Notes on the theory of ship waves

Bengt Joel Andersson

1975

Hydromechanics

The Royal Institute of Technology S-lOO 4 Stockholm 70

Introduction ₁ Velocity Potential for Surface Singularities 5

Velocity Potential for a Source 11

Partition of the Fundamental Solution 14

Surface Elevation and Waves Generated by a Source 18

Surface Waves Generated by a Point Forcive 35

Diagram I ₃₆

Diagram II and III ₃₇

Table 1 and 2 38

References ₃₉

ACKNOWLEDGEMENT

The author is very much obliged _{to Mr OWE SUNDSTRLSM, assistent at the} Division of Hydromechanics KTH, for his careful checking of the

1. mt rod u ct ion

In the linearized theory of ship waves on the surface of an ideal, homogen-eous fluid the ship hull is often represented by a distribution of _sources (sinks) and dipoles with the fundamental solutions of Laplace's _equation adapted to the given condition of pressure at the free _{water surface.} In this paper we shall study the classical steady-state surface _wave

problem for a ship in translational motion at constant velocity _{a in water} of constant depth d < _{The perturbation disturbances due to the ship} motion only are defined by the limit t -, of the solution, corresponding to the fluid at rest for the time t < 0 and a ship velocity o(t) being a "sufficiently good" function, such that c(t) = 0 for t < 0 and c(co) = c (Cf STOKER (1957)). _{In particular, we shall derive the velocity potential} and the surface waves for fundamental solutions, corresponding _{to a} sub-merged source and a concentrated point "forcive" _{on the water surface.}

The diverging angle of the wave pattern _{is determined and we shall find} a small difference from classical results by HAVELOCK (1908). _Asymptotic expansions of the waves at large distances from the disturbances are

deduced and may be considered as generalizations of formulas, given by HAVELOCK (1908), HOGNER (1923, 1925), PETERS (19+9) for a point forcive and infinite depth.

There are well-known general methods for _{construction of long asymptotic} expansions for the wave amplitude. This is demonstrated in this paper.

However, the intention is the presentation _{of a relatively closed theory} which covers a relatively wide class of problems concerning fundamental solutions in the theory of ship waves. Of course, such research in a classical field can, perhaps, be considered _{as variations of well-known} topics.

Boundary Conditions and Basic Equations

e

n

Fig. 1.1 The frame of reference

Let {x} be Cartesian coordinates, fixed to the ship. We write the position vector

where {} are the base unit vectors. We choose and horizontal and e3 vertical and directed downwards. The undisturbed free water surface defines the level S = 0.

We assume that the ship moves with the velocity - relative to the undisturbed water. We presuppose irrotational flow and write the potential of the fluid velocity relative to the reference frame

c(t)x1 + 4(t,) _(1.1)

where is the potential of the perturbations due to the ship. Then, by Bernoullis theorem

2

(1.2)

Si

where p is the pressure, p is the fluid density (constant) and g is the acceleration of gravity.

The potential is a harmonic function. Assuming translational motion, we have the following boundary conditions.:

On the wetted surface of the ship the velocity vector is orthogonal to the surface normal e. Hence

= - c(,1) when IeI

1 _(1.3)

On the free surface 53 = r(t,x1,x2) the pressure is constant, say p = 0, and a point on the surface remains on the surface. After classical linearization, assuming "small" amplitude of the surface waves, this statement is expressed in the boundary condition

c4- gc

0

Ic + cç ₅₃

Hence, if t is eliminated,

(a/at + ca/ax1)2

- ga/ax3 = 0,

S3

-* + 0

(1.5)

3. At the bottom of the sea, x3 d, we have

=0

x3

We shall only treat the case d = constant.

(1.6)

The surface condition (1.4) is modified if, in the mathematical model, singularities are distributed over the plane x3 0 (see section 2). HOGNER (1923) studies a flat ship, assumed to be equivalent to a pressure distribution over the originally undisturbed water surface, and he

uses the linearized boundary condition also on the wetted surface of the ship. Then, the boundary condition (1.3) leads in general to.an

integral equation for determination of the surface pressure. It is not evident that this equation, of singular type, has a realistic solution. For very small Froude numbers (small velocity c) the difficulties _are over-come, in a first approximation, if we set the pressure on the wetted sur-face equal to the known hydrostatic pressure.

It cannot be expected that the three boundary conditions are sufficient for defining an unique solution of the steady state problem. There are

phenomena, not involved in the boundary _{conditions or the equations but} necessary to consider at the evaluation of the wave resistance. For example, at large Froude numbers the impulse _{due to the horizontal} component of the resultant normal force _{upon the wetted surface of a} flat ship (planing boat) _{can be of the same order of magnitude}

as the change

of

momentum in the

_spray,

_{generated along the front part of the} waterline. In the linearized mathematical model, this spray generation is represented by a distribution _{of sources (sinks) along the waterline.} There are phenomenological as well as mathematical reasons for _{a weak} conformity with the three-dimensional theory of thin airfoils, implying a distribution of singularities upon the "trace" of the ship on the free surface.

KOCHIN (1937, 1951) has studied the case of an immersed body, d _He assumes that the velocity potential is _{represented by a source} distribu-tion over the body surface. _{The boundary condition (1.3) gives an integral} equation of the second kind. _{For Froude number sufficiently}

small, this equation is of regular Fredhoim type and, thus, the solving _source distri-bution exists and is unique.

an open question. BRARD (1972) gives a review of the situation from a mathema-tical point of view. GUEVEL, VAUSSY, KOBUS (197L) discuss the necessity _{of a} singularity distribution on the waterline.

The attempts to find solutions of the ship wave problem in the form of distri-butions of sources and dipoles (single- and double-layer) _{are related to the} classical methods, used for solving the problems of Neumann and _{Dirichlet, and} are closely connected to the use of Eulerian coordinates. The alternative _use of Lagrangian variables is discussed by WEHAUSEN (1969) and may be a fruitful way for a further analysis of relevant singularity distributions.

Let the fluid motion relative to the reference frame be _{generated by a} pressure distribution or "forcive" p pq(t,x1,X2) and a source distri-bution k(t,x1,x2) on the originally undisturbed water _{surface. We assume} that q = k = 0 for t < 0.

Then, the linearized surface boundary conditions are

Then

+ = - q

+c

_{=+ -k}

t x

S3 - + 0

Firstly, let q and k be "sufficiently good functions". We use the notation of the two-dimensional Fourier transform

f(x1,x2)

=

JJ

+

Then, also depends on the variables 53 and t. Since _{is a harmonic function, we obtain}

633-

23 _{= 0}

where 2 = +

The boundary conditions (1.6) and (2.1) give

53

= d: ₆₅₃ = 0, 0: +

_icF13

-

=

-+ 1 ₌

Moreover, we have the initial condition ₃

= = 0 for t = 0. By

straight-forward solution of ordinary differential equations, we can now determine the transform _{3 and, using the Fourier inversion formulas, we obtain the}

potential . Set

K(x1,x2,x3,t)

(2)2Jfv/g/tanh

d sin(Vg tanh d t)x

(2.2) + t where

_{y1 _J:t)tv.}

= JdtJJ _{(k(t-c,y,y2)X(x1-y1,x2-y2,x3,t)} 0 + j1q(t-t,y,y2)K1(x1-y1,x2-y2,x3,t)1dy1dy2 (2.1) I

(2.3)

The validity of the solution (2.3) is extended to generalized _functions q and k.

The steady state

solution is defined as the limit $(x) = $(oo,x) for functions

q, k

and c, independent of t for t > 0. The assumption of _{a step change at} t _{0 simplifies the analysis, which can be applied equally well for a} continuous start.

The velocity potential for unit a0ure at the origin,

_{k =}

_{6(x1)t5(x2), becomes}

= j

K(x1-cT,x2,x3,t)dT, c constant

_(2.)

In general, a velocity potential is determined modulo an arbitrary function of t only, which appears in the formula of Bernoullis theorem _{but here, as} usual, neglected in the boundary condition (2.1) as well as in (2.').However, it is convenient to fix the potential level before making _{the limit t -+}

and we assume here that = 0 in the point (O,O,d).

Set V =

gc2

_{and define dimensionless coordinates, depth., time}

and frequencies by

svx1, yvx, Zvx3; Dvd, T-vct;

_a-V1,

8-v2

_(2.5)

and write in polar coordinates

aAcosO

,

BAsine

With these notations we obtain after the integration in (2.&1)

= vG(x,y,z;TID)

7112

(x,y,z;TID) = (2,T2Y1Re fdef sech(AD){cosh(A(D-z) ]e -

i} x

-1112 0 -i(a-A)T i(ct+A)T1 A

l-e

a-A

_j where - -. _IdA

a+ A

(2.6) (2.7) A = VA tanh AD _(2.8)

The integrand is analytic in Re A > 0 for real 8. Applying Cauchy#s _integral theorem to the inner integral, we shall choose a path of integration such that we secure the convergence under the integral sign when T -' .

Firstly, we assume D . Then

1112

_1 _-i(a-V5)T

1

C = (2it2Y'Re _jdOj1 '

_e

_A2+i(D)

VAIl

-21 a -

-jdA -W2 0

Set r and A a+ii, a and

t

real. Then

Re[Az-i(x+y)] > 0 in D = {A;Iarg xi < arctan z/r}

Re

i(ct-V5) >

0 in D {A; T(\/a2+12 + a - -sec2O) < o}

Re i(+VA) > 0 in D2 {x; i < o}

where D is the smallest set, independent of 0 and the sign of x and _y. The boundary of D1 is the real axis and the parabola a -sec20 - t2cos20.

Fig.2.l The regions D, D1 and D2 in the complex A-plane

Let L1 be a contour OABC from A = 0 to A in D1 fl D and throu _the

boundary point A = -sec20 as defined in Figure 2.1 b. Assuming as we may that IdA/dai = 3 on AB and arg A = ± --arctan zir on OA and BC, we obtain the estimate

_Az+i(cix+y)dA 2/c2+y2+z2 _-azI2

da z e A L1

which is integrable over {(a,e); a 101 < ir/2}. Since

I -i(-V)T

< 1 and + 0 as T -

=

for A L1, A * 0, *sec20 the Lebesgue convergence theorem and Cauchy's theorem imply that

i2 1T12 -Az+i(ctx+8y)r

-JJ

t,e_Az+i(ax+y)

1im1d0I

1-e

- - dx Tt' _1112 0 -1T2 L1

Similarly, if L2 is a path in D2fl D as indicated in Figure 2.1 c,

1fl2 lim 1d01 -Az+i(czxty) _i(cz+/X)T]dA _JdOJ _Az+i(ax+8y)

[-e

dA -1112 o _-1112 L2

Applying Cauchys integral theorem to the remainder after the limit T -we observe that the path L1 can be

continuously deformed into

L2

without

passing the pole A

Ac) =

sec2O. We

-f

-L obtain a path of integration L in the A-plane, common for all s, y and z,

12

S..

. if L2 is deformed into the positive real axis with an arbitrary small Fig.2.2 The path L in the A-plane

displacement out into Im A < 0 in the neighborhood of the pole A = A0 as shown in Figure 2.2. Thus

flj2

-Az+i(ct.x+By)

(x,y,z;ccicø)

=

(2,T2)'Re

Idef

cos2O -

dA (2.10)

-'12 L

This function is harmonic and satisfies the equation

G

-_(21TY'z[x2 + 2 + z2]312

(2.11)

We next consider the case

D < .

Omitting space-requiring details of

almost elementary character in the calculus, we shall derive the steady

state solution by the same method as used for

D

Given ,y,z,D and an angle 4, < ¶12 and < arctan zir, there exist finite constants C, > 0 such that

Isech(AD)[cosh(A(D-z)]e'

-ttanh ADJ > C3aD/(1 + aD)

for A = a+it

D()

= {A;larg Al < 4,]. These estimates guarantee the absolute convergence of the integral (2.7) for T < .

The roots of the equation a2- A2 A(A cos2O - tanh AD) = 0 _{lie on the} imaginary A-axis except, for cos2O < D, two roots A ± A0(6), which are real. The positive root A0(e) is an increasing function of sc28.

Set f (A,8) a - A , f2(A,&) a + A and define

D.(e) {A; Re A > 0, Re if.(A,O) > o},

j =

1, 2

The boundary of D2 contains the lower border A a - iO of the positive real axis. The boundary of D, contains the upper border A = a + iO for 0 < a < A, and the lower border for A, _{< a, where AJO) is root of the} equation _{f, /A} = 0. Then 0 . A1 A0 with equality only for D < cos28.

Given 8, the path of integration in (2.7) over the positiv real A-axis is now deformed into smooth curves L.(e) in D.(e) for each of the two terms

J J

in the parenthesis. Then, if D > cos28, L(0) must pass through A on the real axis. Further, the minimum distance from L.(e) to A

may be > C1A01, C being a sufficiently small positive constant, indepen-dent of 8. Applying the Lebesgue convergence theorem, we obtain after a final deformation of the path of integration

(x,y,z;ID) (22)'Re JdeJ [A cos28 - tanh ADY' x

1rj2 L (2.12)

x sech(AD)[cosh(A(D_z))e

-where L is the positive real axis with, if D > cos2O, a displacement out into Im A < 0 at the pole A = X0(e) as shown in Figure 2.2.

In the following we shall also use the brief notations

(x) = G(x,y,z) = (x,y,z;cID) (2.13)

where

=

= s+y2+Z3

is the dimensionless position vector.

The velocity potential for a

delta function pressure distribution

p = (s1 )5(s2) on the surface x3 0, acting at the origin, is derived

from (2.3). The solution for steady motion is defined by

= lim Q(x1-ct,x2,x3t)dt,

Q(,i)

t-*

0

Repeating in principle the deduction for a source, we obtain

v(pc)1(s,y,z)

Hence follows that a pressure distribution on the surface x3 = 0 is equivalent to a dipole distribution with the moment parallel to the

x, -axis.

The functions p(x) and (0cY'(x) are fundamental solutions for the boundary conditions (2.1) at steady motion and appropriate conditions at infinity. We shall here give a general solution, obtained by convolution, in dimensionless form. Besides the dimensionless coordinates we shall

also introduce dimensionless potential, surface elevation, pressure and source intensity

Z(x,y) = v(x1,x2)

(2.

l'4)

11(x) = (pc2Yp(),

_{K(x,y) =}

_ck(x1

_,x2)

and set 11(x,y) = 11(x,y,0). In the steady state problem the boundary conditions (2.1) are formally reduced to

dimensionless form g -q X3 01-cc - =

-k

SI 53 tials as

-We define the two-dimensional convolution (F1

* F2)(x)

JJ

F1

(x',y')F2(x -

x',y - yt,z)dxtdyt

Then we obtain

0

(11* G)(x) +

(K * G)(x)

(2.16)

(2.17)

This formula agrees with the usual presentation of solutions of linear differential equation problems. However, especially the second convolu-tion integral implies harder restricconvolu-tions on integrability of K than are satisfied in many actual problems. If K IS not integrable but has a

Fourier transform (i.e. K =

[a2+x2+y2)

1/2) we can again define the steady state problem as the limit T - at the corresponding initial-value

pro-blem. Then we start again from the boundary conditions (2.1), here in dimensionless form and assuming c constant,

ci) + ci) - Z - TI , Z + Z - 0 = -K for z = +0 (2.18)

T x o T x 2

with the initial conditions

= = 0 for T = 0.

Here fl and may be independent of T. The solution (2.17) shall be interpreted in this wider

sense. The procedure is applied in the next section.

For the initial-value problem the usual Fourier transform methods can be applied without trouble for any finite time interval and the potentials

(t,x) are good functions of x for t < . As t 9 the last property is lost, however, and the Fourier transform at t = c will be a generalized function with a "mass"-distribution on the curve = v tanh D and, if D < , singularity in 0, which corresponds to weak decreasing

poten-ci>

- Z = -TI

5

0

z - 0+ (2.15)

Z - -K

3. Velocity Potential for a Source

We assume steady state conditions. The velocity potential for a unit source at x x 0, x3- x; 0 < x< d, is denoted

4(x) = vG(s,y,z:z') ; z' vx (3.1)

where G is adapted to homogeneous boundary conditions at z = 0 and z D.

Then the dimensionless velocity potential G(x,y,z:z') has the forni G(x,y,z:z') _(Liri' [x2+y2+(z_z)2 ]h12+

regular harmonic function We define the Coulomb potential for unit sources at {(0,0,z'+4nD)},

((0,0,2D-z'i-4nD)) and unit sinks at {(0,O,-z'+nD)}, {(0,O,2D+z'+LnD)}, h(x,y,z:z') -(4Y' (_l)fl{[x2+y2+(z_zt_2nD)2 )-t/2

- [x2+y2+(z+zI+2flD)2 j_l/2j

Using the Fourier transform formula

(x2 ₊ + z2)_1/2 (21T)_1JJ e dcsd8; 2

we obtain after summing

h(s,y,z:z') = (2w)_2JJ e

AlZZl

-e

-A(z+z') + -AD

1- 2e

_{sech(AD)sinh(Az')sinh(Az)]e'/2A dad8}

Set

G(x,,z:z') = h(x,y,z:z') +H(s,y,z:z')

(3.3)

The function

H is

harmonic and regular in 0 < z < D. Since h for a 0,

we obtain the boundary conditions for H,

H

-H -h

forzO;

H =0

forz

xx z a a

where

= _(21r)_2JJ

_{sech(AD)cosh[A(D_zt)]e8dad8}

Then in view of (2.17) we obtain formally H(x:z') _{(K * G)(x) with}

K(x,y)

= -

h(x,y,O:z'). However, for D < the integrability is insuffi-cient. Then we define the convolution integral as the limit of the solution of the corresponding initial-value problem when t

(3.2)

Let (T,x) Gt(T,x:zt) be the velocity potential in the initial-value problem. Then satisfies the boundary condition (2.18) for K _ll = 0 and thus, after eliminating the surface elevation Z,

Gt +2Gt

_+G

_Gt0

_forz+O

TT xT xx z

We have the initial condition = _{= 0 for T < 0 and the bottom}

condi-tionGtoforz=D.Set

Gt(T,x:zt)

_{= h(x:z')E(T) +}

Ff(Tx:zt)

E(T) = 0 for T < 0 and

E(T)

= 1 for T > 0

Then is harmonic and regular in 0 < z < D. We obtain Ht(T,x:zt)

_(2w)JJ

sech2(AD)cosh(A(D-z')]cosh[A(D-z)] x (3.4) 1 -i(a-A)T _1.(u+A)T1

eir)dad

1-c

1-e

+A

_J

For z' 0, we obtain by comparison with (2.7)

Ht(Tx.o) - Ht(T,xo:O) _(x,y,z:TID)

where = (O,0,D). We now define

H(x,y,z:z')

urn

[Iit(T,x;zI)

-

Ht(T,xo:zI)]

and obtain

2

H(x,y,z:z') = (2112Y'Re JdOJ [A cos2O - tanh AD]' x

(3.5)

_7112 L

x sech2(AD)cosh[A(D_zt)][cosh[A(D_z)]e D13y) - 1)dx

with a pole on the real A-axis for cos28 < D. Hence, for D =

1T 2

-A(z+z' )+i(ctx-i-y) e

G(x,y,z:z') -

()'(r1

r'] +

(22)'Re

fdef

Ao28

- dA

_1112 L

(3.6)

For D < , the solution is often given in the form

G(x,y,z:z') = _(41tY'[r;l + r' _{- H'(x,y,z:z')]}

r x2 + + (2D-z-z')2

and we obtain from (3.3) and (3.5)

1TI 2

2

Re [A cos2O - tanh AD]1 x

_(3.7)

-I2 L - AD

x{e _{sech(AD)cosh[A(D-z)]cosh[A(D-z')](l+A} cO520)ei Y) _{- C(A)}dA}

with C(A) = sech2(AD)cosh X(D-z').

Making z' 0, we obtain

G(x,y,z:0) (x,y,z)

G(s,y,z:0) (x,y,z)

(3.8)

Thus, the fundamental solutions for surface singularities are simply derived as particular cases of the solutions for submerged singularities.

KHASKIND (1945) and LUNDE (1951) have derived formulas for the velocity potential for a source, which in principle are identical with (3.7) for

C(A) 0. Therefore their integrals are divergent. WEHAUSEN and LAITONE (1960) give a revised but incorrect form of Lunde's formula. This is later rectified by WEHAUSEN (1973) to a formula of the type (3.7) with C(A) 1. Then the convergence of the integral seems to be at least conditional.

However, a formula given by KOSTYUKOV (1959, 1968) is easily transformed to (3.7) with C(A) = sech AD, and the difference is an insignificant constant.

Set

x = r cos p, y r sin p (L.l)

Let

r

be a contour in the complex A-plane and set for the sake of brevity

F(6,A:r,W,z,z') = cosh[A(D_zt)]sech2(AD)[coshtA(D_Z)]e'

cos(O-)

I(0:r,,z,zt)

(27T2)1Re F(O,A:r,(.2,z,z') _dX

A -

sec2O tanh AD

(4.2)

= 2A0D

csch 2A0D

(4.9)

In (4.8) we set A0 E 0 for cos28 > D.

Then, by (3.3) and (3.5)

ITI 2

G(x,y,z:z') h(x,y,z:z')

_{JIL(e:r(2,zz)sece}

do

(4.3)

where L as before is a path of

integration from A

0 to A

o

¼

as shown in Figures 2.2 and 4.1.

L

The singularity to be avoided is

the eventual positive root

Fig.'4.l A A0(e) of the equation

A -

sec28 tanh AD

= 0 (4.4)

By (4.3) we obtain

?r12

G(-x,y,z:z') = h(x,y,z:z') + JIL (O:r,,z ,z')sec20 dO (4.5)

- 2

where L is the image of L at reflection in the real axis. Then I - I L

is reduced to the integral round an arbitrary small circle with centre A0.

Hence, by the residue theorem

G(x,y,z:z') G*(x,y,z:z') + Q**(x,y,z:z') (4.6)

Q*(x,y,z:z') = G(-IxI,y,z:z')

(.7)

G** 0

for

x

< 0 and for x > 0 itf 2 G**(x,y,z:z')

_LIm

J

F(O,A0:r,p,z,zt)(l - Y'sec2O dO (4.8) -7r12 where

In particular we obtain for D = , x > 0

1T12

G**(x,y,z:zt)

= - 1-

Im

_f

e_SecO(z12_

CO5(0_]sec2O

dO

2

We shall derive useful expressions for G* only in the case D = . The methods are readily generalized to the case D < , and we shall give

the results, when we need them.

Because of symmetry we may assume y 0. For D

= =

we have G*(x,y,z:zt) = h(x,y,z:z') + (-lxI,y,z+z')

where, for x > 0

¶12

-A[z + irlcos(O+p)I) (-x,y,z) =

(2n2YRe

e _{A cos2O - i} dA +

7112 L _1I+ , f 2 -sec2O[z - ir

_{cos(8_p)]sec2O}

dO +-i-Im e in - 1!12

The path of integration L may be deformed into the half-line arg A = arg(z - irlcos(ep)!] in Im X< 0. By an obvious change of

variable, interchanging the order of integration and introducing the complex variable w =

pe8,

p > 0, we obtain

-s

G = (712i)_1J e ds

8w+2y ix)w3+2

2z)w2 2(y+ix)s

(4.13)

The

denominator has exactly two zeros in

Iwl < 1,

and the inner

integral

may be evaluated by the residue theorem. In particular, for y 0 we have ₂ = 0 and obtain

IxIe

de

(4 14)

4x2+(z_s)2](x2_2sz+2s22av/x2+(z_s)2]

We note that the form (4.12) with (4.13) of the expression for Q* may in its details depend on the introduction of polar coordinates according to (2.6). If our aim is to obtain a theoretical description of the surface wave phenomena, this choice of variables is natural since it is closely connected with the use of cylindrical coordinates r, p, z. However, the surface waves are essentially located to the region x > 0 and we may consider the term G* in (4.6) as negligible compared with G**. Parallel to the use of cylindrical coordinates we are also interested of descrip-tions and estimates of the perturbadescrip-tions in terms of cartesian coordinates.

(4.10)

(4.11)

PETERS (l9L9) has given an alternative formulation of the solution of the surface wave problem for a concentrated surface pressure at steady motion and

D , corresponding to the fundamental solution _{The surface elevation in} front of the singularity is derived in cartesian coordinates, and Peters has also shown that his formulas are not obtained by trivial transformations _of classical formulas, based on the use of cylindrical coordinates.

We shall now state that PeterS' solution lies within the range of the initial-value problem and may be generalized to the problem for a source. We write (2.9) in cartesian coordinates

(x,y,z;TIct) =

(2ir2)

1ReJ

e1a8J

e . .]dct

where (...] stands for the parenthesis in (2.9). The inner integral may be considered as a line integral in the complex ci-plane. When T -* , we obtain by arguments parallel to those in section 2

- Az+z a. e (x,y,z) (2n2Y'Re

J

8YJ

2 dci

ct-A

- _L

where L is a path of integration as defined in Figure 2.2, here avoiding the

pole % vi +

. on the positive real axis.

For x < 0 we may deform the path of integration L into the negative imaginary axis and obtain, with ci -it and obvious changes of variables and path of integration, the identities

izv't2_82_ tIxI e (-IxI,y,z) = _(21r2Y'Im

f

eJ

2 1t282 dt - tI - ,x1V12+82 -Ydy , I e

_(22)'Im

f

e dBJ y2 82 - iy ₂₊₈₂ - 0 iT COSO dO (12i)_1J _{e_SdsJ}

s-(y

sinG +z cosO )cos O-ilxlcosG 0 2

l+W

_dw =(2n2)_1f

_e8ds

(z_iy)w+2iIxIW3+2(Z-2S)W2+2iIXIW+Z+iY 0 _fwf=l

For y = 0, we obtain simply the result (L.lL). Since C is an even function of y, it follows immediately from Schwarz reflection principle that the formulas (L.12) and (L.l5) define the saie harmonic function.

We have

(4w)_1(r -

r'] =

(22)1f

cosBydJ e_22 .

SI_fl SI_fl. Z d

0

Then from (4.11), using the second formula (4.15) and introducing the harmonic function

M(x,y,z:z') =

_(22)_1f coseyd8j e_t22

x

-w 0

(y2+82)sinya' + YCOSYZ' . dy x

(y2+2)2 + SIflYZ /i2+B2

we obtain

G*(s,y,z:z') = M(x,y,z:z') + M(s,y,z:z') (4.16)

The function M is a Fourier sine transform and the function M is a

xx z

Fourier cosine transform with respect to the variable z. This formula is arranged for an immediate application of Parsevals theorem to the evalu-ation of integrals of quadratic forms of the velocity components.

For x > 0 we may deform the path of integration L into the positive imaginary axis in the ct-plane. Then, by the residue theorem we have a contribution from the pole a = a0 and obtain the partition formula (4.6), and making the simple change of variable = sinO sec2e we obtain the function G** as defined by (4.10).

Differentiating (4.16), we obtain an expression for G*(x,y,z:O), which, disregarding the notations, is identical with Peters' fori4a.

It should be mentioned here that Peters' transformation is also proposed by HOGNER (1925), who found it "probable for physical reasons".

5.Surface Elevation and Waves Generated by

_{a Source}

By (2.15) we obtain the surface elevation for a-source in (O,O,z')

Z(x,y:z') _{= G(x,y,O:z')}

Following the partitioning in section 4, we write

z z* + z**

where

Z*(x,y:zI) _{G(x,y,O:z'); Z**(s,y:zt) = G*(s,y,O:zt)}

Then Z is an odd function of x,

Z*(_x,y:z) = - Z*(s,y:z?)

and may be discontinuous at x = 0.

For x < 0 we have Z _{0. By (4.8) we obtain for x > 0} 1112

--

J

(1-riY'sec0 Aosech(X0D)cosh[Ao(D-z')]cos[w(e,u?)r]de

-1112

where we have introduced the frequency

A0cos(0-)

₍₅₄₎

For large values of r, Z is the dominant part of Z and contains in its

analytical structure the requirements for an explanation of the characteristic wave motion behind the singularity.

Let us firstly estimate the magnitude of the surface elevation in front of the singularity for the case D = co. In the two last formulas (4.15) the inner

integral can be evaluated in a straightforward manner using residue calculus. The poles in < 1, or, after the change of variable u tan , in the half-plane Im u > 0, are obtained by solving an equation of the fourth degree. Then

11

cos e

do

(w2i)UJ

s-(.ysin 8+ zcos @ )cos 0-ilxlcos 8 -11

(2112)'J

IxI

(a(1tu2)-(z+yu)]2+s2(1+u2)

du = f(s/r,z/r:.p)/r

where f is analytic and regular in s/r zir = 0. We obtain the power series

(5.1)

(5.2)

Hence

G(-Ix,y,z) =

(2,r)

₊

_{(1 - 2cos)r2}

+ O(r3)];

_r

Then, from ('4.11), we obtain for D =

1 S

Z*(x,y:z') - (277)

(x2y2)3'2 ; r

-*

f(a/r,zlr:p) = (2JT)(1 + (1 - 2cos2y)slr - z2/2r2 +

+ (2 - 3sin2p)sz/r2

+ (4cos'4cp + 3.5sin'4Q -

2)s2/r2 + ]

For D 00, we obtain from (4.3)

Wj2

cosh[A(D-z' ))sech(AD)i(czx+By)AdA

G(x,y,O:z') = _(2it2)'ITn

J

sece def A - sec2O tanh AD

-1112 L

_(2,i2)11m

J

ieiJ

ci cosh(A(D-z')]sech(AD)

icix

ci2 A tanh AD

e dci

- L

If B is real, then the non-real zeros of the denominator ci2- A tanh AD lie on the imaginary

cx-axis. If x < 0, the path of integration L in the ci-plane may be deformed into the negative

imaginary axis, avoiding the poles by following small semicircles in Re a > _{0 (Fig.5.l). The}

poles lie on Im ci<-iBI. After this change of the path of integration, only the semicircles give contributions to G5. By the residue theorem, using the variable y V'1a12_62, _{we get}

G5(-Ixl,y,0:z') = -(2w)'Re

Jysec(yD)cos[y(D_zt)]e_V82jdy/B

E

(5.6)

B = (y tan yD 2)l/2 ; E > 0, tan yD - y

> oJ

The set E is the union of intervals = ]y,(2n_l)W/2D(, n = 1, 2,

where y = tan (n-1)ir/D y < (2n-1)w/2D. Then 0 for D 1 and 0 _< < w/2D for D < 1. Further

(2n-l)it/2D - y - 2/(2n-l)n; n - 00

Fig. 5.1

For n we have the inequality

JIsec(yD)Ie'ydy/8

E <

n

Hence follows that the integral (5.6) is absolutely convergent. The asymptotic behaviour is essentially determined by the integral over the first interval E1, and for this evaluation we apply the classical method by Laplace. Then, we introduce (y-y1)/r as variable and make the limit r -' . For this technique we

refer the reader to WIDDER (l96). Thus, for D > 1 Z(x,y;z') (21r)_1

Dx2+(Dny2

r and for D < 1 1 / Z*(x,y:z?)

_sgn(x)(2n)

Dl+-

(cos + y1sin y1z')

-1/2

[

D(l+y)(2x2i-y2) + (2s2_y2)1 X exp 2txI(D(l+y) + 1)

j

; r

+

When x = 0 we obtain

Z(O,y:z') = Z*(_O,y:z') =.Z**(+0,y:z1)

in view of (5.1) and (5.2). Hence, if D

Z(0,y:z')

= -

-z' sec26cos(y sec2O sinO )sec3O dO

When z' = 0, this integral is identified as a Hankel function

Z(0,y:0) =

Hence

Z(0,y:0) .- -

_e

_'2/vTii

_{; lyl (D}

_{= o)}

(5.10)

In the general case a' 0, we rewrite (5.g) as an integral with respect to the variable A = A0(e) = sec2O,

Z(0,y:z') = -(21T)1Re

f

_Az'+iVA(A_l)JYIAdAIVA(A_l)

1

(5.7)

Fig. 5.2

A

=

4 + iT

1/2

We apply Cauchys integral theorem and choose the path of integration in the complex A-plane as shown in Figure 5.2. Then, we have no contribution to the integral from the segment

[f,

1] on the real axis. We also notice that the half-line

(3,+

i] can be deforned into contours from A = to A within the region

{ A; Re A > 0, Im[A(A-1)]'

o}.

Thus

Z(0,y:z')

=

-(2)'Re

J

0

Making the change of variable t = TVii, we obtain

Z(O,y:z')

(2j1)_1_(Z+tYI)/2

J

e dt 0 -(z'+I - - e

'12/i-i

; iyi-in agreement with (5.10).

Similarly, if D < , we integrate (5.3) with respect to the variable A0

over the interval 9 < A0 < , where tanh 2D = 9 > 0 for

D > 1.

_Repeating

the evaluation technique as demonstrated for D = , we obtain for

D > 1

Z(0,y:z')

-

(l+v)3tanh A'D 1I2 -1/4

k Il-vt + 2v2sinhLA?DJ I1-v2J x

x

sech(AD)cosh(At(D_zt)Je_th1h1_V21

_IYI/2/V

where A' is the positive root of the equation d(A tanh AD _{- A2)/dA =} tanh AD + AD sech2AD - 2A = 0 and v = 2A'D csch(2A'D).

If 0 <

D < 1,

_{we obtain the asymptotic formula by replacing A' with the} root -' [0,

_n/2D]

of the equation d(y2 - y tan yD)/dy = 0, replacing tanh A'D with tan y'D, sinh2A'D with sin2y'D,

sech(A'D)cosh(A'(D-z')] with sec(y'D)cos[y'(D-z')] and we set v 2y'D csc(2-y'D).

In particular, if D-1 = o(1) and D = o(1)

Z(0,y:z')

__(4 tD_lI_1/2e_ID_1WIYI/2//lliyI

1

Z(O,y:z')

--(2D)[cos(z'/2D)+/2/Dsin(irz'/2D)]

e

tyh12D -

_/Vrrlyt; (5.11)

We next consider the elevation component Z**, defined by (5.3).

Set C = p 2 0. Then, by (14.4) and (5.14), we obtain for p = 1, 101 < w/2

and these formulas define analytic continuations of the functions A0(C) and w(C) to the whole C-plane. Then (the bar denotes complex conjugation)

= A0(-C) = A0(C')

A0()

(5.13) w(C)

= - w(-;) =

Since dO = (iC)'dC dA0/dC

= -

2A0(l-n)'(c+C'Y'C1(c-c1)

we can write (5.3) iwr =

-1Re

ii

(C-C')'sech(A0D)cosh[A0(D-3')]e (dX0/dC)dC JI,

c0 = {c; C =

e0,

it < ir/2, cosO <

V}

Using Cauchy's integral theorem we can deform the path of integration into a contour, convenient for estimates when r - . We shall elucidate the evaluation technique in the case D = . Then

=

w'Re

i( 8(C+C_1)_3eAoZdC/C

0

_{(5.15; D)}

A0 14(

+ c')2

w =

2(C + C1Y2(Ce

+ C1e)

To simplify the statements of absolute convergence the alternative paths of integration are chosen in the region {ç; Re A0 0, Im w ? o}.

The Riemann surface R over the A0-plane of the inverse function ç(A0) has

A0

four sheets and the singular points over A0 = 0, 1 and . The image of R

A0

under the mapping C(A0) is shown in Figure 5.3. The notations are self-explana-tory.

X1tanh(X0D) = - (r +

-iQ

w

f

A0(Ce

+ Cle)

In particular, if c.p 0,

1 Z**(x,O:z') = ir1Re 1-I

w e

JO

0

w = 2(c +

-A0-plane (One sheet of RA)

Fig.5.3 The mapping of R

A0 onto the c-plane.

(5.16; D)

Then Im U) > 0 in {; (1-1r12)Im

>o}.

The path of integration may be deformed into a contour C through = 1 as shown in Figure 5.'.

In = 1 we have

w = 1, dU)/dC = 0, _{d2w/dC2 = - 1} Thus

w 1 - - (_1)2 t O[(c-i)3]; -1 = o(l) Making the change of variable

8 =

(-i)vc7

we obtain when x

-Z'+lS

J

3 -(w2-l)z'+i(w-l)x

i.e

we

Z**(x,O:z') = w1Re (5.17) -z, _zt+i(x+7t/t),/2i

I

e2ds = -

/2/irx

e

cos(x + w/&)

With the change of variable ç = ç(w) in (5.16), we obtain

Z**(x,O:z') = - 2n 1Re e 1

Thus, the spectral function becomes infinite when w and, therefore, the frequency w = 1 is dominating as shown by the formula (5.17).

The Fourier transform representationis suitable for deriving a refined asymptotic expansion.

Set w-1 = t2/x and integrate along an equivalent contour in the sector 0 < arg t < ir/8. In the neighborhood of w 1 we have the power series

e_(W2)Z'2(w21)_h/2dw/dt

_{= V7}

_c(t2/4'

Then, integrating formally term by term, we obtain the asymptotic formula

cr(ni-) inw/2 -n

Z**(x,0:z')

_{- wRe}

V7

_{e - e x (5.18)}

where r(n+}) = ./(2n)!/4n!. We obtain the first coefficients

co= 1,

C1 =-

2z', C2 =

- -

' +

2z',

The first term in (5.18) is identical with (5.17).

Mow we assume 0 < (P < ir/2. The frequency w is real on C0 and

w 0 for ±e'80,

00=cp- ,/2,

and, C= 0,.

Then eZO0ECo, -ir/2 < ₈₀ < 0. Further

dw/dç 21(C

+ C1)3C1(C2e6o + C2e80 -

6 cos

hence, the zeros of dw/dr lie on C0 if and only if cos O, sin (P 1/3, i. e.

0 p c sin 11/3 = 1947.

If sin c.p > 1/3, the equation dw/dC 0 has the four roots

[v'l

t

sin

q -

iVL - sin -p ][v'3 sin

+ 1 - V'3

sin p - 1 ]

and C2 = c3

- C, C.

= -

C The corresponding values of w are

WI = - csc p[(l - sin .p)"(3 sin + i)" + i(l i- sin

(p)/2(3

sinp -

_1)3'2]

andw2

w3

-wi, w

=

-w2.

Fig. 5.5

The Riemann surface R of the function

1(w)

has four sheets, each

containing two of the four singular places (S,,w,). The cuts are assumed to be parallel to the real axis. As illustrated in Figure 5.5 the path C0

is represented one-to-one on the real axis of one sheet R (1) with the singular places (,w1) and (r2,w2). The image in R of an alternative path of integration C may pass the cut, ending in (1,w1),and continue on the sheet R(2). The singular places on R(2) are (c,w1) and (r31w3), and it is easy to see that the image of C is restricted to the upper halfes of R(l) and R(2) and, hence, the contour C lies in PcI < 1. We notice that Mm Im ü(r) Lii for each contour C, equivalent to _CO3

with equality only for a class of contours through

= . We can choose a contour C such that Re A0 > 0 and Im w Lu + kiw-w1i, k > 0, on C. After the change of variable t = (c-c)V and proceeding as in the case

= 0, we

obtain when r

-'/

-i Z'+j

r Z**(x,y;2') _{- Re [LiL/(9}

_sin2p

_-

_1)]

_e 1 1

/VF

where

= = 4j cscp(1 + 3

sin2p

-

i

cosp

V

sin2p - 1 ]

Since Im > 0, it follows that Z, in view of (5.5), is do.inating over

Z for large r in the sector sin 1113 < _{< ir/2.} c-plane

For sin p < 1/3, the zeros of dw/dC lie on the unit circle. We obtain

C1 = --

(Vi +

sin .p - iV'l - sin '.p + 3 sin

cp - iVi -

3 sin p ] = e0oC_1 _{C3 = - C1, Ck}

- c2 (5. 20)

Then CO3 C2 C0. The corresponding values of w are

csc q,[(l -

sin)"2(l

+ 3 sin p)3"2 ₊ (1 + sin

cp)"2(l

- 3 sin

ip)2]

CSC ( w11, w3 - Wi, W

-We shall give a parameter representation of these formulas, useful for generalization to the case D < .

Set iO

1=e

1, _C2 e 2 Then _{0 > 02 > 0} > > - n/2 and Oi + 02 = (5.21) (5.22)

By (5.19) it follows that the arguments 01 and 02 are roots of the equation

cos(28 - 0) = 3 cos

Hence

tan (P = - sin Ocos 01(1 + sin20) (5.23)

Set -

sin e. Then 0 <

< 1. We

can write (5.23)

tan (P = (5.23 a)

To each (P we obtain two roots = - sin 8 and

2 = - sin

tan cp > 2 Then 2

1// for

tan 1/2VT(sin tp 1/3).

-

-T---We have an one-to-one

correspond

--- ence between the values of and

and2for<l/\/.

Given r as a zero of dw/dC, we now obtain the frequency

1/2

w secO (1 + 3 sin 8) =

[(l_2)(l+32)J2

and the second derivative

d2w/d2

-- 2i 0

= - e -2i8

e sec 0 (1 - 3

sin28)(1 + 3

sin2e)_t2

When varies along CO3 the function is increasing from w = - to a maximum w = in the interval (-n/2, er], decreasing to a minimum w = W2

Ifl

_[e',

_02]

and increasing to .Li

= +

_{in [82,}

w/2].

Evidently, when varying along CO3 the variable takes its values from three different branches of the four-valued function _{(w). The situation} is illustrated in Figure 5.7 by the image of the Riemann surface R on the c-plane. In the figure, R denotes the image of the sheet R(v).

t-plane

(5.24)

(5.25)

The four sheets of R with

(1)

the cuts in the real axis.

Fig. 5.7 The mapping of R onto the -p1ane when sin p < 1/3, D = co.

An alternative path of integration C in the integral (5.15) shall _pass through the points z and and is obtained by a continuous deformation of the arcs {e0; 0

(-ir/2,

_{O) U [82, 1T/2J} into paths in IçI < 1 and} of the arc 0 _[0k,

_e2]}

into a path in 1r1 >

CL)

-1411

U

R('4)

12 SI:'

-Set C {r; 1zI = p(0), -i/2 < 0 < t/2}. We can assume that p(8) is single-valued. Then p > 1 for 0

_(er,

02] and p 1 elsewhere.

Let 01 < 8' _{< 82} With the path C0 in the integral (5.15) deformed into C, we break the path into two parts C1 and C2 corresponding to the intervals

[-ir/2, 0') and [0', n/2] respectively. If r is very large, only the parts of the integral taken over small arcs {; =

pe0,

< 0 <

e,,+c, v = 1, 2

}

are significant. Hence, using the variable

_{8 =}

_{(_r)Von C, we obtain}

when r -' =

where

2. A0(e8"), w00(9) = d2w(e8)/d02

When

e0V

we obtain w09:0) = - e28d2w/dC2. Then, by (5.15), (5.25) - 2/wr sec3e 1w (0 ) -1/2 -2.

Z'

I e

'

cos[w r

(-l)'w/L]

V 80 V sec30 1w (0 ),_1/2 = v 08 \,

(l)

If sin p = 1/3, then 01 = 02 w1 w2, = 2.2 and

-3i0 iO

dw/dç d2w/d2 = 0, d3w/d3 = i.e w000(61) for

= = e

where w068(8) = dw(e8)/d8. We obtain the values

Fig. 5.8 Modification of Fig. 5.7 when sinq = 1/3.

(5.26)

,

= V (5.27)

sin 01 = -

i/V5,

U.)1 = = 3/2, w800(01) =

3v7

The alternative path C for the integral (5.15) shall pass through

= After a choice of C such that Re A0 > 0, Im w > klw-w11, k > 0, on C, we find that the asymptotic behaviour of Z** when

r is essentially determined by the contribution to the inte-gral from the part of C in a small neighborhood of

A simple path C, useful for estimates, is obtained if p(0) = trI is chosen as solution of the equation = 2 cos Osec ₈₁ sinl0-011.

Setting 8 (-1)r'/ and making r we obtain

Z**__,t

-1

2 3 F

1/3 -1/6

(1/3) sec381u088(01)-1/3 -1/3r e

-

1 '_cos(w1r)

= -

(2)36F(l/3)

-1/3 _3Zt/2 -r e cos(W3r/2)

As shown in the case tp = 0, we can derive asymptotic expansions by using the Fourier transform representation and, thus, integrate with respect to the variable w. We shall here only present the next term in the

asymptotic formula when sin c.p 1/3. Then

1 -3z'/2

- (2n)

e [35/6r(l/3)r1/3cos(Vr/2) +

-2/3 . -4/3

'+

3F(2/3)(-

-

z')r sin({3r/2) +

O(r

_{) ]} Summing up for D = co, we have Z O(r2) when r in the sector sin 1113 < < it (formula (5.5), sinl/3 i97). On the straight lines kpl = sin'l/3 we have Z = O(r) and by (5.28) it follows that

the surface elevation has a wave character.

Finally, when 1w! < sinl/3 the

surface is essentially determined

-- by the two terms in (5.26),

- - - defining the diverging (v

1)

v =

1--

= 2 and the transverse (v = 2) wave

0

-

systeras.

The

curves of constant

S

phase are determined by

constant = K Fig. 5.9 Curves of constant _Settings

=

r cos, y

r sinq

phase,

D =

we obtain by (5.23 a) and (5.2L)

r = K(1+2)\tJ_F2, KF(l-F)

Hence dy/dx

=

vL_2/

= -

cot 8, and the parameter 0 has a simple geometrical interpretation (Figure 5.9, Cf STOKER, Water waves (1957), p, 238). The two wave systems have a re.ative phase difference = it!2.

Finally, we consider the case D < . Then the analytic function A0(C), defined

by (5.12), is multiple-valued with an infinite number of branch points in the c-plane defined by the equation r 1. However, when deforming the path of integration C0 in (5.1L4), we can avoid these singularities if the alternative paths are chosen in the region {c; arg < rr/6, mi > o}. Then the

methods used for D = are still applicable.

Thus, if, for a given c.p [0, ir/2],we have dw/dC * 0 on CO3 then the contribu-tion from Z** to the surface elevacontribu-tion when r - is dominated by a simple

wave with exponentially decreasing amplitude (cf. D = , sinq > 1/3).

We obtain the wave phenomena when the function dw/dc has zeros on tc = 1. On C0 the function A0(e) a A0(0) is defined by (L.4), (5.12), and we obtain

A0 0 where ' tanh 9,'D

Then t' > 0 for D > 1 and £' = 0 for D 1. The variables

- sin 6, n = 2A0D csch 2A0D

are suitable for para*etrization, and we also introduce the function

T()

= tanh A0D

which is decreasing in 0 < ii < 1; T(0) = 1, T(l) = 0. The inverse is explicitly

=

f

(T1- T) ln[(l+T)/(1-T)J

We obtain

D = A0 =

T(l-2Y'

The graphs of D = constant and A0 = 1 are given in Diagram I. We notice that 0 < r' 29.'D csch 22'D < 1 for D > 1 and 2 _{1-D for D}

< 1.

For fixed D we obtain the derivatives with respect to the variable 6

=

constant and i.p = constant

< (

< o and no inter-function of D; tp0(lsO) = = 0. If w0 = 0, then T(1_n){(l_2)[(l_n)2+(3+2r,_r)2)2]}h//2 _(5.30) (U90 (5.31)

where N = 2n2(l+nY1(1-T2Y. Diagram II is a graphical representation of

w, defined by (5.30). By (5.31) we obtain the locus of cp =

= (_n)2(3_8n+n2+IN)' (5.32)

The graph _SG is given in the Diagrams I and II. It is easily shown that S0 lies to the left of D = 1. If then (F1,n1) lies to the right of S0 W88 < 0, and we shall see that this point cotesponds to the diverging wave system. The point (F2,r2) lies to the left of S0, w00 > 0, and

corresponds to the transverse wave system. When sp ₌ p0, we obtain the third derivative

We00 = _{4W[,Vc]3(1+fl)(1_fl)_(3(l_5fl2fl2)(15_l1fl)N+2(2_l,fl)N2]} (533)

where I > 0 is given by (5.32).

When D < 1, we obtain only one intersection (F,1) for 0 _{< GPo,}

sin/

(5 34)

and no intersections for p > _p0. In this case we have only the diverging wave system.

The derivative of

= ).0cos(O-p) = T(1_2)-1(V

cosp -

sinç]

with respect to 0 for fixed D and p becomes

we cos + (l-r+(l+)2]sin i:p}

Hence = ie0dw/dC 0 on C0 for

tan = (l+i,)V1_/(1_n+(1tn)2] (5.29) This formula is the generalization of (5.23 a). The graphs of p = constant

are

given in Diagram I. The curves are convex. Given D > 1, there exists a PQ(D) such that D = have two intersections (,n) and

(F2,n2)

for sections for p>p0. Then ip0(D) is a decreasing (p0(aD) sin 11/3. _{For P we have}

Consequently, we set

sinO

; -ir/2 < e

<a

V V

= A0(8)

T(n)(12)'

w = 2 cos(6 -tp) = w( ,n ), ef. (5.30)

\) V V V V

We repeat the evaluations, demonstrated for D , and obtain for 0 < <

when r - Co

- -

2/itr E(F ,n )sech(t D)cosh(9 (D-z'flcos(w r +

(-l)"w/Li]

V V V V V V

E(,n)

= Th/2(l+n)_h/2(l_2) 3/4t(l_y1)2_(3_8n+fl2+t4N)211/2 x ₍₅₃₅₎ x[(1_n)21(3+2n_n2)F2]"4

This is the generalization of (5.26). Each point (E,n) corresponds uniquely

to a couple (D,p), and we may write E(,n) E(D,(P). If D < 1, we obtain only one term (v = 1).

In particular, setting = 0, we obtain for the transverse wave in the plane y = 0 ( 0, D > 1),

= T, D = -} T11n((l+T)/(l-T)], E = (T/(l_n2)]1.2

We see (table 1) that 1 and E 1, except in the neighborhood of D = 1 where

'2 v3(D-l) and E 3"/2(D-l)". When (P = D > 1, set

= ( w2). By (5.32) we eliminate

in the formulas and obtain D, 2() and as functions of the parameter r,

D = 2ri(l-T2) 1(l_3n+2N)(3_8fl+n2+&NY' tan = 23/'2(l+fl)(l_3ri+2N)1'2(l_2r,+N)l

nT(l_T2)'D'

(5.36) 1/2 -1/2

+

T(3-8+2+4N)(l-3ri+2N)

(3(l-)+2N]

Table 2 and Diagram III give the values of and as functions of D and

For p p, D > 1, we obtain the generalization of the evaluation (5.28)

when r

-- (21r)_135/6F(l/3)E0(D)

(5.37)

where, expressed in the parameter ri,

T2I3(l_n)2/3(1+)31(3_8n+n2+4N)7/6(i_3n+2N)6x

x[3(i_n)+2N]116[3(l_5n+2n2)+(l5_lln)N+2(2_1/n)N2]_h/3

The function E0 is presented in table 2. When D -

1+0,

then cos - /8(D-l),

For D < 1, we find that E1(D,(p) + as - In this case, the path

of integration is broken into two parts by the condition cos 8 < For

= defined by (5.34), we have u = = 0 at 6 = eo; cos 80 =

sin 00 - Vl-D. If r is very large, only the part of the integral (5.3) taken over a small interval [80-c, 0] is significant. Using the estimates

60-0 = - D52(l_D) V2A [i+o(i)], w = A0(6-60)[iso(l)), we obtain

- 1 - 1/6 - 5/6 -1/3 -1/3

- (2w) 48 r(l/3)D (1-D) r (1p

_{= %,}

D < 1)

We notice some general properties of the wave pattern, which is illustrated by curves of constant phase in Figure 5.10

By (5.35) it follows that the transverse and the diverging wave systems have a relative phase difference = /2, independent of D.

Given D, then the frequencies w are decreasing functions of . Further,

given (p and

D > 1,

then > > w0. The proof is elementary.

The geometrical interpretation of the parameter 6, shown for V , is true for all values of D.(Figure 5.g).

The diverging angle of the wave pattern p0 can equally well be derived by the use of the principle of stationary phase. HAVELOCK (1908) proposes

cos2q0 = 8(l-n)/(3-n)2 for V > 1. This is the relation between p arid r

when r, defined by (5.29), has a minimum for given (p, thus 0. His illustrations of the wave pattern for D < 1 are strange.

INUI (1957) presents a graph of (p0, which obviously is identical with

Fig. 5.10 Curves of constant phase, wr = 1, for D ,

1.1, 0.5

and 1/9. When

D < 1

and x is large, we obtain

y

¶/D/(l-D) x

32/3(Q3

+O.2D)(D/(l_D)]s/6xV3

+ 0(xh/3)

The asymptot, defined by the two first terms, is indicated in the figure.

6.Surface Waves Generated bya Point Forcive

By the deduction in section 2 and formula (3.8), we find that the dimensionless velocity potential (see (2.l)) for a

point

forcive P, i. e. a surface pressure distribution p = P6(x1)cS(x2), becomes

= Pg2p1c'6G(x,y,z:O)

Hence, for P = pc6g2 we obtain the dimensionless surface elevation

Z(x,y) Z(x,y:O)

Setting Z = Z

Z, where Z**

0 for x < 0 and Z is an even function of s, we obtain by (5.5) and (5.7) after differentiatioii

- 3 sin2p)r3 for D =

-(2w)1fD/(D-l)(D - (2D-l)sin2p](D - sin2(P2r2 for D > 1

when r - . If D < 1, it follows from (5.8) that = O(e°1'/V'r), > 0,

when r

+

. W can show that c > (D-l)2/4D.

If < r.p _< w/2, it follows immediately from the definition Z** =

that Z = O(e_81'), B > 0, when r is large. For 0 < < P, we obtain

-

I

E( ,n ) sin(w r + (-l)"w/4]

v_

V V V

E(,n) =

T(l_2)l/2E(,n)

(6.1)

When p 0, D > 1, we obtain the transverse wave in the plane y = 0,

Z**(x,O) - V2/irx 2(D,0) sin(w2x + = T

where (D,0) = TE(0,n) = TE2(D,0). Computed values are given in table 1. If p = (p0, P > 1, then

Z**.wl2_3/234/3r(l/3) E0(D)r'sin(w0r)

(D) = T((3_8n+r)2+LN)/3]1/2(l_3n+2N)1/2 E0(D)

(6.2)

The function is computed and given in table 2. Finally, if p p0, D < 1

-

6/r(l/3)1

Dh/6 (l_D)1/6

-2/3

r

0

Diagram .1 Graphs of D constant and ,p constant at the frequencies of the dominant waves (w0 = 0).

Transverse waves to the left and diverging waves to the right of S0.

The curve through the minima of p constant corresponds to Ravelocks relation cos2cp = 8(l-)/(3-)2.

Diagr'aa II

0 .5 10 15

Diagrar III

90 60 30

Table 1. Data of the transverse wave in the plane y 0.

Table 2. Characteristic data at the diverging angle of the wave pattern for V > 1 (

< v).

8o is defined in Diagrai III.

D 1.0 1.0137 1.0591 1.1552 1.3733 1.6357 1.9282 2.3444 2.6734 0.0 .2 .4 .6 .8 .9 .95 .98 .99 1.0 1.9417 1.3851 1.1504 1.0290 .9981 .9924 .9924 .9963 1.0 0.0 .3883 .5504 .6903 .8232 .8983 .9428 .9744 .9863 1.0 T D p0 _wo S0

0.0

1.0

0.0

1.0

90°

00

0.0 0.0 .2 .973823 .197375 1.00888 75.03 3,78 .038466 1.63617 .26425 .4 .900793 .379949 1.03540 61.67 7.38 .138153 1.07984 .33779 .6 .794986 .537050 1.07906 50.75 10.65 .266128 .88350 .39420 .8 .673523 .664037 1.13871 42.33 13.54 .394184 .79452 .44309 1. .551441 .761594 1.21244 36.03 16.07 .506709 .75402 .48794 1.2 .439060 .833655 1.29763 31.40 18.29 .598263 .74065 .53098 1.4 .341800 .885352 1.39126 28.03 20.28 .669198 .74484 .57403 1.6 .261312 .921669 1.49026 25.61 22.10 .722420 .76146 .61847 1.8 .196878 .946806 1.59190 23.88 23.78 .761531 .78719 .66504 2.0 .146574 .964028 1.69408 22.64 25.36 .789922 .81924 .71360 2.2 .108057 .975743 1.79549 21.77 26.83 .810422 .85469 .76305 2.4 .079011 .983675 1.89571 21.14 28.18 .825235 .89053 .81143 2.6 .057374 .989027 1.99504 20.69 29.41 .835996 .92379 .85633 2.8 .041417 .992632 2.09430 20.37 30.49 .843877 .95215 .89560 3.0 .029745 .995055 2.19453 20.13 31.44 .849696 .97437 .92560 3.5 .012766 .998178 2.145540 19.77 33.19 .858522 1.001439 .97821 14.0 .005367 .999329 2.73677 19.60 34.22 .862704 1.01154 .99815 '4.5 .002221 .999753 3.03767 19.53 34.76 .864609 1.00940 1.00300 5.0 .000908 .999909 3.35261 19.50 35.03 .865439 1.00617 1.00323 CD _0.0 1.0

1947 3526

.866025 1.0 1.0

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