 n

cos [ icr,(yzk)sio 0se20 ]sec50d 0

Expanding the two cosines we find that this expression may be written 2 K , 2 2 .

R.----P--

_{(P + Q ) Set Od 0} 2 41 71.1) C s cos (

Ko (,X

cos e + y sine ) sec u)

2 ^. ) _cbcdy

sin

= 55 .ps (h,.k)dhdk 5 d 0 5 -,cosh.K(z+d,) e

.?-7T °

i K

Substituting this expression and (1.18) in (1. 14) we get

and (4. 10) ,

- Re

7T _{cn cosh K(z+d,)sech Kd} i Kw 2 psdhdkf sece; de f _KdK 77 pC ; ° K-Kosec20 tanh Kd,

with io = (x-h) cos 0+ (y-k)sin . It will easily be seen that (4. 10)

satisfies the conditions expressed by (1. 14) (1.15) and (1. 17) and give zero normal velocity at the bed.

In order to consider the condition given by (1. 16) we write (4.10) as

1 7T/2 CO cosh K(z+d,) secli Kd,

(4.11) q5 = - psdhdk f

27r pc sec Od 61m fo K-K0sec Otanh Kdi

iK(Lcos 9 + F sine) iK(Lcos 9 - F sin0)

( e _{e KdK}

It does not appear practical in this case to make use of the substitution

employed when considering (1. 21). The integrals in K in (4. 11) are there-fore trandformed as they now stand by contour integration. Assuming that the pressure distribution is symmetrical about- the axis of x it will only

be necessary to consider that part of the icy-pla.ne for whichy .0, thus we have two conditions to take account of, namely, L >

o, F > o

and

L >

o, F <

o.

The integrand in (4. 11) has poles given by the roots of (4. 12) K-K0 sec 2e tanh Kd, or cos 26

-Kod, _Ko

Supposing K0d, _we note that (4.12) has a pair of real roots ± K only when cos NI 1c0d, e 77/ 2

, remembering that o

(tanh Kdi)/Kd,

We note further-more that (4. 12) has an infinite number ofpure imaginary

roots of type -± iJ3n where

< 1-l-<

I ₂ I.< I fi I <

_1

the first root ,80 disappearing when cos

JKd,

If we now suppose Kod, > 1, then (4. 12) has a pair of real roots

± K and an infinite number of pure imaginary roots ± i An where

there being no imaginary root corresponding to0

tanh Kdi

8 4 7T/2.

31

-The integral in K is now transformed by contour integration. For 'cod, <- 1,

o 4 0 < cos

14g, and Lcos 9

Fsin 6 > '0 we

use the contour indicated On fig. 3 consisting of the positive half of the

real axis, a quadrant whose radius tends to infinity, and the positive

half of the imaginary. axis. As the integrand has an infinite number of poles on the imaginary axis we cut out the part of the path within a small

circle of radius 7 about each pole, and then make y tend to zero, thus

taking the principle value of the 'integral in the usual sense., :It will read-ily be seen that we get no contribution from The quadrant when its radius'

is made infinite large and Lcos 0 ± Fsin > 0 . With the same con;.-.

ditions on Kod, and 9 but with Lcos 6 Fsin 9.<.O w'e uSe

a contour which is the image in the real axis of the contour indicated I., On fig. 3. Again we get no contribution from the quadrant when its radius is made infinite large.

-1

For K0 d, < 1, cos _Kod,

_0$

Tr /2 and Lcos 9 Fsin 6 > o

or Kod, > 1, o 4 6 4 7r/2 and Lcos 9 ± Fsin 0 > o we use the contour indicated on fig. 4 consisting of the positive half of the real axis intended, at the pole, a quadrant whose radius tends. to infinity; and the positive half of the imaginary axis intended at all the poles on the axis.

When Lcos 6 Fsin 0 < o we use a contour which is the image in the

real aids of the Contour shown on fig. 4. In these cases 'we'get'n.o contri-bution from the quadrants when their radius are made infinite large.

Performing the contour integration we find that the Contribution. from the imaginary axis and from the small half Circles about all the poles on this axis gives a non-oscillating function which is exponentiallyvan-

-ishing with increasing distance from the pressure distribution, and which

really represents the local disturbance of the wave profile,' Whilst the con-tribution from the half circle about the pole on the real axis gives the dominant part, Which is really the periodic part of the wave profile.

Before the results of the contour integration may be substituted in

(4.11), however , we note that a factor such as for -exaMple Lcose

will for L > o and F > o change sign within the range of integration With

respect to thus the range of 0 in (4.11) may have to be split into shorter ranges within which Lcos 9 -Fsin 0 for example'-is known to be

p i the r e or negative. Doing this we now End for large values of

I x-h I or I L (4.13) Of" L >0 L > o Tr/2 coshK(z+d,)sechKd

If psdhdkf

X n 0_0

1-k0dtsec29sech2Kdi.-cos(KLcos 0)cos (KFsin 0) Ksec6 d 6

1

1 _{TV2cosh K(z+d ) sech Kd,}

4. 14) 02(14< _{JJ p-dhdk '}

o 7rpc

f2 _{00 1-K0d sec20 sech2 Kd,}

X cos (KLcos 0)cos(KFsin e) Ksec ed 9

*here' 0 ..o for c 2< gd" 60 = cos-14g71,7C2 for c2>gdf and where K is the positive real root of (4. 12). It will be seen that (4. 13), (4. 14) reduces to (1. 24), (1. 26) respectively when dl, the uniform depth of the

water, is made infinitely large. _{From (1. 12), (4. 13) and (4. 14) we find}

that the wave profile for large values ofl x-h I or IL:I approximates to

(4. 15) tank ? 0 L >o , (4:16) e(31 - L< o L< o 7T/2 1 r r sin(KLcose )cos(Krsin0 ) J J psdhdk f K 2d0

irp g f2 90 _{1-K0d,sec 28 sech2 Kdo}

1 s Tr/2 sin(KLcos0 )cos(KFsin6)

ff

f

irpg n 90

1-Kod1sec 20 sech2 KdI

These expressions reduce to (1. 28) and (1.30) respectively when d, is made infinitely large.

From (4. 15) we see that the condition expressed by (1.16) is not

satisfied, and it becomes necessary to superpose a freely moving wave pattern in such a way as to cancel out approximately the free surface disturbance at a long distance in front of the pressure distribution, and

double the waves at a long distance to the

rear.

The term to be added to (4. 10.) must further-more by itself alone satisfy (1. 15), (1. 17) and

give zero_normal velocity component at the bed. A suitable term is

ac-cordingly' (4.14) with no limitation on the value of L. The velocity po-tential, may now be written

77/2 a)

Si' p dhdk S sects+ (19

f

cosh K(z+d ) sech Kd,

c Cl 0

2

K-K0sec0 tanh Kd, X sin( K(x-h)cos9 ] cos [K(y-k)sin KdK

r s 77/2 coshK(z+do) sechKd,

jf p dhdkf

1-K0d, sec2Osech2Kd

X cos [K(x-h)cos e]cos [K(y-k)sine] Ksec ede where K in the last term is the positive real root of (4. 12).

K2d 0

(4.17) ck.=

-2

1

- 33

It Will be seen that (4. 17) satisfies all'conditions and reduces to (1. 31) when d, is made infinitely large. A different method by'which the same velocity potential may be established i s shown in appendix III.

The corresponding waVe profile-for large values of I x-h I is now

o o and ff psdhdk 5 (4. 18) C (x-h) <73 -rrp g n 00 1-K a sec20sech2 Kdi o 1 ,

In order to find an expression for the wave resistance we make use of (2. 19) and (2. 24). We notice that (4. 18) may be written in the form of (2. 19) if we put

Qi .= -2

ff

s cosA' cos B'

-- - - K 2dhdk

-rrpg 0 2 , 2

1-K0disec usech Kd,

2 7T/2 _{sin [K(x-h)cost9kos [K(y-k)sinei K2 de}

Q2 2

jj.ps

einA' sin B'

7rpg n

1-K0d1.sec261 sech2

2ff s

sin A' cos B

Q3= - p

irp g _{1-Kod set. esech Kdi}

K 2dhdk dhclk -2 _{cos A' sin B'} 2 Q4

if

Ps _{K dhdk} 7rp g _{1-K disec20 sect'? Kd,}

where A' a Khcos 9. Khsin0 Substituting the expressions for the different Q -functions in (2. 24) we find that the wave resistance' expression may be written 1 7r/2 2 K cos

I

(PII+QII)

2d

g ea - 1,-K disec 0 p,ech2 Kd -(4. 20)

ifljj

.ps mc'13.:s1 .K(xcos _{0) dhdk} 'QII) _. S.111 J , ' (4.19) R= With

(4.22)

where IC is the positive real root of (4. 12) and where 0

o = cos

-11-7377

for c25 gd, and 0 9 for c2< gd, .

It will readily be seen that (4._19), (4. 20) reduces to (3. 10), (3. 11) respectively when d, is made infinitely large.

Both (4.19) and (4. 20) may he put in a different form when we

con-sider a pressure distribution which is symrnetrital about the origin. Let the pressure distribution be given by p p5(r) where r2= x2+ y2.

Substi-tuting in (4. 20) we find as in section 3 that Q. o and co

P = 2 7r 5 ps(r)rj(K0rsec2etanh Kci,)dr = 27r E(K0sec2etanh Kd ) where,_

E ( y) = f

p(r) rj(r y ) dr

0

The expression of the wave resistance -may now be written

4 77 72 [E(K sec2Otanh Kd,) ] 2

I

° K 3 cose d 0

g p 8= _{1-K0d, sec20s_ec1i2_K4,}

or by making use of (4. 12)'

477 e2 [E(Knsec2e tanh Kdo) )2

(4. 21) R J

sece d

'2 2 2 2 4

gsec 0 (c -gd sec OPEK c d,

Considering as a particular case the pressure system given by

(3. 20) we find that

-kofsec 2.61tanli Kf

E(Kosec2etanh Kd ).. Be

which when subs,tituted, in, (4.21) gives

417 B2c 77/2

e 2Kf K3secO

2 4 dO

gsec2 0 (c 2-gcl,sec219) + K

c d

This expression reduces "to (-3. 21) when d, is made infinitely large. The integral in (4. 22) does not appear to come out in tabulated functions and its value for different c 2/gclo and di/f ratios has

ac-cordingly to be found by direct,Au.adrature. Determining the wave

resistance R it will be found, in agreement with experimental re-sults, that the curve of R to a base of speed increases to begin with at a higher rate when the depthof the water decreases. A maximum in the different wave resistance curves occurs at speeds slightly

35

-the least depth giving -the highest maximum value. When -the depth of

the water i s fairly large there is a distinct seccind,maximum

corre-sponding to the deep water maximum. This latter hump always occurs at a lower speed than the maximum corresponding approximately to c= 1 gd, . As, however, the depth becomes lest the maximum at

ap-proximately c = 'E F1, becomes of increasing importance and the second hump can now only be detected by a slight flattening of the

re-sistance curve. It will also be found that the wave resistance decreases again after the velocity c = 4g-Cri has been passed, the rate of decrease being faster for the shallow water. This in turn leads to a smaller

re-sistance for the shallowest water and is partly connected with the lower limit of the integral in (4. 22) which is given by

-1

= cos rica77 for c > 4 gd,.

As the depth di

de-creases 00 inde-creases at these high speeds, its maximum value being

7T /2, thus reducing the range of the integral. For the two dimensional

case with Only transverse waves we will.accordingly get zero wave

re-sistance at speeds higher than rg7it due to the disappearance of these

waves.

From (4.18) we see that the wave pattern a long way away from the pressure distribution may be assumed to be built up of plane com-ponent waves as in the deep water case thus justifying the use of (2. 24)

in order to find an expression of the wave resistance. In order to show that (4.18) exhibit the usual features of the waves behind a body moving on the surface of fluid of finite depth we consider the particular case of

a pressure point of magnitude P situated, at the origin in a tniiform

-stream. Expression (4. 18) reduces now to

2

ZPK0 7T/2 sin(K I x tanhKd, sect? )

(4.23) -

f

o

(lx I,y) >o

Trp g 90 1 Kod, sec28 sech2Kd,

X

>ecos(KoytanhKdisin8 sec20 )sec4 9tanh2 Kdr de

This expression may be written in a similar manner as (1. 39) with w ( = i sin a (cot a cosh - sinh Zr )tanh_Kd, for the shallow water

case. Noting from (4.12) that

Ko sinh ZT tanh Kd, 1-K0df cosh2T sech2 Kdi

we find that the saddle points w1( T) = o are given by (4. 24) tane

nil

cot±,I( "1

a cot2a - 1-n

4 ₂

where n =2Kddsinh2Kdi . It will be seen that 4. 24) reduces to (1.40)

when d, is taken infinitely large. Again we get two values of e the dk

direction angle of the plane component waves, provided

cot/(1

2 4 >8(1-n) + n2 , the smaller root corresponding verse waves and the larger one to the diverging waves. line of cusps where the direction angle of the divergent cides with that ,of the transverse waves we have

2 2

(4. ) cot a 7. 8(1-n)/(11-n)

or cos

8(1-1')/(3-0 2

As o< n <1

whatever value K may take we must have o .4 cos 2a4 8/9

or ± 19° 28' a 17/2. The smaller value oflal is of course the limiting

angle for deep water, that is for d, co giving n o. Thus the two lines of cusps may for c 2 < gd, make any angle between 19° 28'- and 90° with the line of motion depending on the speed of advance and/or the uniform

A

depht d,_. Within the area bounded by these two lines the main part of the wave pattern occurs. Along the lines of cusps we have from (4. 24) and (4.25) ta.n0 =(---272 ) which for n= o. that is deep water, gives tan0 = 12/2 or 0 =350 16' and Which for n.1 gives 8=0

corresponding to C 2= gd, . Thus at the very lowest speeds the

trans-verse Wave systems are made up of plane component waves whose direction angle in the plane bounded by the negative half of the axis

of X and the positive half of the axis of y varies between 00 and

35°16' . As, however. the speed increases, this latter limit of 0

decreases continually until the direction angle 0 becomes zero when

c 2= gd we shall, however,

return to this point later.

The expression given by (4:24) represents really the envelopes of the straight crests (or troughs) of the plane component waves which makes up the wave pattern. The family of straight lines, corresponding

to the crests of the plane component waves of the travelling wave system

may in this case be represented by

(4.26)

x dose -

sine?

where K is the positive real root of (4.12) and N

1,2, 3 ...

. Finding

the envelopes in the usual manner by differentiating (4.26) with respect to 0 and equating to zero we find that we may write

(4. 27) x = N ZIT.sec0 2-(1+n)cos203 N 277- 2-qm(1 +n)-K _{1-n K} crFn 1=n

127r

sin 0 N 4-7:7Fri- (_1+n)

1-n-

- - - K 1-n of course to the trans-Along the waves

coin-where m =tanh Kdy/Kdi and q= Kod, . It will readily be seen that (4. 27)

and (4. 28) saiisfies (4. 24), and when d, is made infinitely large reduces

to the expressions for x, y given in section 1 for deep Water. From (4. 27) and (4. 28) we find that

(4. 29) cos2

a -

(1- 1-qm (l+n) ]2 1- 1/4 qm(l+n)(3-n)

for any point on the envelopes. Substituting (4. 29) in (4. 24) we find that the direction angle of the plane component waves building up the divergent

-and the transverse are given respectively by

(4.30) tan

(4.33)

1 1-qm qm (1-qm) and "(4.31) tan 8 2 - cim (1-n) qm(1-qm)

Along the line of cusps where the divergent and the transverse waves coincides we have 01 _{= 02} or

(4. 32) m(3-n) 2/q

As, however, the greatest possible value m(3-n) can have is 2 we must

have q > 1 and only in this case is there a double wave system consisting

of transverse and divergent waves with a line of cusps. .

From (4.32) we note that when q decreases to unity, that is c increaseS

to _{gd, , both in and n approach their. limiting value of unity along -the line} of cusps. As, the cusp _{angle is given by (4. 25).-we see that this angle widens}

out and reaches its limiting of n/2 when c= IF thus the waves at this critical speed are situated on the, whole of the half.plane to the rear of the

-.disturbance._ For those component waves whose direction angle 0 iS zero we have from (4.12) K= Kotanh Kdi or m = 1/q = c.2/gd, .- As the wave

,lenght is giVen by-k = 2 Tr/K. we-have for. these cOmlionent waves

tanh 277di/X Zwd,/k

37

-, sin2a =(1+n)2 (1-qm) qm

. 4-qm (l+n)(3-n)

Thus as q decreases to unity, the wave lenght A increases indefinitely. We have seen before that the direction angle 0 along the line of cusps is zero