International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 4

Number 4

December 2010

415

1 INTRODUCTION

The intensification of shipping gave rise to compli-

cation of navigating conditions at sea roots and in

recent years over 75% of navigation accidents oc-

curred in restricted waters and bounded waterways.

The growth of merchant ship dimensions during the

last decades led to the situation in which vast regions

of oceans and seas become comparatively shallow.

The wrecks of modern large vessels getting

stranded or collided are accompanied by serious

economical losses and negative ecological conse-

quences.

The estimation of ship motion characteristics in

restricted waterways is necessary not only for elimi-

nating the possibilities or minimizing the number of

accidents, but for substantiation of sea routes dimen-

sions in the proximity of ports as well.

Modern theoretical and experimental hydrody-

namics provides us with a great amount of infor-

mation for predicting seakeeping qualities of ships in

open deep sea. On the contrary such an information

for a vessel sailing in shallow water is comparatively

poor and the proper methods are not widely devel-

oped. Such a situation may be explained by virtue of

the additional difficulties arising in the theoretical

investigation of the potential boundary value prob-

lems for a ship propagating in shallow water condi-

tions. First of all the complicacy of the singularities

method is ten times higher for shallow water poten-

tial problems in comparison with the unbounded sea

ones. Then the strip method widely used in practical

calculations is inconsistent with the physical reality

and often causes insoluble problems when using in

shallow water cases with clearly expressed three di-

mensional water flow phenomena.

Thus a new approach for investigating ship hy-

drodynamic problems free from difficulties of clas-

sical method of singularities and shortcomings of

strip method is vital. Such approach is demonstrated

in this paper.

2 BOUNDARY VALUE PROBLEM FOR

VELOCITY POTENTIAL

If the water around a ship is considered as inviscid

incompressible fluid the important hydrodynamic in-

formation is derived from the solutions of boundary

value problems for the velocity potential. Founded

on basic physical principles the nonlinear problems

with apriori unknown boundaries are simplified by

linearization and the solutions of corresponding lin-

ear problems are practically used. Consider a vessel

floating with a zero forward speed in shallow water

with the depth

H

under the action of regular waves

ti

вв

er

σ

ζ

=

,

в

r

and

σ

being wave amplitude and cir-

cular frequency accordingly.

The longitudinal

x

and transverse

y

axes of the

Cartesian coordinate system are taken on the free

surface of water and the vertical axe

z

is pointed

downward.

The potential function

( )

tzyx ,,,Φ

can be divided

into cosine

ñ

Φ

and sine

s

Φ

parts

( ) ( ) ( )

[ ]

tisc

ezyxizyxtzyx

σ

,,,,Re,,, Φ−Φ=Φ

(1)

Asymptotic Theory of Ship Motions in Regular

Waves Under Shallow Water Conditions

Y.L. Vorobyov

Marine Engineering Bureau, Ukraine

M.S. Stasenko

Odessa National Maritime University, Odessa, Ukraine

ABSTRACT: The hydrodynamic theory of ship motions in shallow water under the action of regular waves is

discussed. The boundary value problem for velocity potential is solved using the matched asymptotic expan-

sion method (MAEM). The solution is based on Fourier – Michell integral transformation technique and char-

acteristics of Helmholtz and Klein – Gordon equations. Using the obtained results formulae for hydrodynamic

characteristics are derived. The application of these formulae demonstrated good coincidence of the results of

calculations and model experiments carried out in towing tank of Odessa National Maritime University.

416

It is systematically demonstrated in investigations

of Y.L. Vorobyov (Vorobyov, 2002), that estimation

of added inertia, damping, coupling coefficients and

exciting forces can be done using asymptotic values

of radiation potential. So we can avoid the necessity

of treating the wave scattering problem and difficul-

ties of integration in the hull proximity.

Consider the ship performing longitudinal har-

monic oscillations with circular frequency

σ

. The

potential functions

( )

zyx

sñ

j

,,

,

Φ

,

5,3,1=j

must satis-

fy the following differential systems

( ) ( )

;,,,0,,

0

,

2

2

2

2

2

2

Ezyxzyx

zyx

sc

j

∈=Φ

∂

∂

+

∂

∂

+

∂

∂

(2)

( ) ( )

0

,

2

,,00,, Σ∈=Φ

+

∂

∂

yxyx

gz

sc

j

σ

; (3)

( ) ( ) ( )

;,,,,,,

,

Szyxzxuzyx

n

j

sc

j

∈=±Φ

∂

∂

(4)

( ) ( )

∞<<∞−=Φ

∂

∂

yxHyx

z

sc

j

,,0,,

,

. (5)

If the ship is performing transverse harmonic os-

cillation with circular frequency

σ

, the potential

functions

( )

zyx

sc

j

,,

,

Φ

,

6,2=j

must satisfy the dif-

ferential systems (2), (3), (5) and hull conditions

( ) ( )

( )

( )

.,,

,0,,,,,,

Szyx

zyx

n

zxvzyx

n

s

jj

c

j

∈

=±Φ

∂

∂

±=±Φ

∂

∂

(6)

Both systems must satisfy radiation conditions in

the infinity.

If the velocity of oscillations is taken to be unity,

( )

zxu

j

,

and

( )

zxv

j

,

are for longitudinal oscillations

5,3,1=j

( ) ( ) ( ) ( )

( ) ( ) ( )

,,cos,cos,

,,cos,,,cos,

5

31

zNxxNzzxu

zNzxuxNzxu

−=

==

(7)

and for transverse oscillations

6,2=j

( ) ( ) ( )

( ) ( )

yNxxNy

zxvyNzxv

,cos,cos

,,,cos,

62

−=

==

(8)

Now let us consider ship as slender body, suppos-

ing that

( )

ε

OLB =

,

( )

ε

OLT =

,

TBL ,,

being her

length, beam and draft,

1<<

ε

and the hull varies

slowly along the longitudinal axe. Under this as-

sumption matched asymptotic expansion method

(MAEM) is used for solving the potential problems.

According to MAEM the flow field is divided in-

to two zones: far field zone where

( )

1OLy =

and

near field zone in which

( )

ε

OLy =

. The condition

along the boundary between the zones are not for-

mulated and satisfied in the process of matching the

solutions in far and near fields along the their

boundary.

3 FAR FIELD SOLUTIONS

If the observation point is located in the far field as

0→

ε

the hull degenerates into a cut

22 LxL ≤≤−=

δ

of free surface plane

0=z

. The

potential functions

sc

j

,

Φ

are harmonic (2) in the

layer

Hz ≤

≤0

with ship centerplane

0=y

ex-

cluded, satisfy free surface (3) and radiation condi-

tions. The boundary conditions on the centerplane

0±=y

, that is on the inner boundary of the outer

zone are not formulated as soon as the hull (with its

centerplane) belongs to inner zone. The only identi-

ties come from the physical considerations

( ) ( )

( )

( )

;5,3,1,0,,,

,,;,,,,

,

,,,

=>+Φ

∂

∂

−=

=−Φ

∂

∂

+Φ=−Φ

jyzyx

y

zyx

y

zyxzyx

sc

j

sc

j

sc

j

sc

j

(9)

( ) ( )

( )

( )

.6,2,0,,,

,,;,,,,

,

,,,

=>+Φ

∂

∂

=

=−Φ

∂

∂

+Φ−=−Φ

jyzyx

y

zyx

y

zyxzyx

sc

j

sc

j

sc

j

sc

j

(10)

In accordance with (9), (10) the boundary condi-

tions on the centerplane

0±=y

are taken in the

form

( ) ( ) ( )

;5,3,1,0,0,;,,0, ==±Φ

∂

∂

±=±Φ

∂

∂

jzx

y

zxfzx

y

s

jj

c

j

(11)

( ) ( ) ( )

6,2,0,0,;,,0, ==±Φ±=±Φ jzxzxgzx

s

jj

c

j

, (12)

where unknown functions

( )

zxf

j

,

and

( )

zxg

j

,

are

taken as known ones for a moment.

Let us find the solution of the outer problem (2),

(3), (11), (12), (5) for cosine amplitude

( )

zyx

c

j

,,Φ

of velocity potential

( )

tzyx

j

,,,Φ

. Using the Fourier

method for the outer differential problem we find the

expansions for

( )

zyx

c

j

,,Φ

( ) ( ) ( ) ( ) ( )

∑

∞

=

+=

Φ

1

0

0

,,,,

m

m

m

jj

c

j

zZyxFzZyxFzyx

; (13)

417

( ) ( ) ( )

( )

( ) ( )

.,,

1

,

;,,

1

,

0

0

0

0

zzZzyx

H

yxF

zzZzyx

H

yxF

m

H

c

j

m

j

H

c

jj

d

d

∫

∫

Φ=

Φ=

(14)

The eigen functions

( )

zZ

0

,

( )

zZ

m

form a com-

plete orthogonal set in

[ ]

H,0

with mean square val-

ue of 1:

( ) ( )

( ) ( ) ( )

( ) ( )

( )

,

2

2sin

1

2

1

;

2

2

1

2

1

;

,

0

0

0

2

1

0

2

1

00

+=Ν

+=Ν−×

×Ν=−Ν=

−−

H

H

z

H

Hsh

zHzch

zZHzchzZ

m

m

m

m

mm

α

α

α

α

α

α

(15)

where

0

α

= real positive root of the equation

Hth

g

00

2

αα

σ

=

(16)

and

<<<

321

ααα

= subsequence of real positive

roots of the equation

0

2

=+

g

Htg

mm

σ

αα

. (17)

As soon as

( )

zyx

c

j

,,Φ

is harmonic and the eign

function system is orthogonal,

( )

yxF

j

,

0

and

( )

yxF

m

j

,

satisfy the Helmholtz and Klein-Gordon

equations

( )

( )

.0,

,0,

2

2

2

2

2

0

2

0

2

2

2

2

=

−

∂

∂

+

∂

∂

=

+

∂

∂

+

∂

∂

yxF

yx

yxF

yx

m

jm

j

α

α

(18)

Taking in mind, that for

5,3,1=j

( ) ( )

zxfzx

y

j

c

j

,,0, ±=±Φ

∂

∂

, the boundary conditions

for equations (18) according to (14) are to be taken

in the form

( ) ( )

( ) ( )

,22,

2

1

0,

,

2

1

0,

00

LxLxxF

y

xxF

y

m

j

m

j

jj

≤≤−±=±

∂

∂

±=±

∂

∂

γ

γ

(19)

where

( ) ( ) ( )

( )

( ) ( )

.,

2

;,

2

0

0

0

0

zzZzxf

H

x

zzZzxf

H

x

m

H

j

m

j

H

jj

d

d

∫

∫

=

=

γ

γ

(20)

According to Green theorem and conditions (19),

(20), after using radiation conditions we find for

5,3,1=j

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

,,

1

,

2

1

,,

0

2

2

0

1

0

0

2

2

000

∫∫

∑

∫∫

−

∞

=

−

Κ−

−Ν=Φ

H

mj

L

L

m

m

m

H

j

L

L

c

j

ddZfRzZ

H

ddZfRzZ

H

zyx

ζξξζξα

π

ζξζζξα

(21)

( ) ( ) ( ) ( ) ( )

,,

2

1

,,

0

0

2

2

000

∫∫

−

−=Φ

H

j

L

L

s

j

ddZfRJzZ

H

zyx

ζξζζξα

(22)

where

( )

RJ

00

α

,

( )

R

00

α

Ν

,

( )

R

m

α

0

Κ

=Bessel func-

tions,

( )

2

2

yxR +−=

ξ

.

The last formula is based on Green theorem,

equations (18), boundary condition

( )

0,0, ≡±Φ

∂

∂

zx

y

s

j

and radiation conditions.

Now returning to (18) we find that for

6,2=j

( ) ( )

zxgzx

y

j

c

j

,,0, ±=±Φ

∂

∂

, the boundary condition

for equation (18) must be taken in the form

( ) ( ) ( ) ( )

xxFxxF

m

j

m

jjj

γγ

2

1

0,,

2

1

0,

00

±=±±=±

. (23)

where

( ) ( ) ( )

( )

( ) ( )

.6,2,,

2

;,

2

0

0

0

0

==

=

∫

∫

jzzZzxg

H

x

zzZzxg

H

x

m

H

j

m

j

H

jj

d

d

γ

γ

(24)

Now taking solutions of (18) that satisfy bounda-

ry conditions (23), (24) along the cut

2Lx ≤

after

using radiation conditions we have

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

,,

1

,

2

1

,,

0

2

2

0

1

0

0

2

2

000

∫∫

∑

∫∫

−

∞

=

−

Κ

∂

∂

−

−Ν

∂

∂

=Φ

H

mj

L

L

m

m

m

H

j

L

L

c

j

ddZgR

y

zZ

H

ddZgR

y

zZ

H

zyx

ζξξζξα

π

ζξζζξα

(25)

418

( ) ( )

( ) ( ) ( )

.6,2,,

2

1

,,

0

0

2

2

00

0

=

∂

∂

×

×−=Φ

∫∫

−

jddZgRJ

y

zZ

H

zyx

H

j

L

L

s

j

ζξζζξα

(26)

4 NEAR FIELD SOLUTIONS. MATCHING

To study flow phenomena in the near field the trans-

verse coordinates are stretched

εη

y=

,

εζ

z=

and as

0→

ε

omitting terms of

( )

2

ε

O

we obtain the

totality of two dimensional boundary value problems

in

constx =

planes for

( )

ζη

,

c

j

Φ

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

.6,5,3,2,1,;0,

;,;,cos,

;,;,cos,

;

2

1

,;00,

;,,0,

3

2

2

11

2

2

2

2

=∞<=Φ

∂

∂

∈=Φ

∂

∂

∈=Φ

∂

∂

>==Φ

+

∂

∂

∈=Φ

∂

∂

+

∂

∂

+

+

jh

xLN

N

xLN

N

xb

g

xe

c

j

c

c

c

j

c

j

ηη

ζ

ζηζζη

ζηηζη

η

σ

εκηκ

ζ

ζηζη

ζη

(27)

Here

( )

xe

= a fluid domain in a form of strip

{ }

h≤≤∞<<∞−

ζη

0,

with frame contour

( )

xL

+

excluded,

( )

xb

= the width of this contour.

The problem for

( )

ζη

,

s

j

Φ

is uniform and has a

trivial zero solution. The boundary value problem

(27) has to be discussed keeping in mind that for

matching procedure the asymptotics of solutions

when

η

tends to infinity are needed. These asymp-

totics are found using specially worked out proce-

dure. In addition to harmonic potential

( )

ζη

,

c

j

Φ

let

us introduce the conjugate harmonic stream function

( )

ζη

,

c

j

Ψ

and multi-valued analytical function

( ) ( )

+Φ=+=

ζηηχ

,,

c

jj

iyU

( )

ζη

,

c

j

iΨ+

being de-

termined outside the close contour

( ) ( )

xLxL

−+

U

.

Various branches of

( )

χ

j

U

differ one from another

by function

[ ]

( )

( )

=

=

+

−

=∆

,5,3,1,

;6,2,

2

000

00

jB

jA

hchshh

hhchch

U

j

λλλ

ζλλ

(28)

where

A

=

( )

ηλλ

00

cos,xP

j

,

B

=

( )

ηλλ

00

sin,xQ

j

−

,

0

λ

= real positive root of the equation

hth

g

00

2

λλ

σ

ε

=

. (29)

We notice that outer boundaries of inner zone

±∞→

η

are at the same time the inner boundaries

of the outer zone

0±=y

. We have from (28) and

(29)

( )

( )

( )

,5,3,1,,

2

1

,0,

000

000

0

=

+

−

±=± j

HHchshH

HzHchch

xQzxf

jj

ααα

ααα

α

(30)

( )

( )

( )

.6,2,,

2

1

,0,

000

00

0

=

+

−

=± j

HHchshH

HzHchch

xPzxg

jj

ααα

αα

α

(31)

Functional coefficients

( )

0

,

α

xQ

j

and

( )

0

,

α

xP

j

are determined in the form

( ) ( )

( )

( ) ( ) ( )

[ ]

( )

( )

,sincos

exp4,

0

0

000

2

1

0

0

2

03

dtt

dt

tdZ

tAttstBtA

tZ

g

xQ

xb

−+

×

−=

+++

∫

αααα

σ

α

(32)

( )

( )

( )

( ) ( ) ( )

[ ]

( )

( )

,cossin

exp4,

0

0

000

2

1

0

0

2

03

dtt

dt

tdZ

tAttstBtA

tZ

g

xP

xb

−+

×

−=

+++

∫

αααα

σ

α

(33)

where

( )

tA

+

= values of potential function under de-

termination on the contour with the equation

( )

tZz

0

=

and

( )

( )

2

0

1

+=

dt

tdZ

tS

.

As soon as

( )

tA

+

is unknown, it is proposed to

take its approximate value when the frequency of

oscillations tends to infinity. Values of

( )

tB

+

are

values of normal derivative of potential function ac-

cording to the hull boundary conditions. The value

of

( )

tA

+

can be easily found using the standard inte-

gral equation procedure.

For

( )

05

,

α

xf

and

( )

06

,

α

xf

we find

( ) ( ) ( ) ( )

02060305

,,;,,

αααα

xxfxfxxfxf −=−=

(34)

Inserting (30)-(34) into (21), (22) and (25), (26)

we actually performed matching of solutions in far

and near field zones upon their boundary and get an

approximate solutions for five radiation potentials

uniformally valid in the whole water domain.

5 HYDRODYNAMIC COEFFICIENTS OF SHIP

MOTIONS

It is convenient to find damping and exciting forces

according to Haskind-Newman approaches where

419

asymptotic expansions of radiation potentials are

used. Thus we avoid the necessity of solving the dif-

fraction problem and simplify calculations because

of the simplicity that asymptotics of potential func-

tions have far from ship hull. According to

(Haskind, 1973, Newman, 1961) wave exciting forc-

es and moments acting on a vessel may be calculated

using such expressions

( ) ( ) ( )

[ ]

5,3,1,2

0

=−=Χ jiFFer

sc

ti

j

ββγβ

σ

. (35)

( )

( )

( )(

)

( )

( )

ξζ

βξα

βξα

ζαα

ζαζξ

β

β

ddHshth

chf

F

F

L

L

H

j

s

c

−

−=

∫ ∫

−

cossin

coscos

,

0

0

00

2

2 0

0

(36)

( ) ( ) ( )

[ ]

6,2,sin

2

1

0

=+=Χ jiFFer

cs

ti

j

βββγβ

σ

. (37)

Functions

( )

β

sc

F

,

are determined by (36), but for

6,2=j

instead of

( )

zxf

j

,

( )

zxg

j

,

is taken. Func-

tions

( )

zx

f

j

,

5,3,1=j

and

( )

zxg

j

,

6,2=j

are giv-

en by (30) - (34).

In expressions (35) - (37)

0

r

- incoming wave

amplitude,

β

- angle between longitudinal axe of

ship hull and vector of wave crests propagation.

The real parts of (35) and (37) must to be taken

into account.

Damping forces and moments are calculated ana-

lyzing the energy flow carried of to infinity from

ship hull by outgoing waves. According to (Haskind,

1973, Newman, 1959) damping coefficients

ij

µ

are

given by formulae

( )

( )

( )( )

( )

( )

.5,3,5,3,,

,

16

000

2

2

2

2

2

3

0

2

3

00

==−×

×−−×

×

Ψ

=

∫ ∫

− −

−−

jidxdxJQ

xQx

q

j

L

L

L

L

j

i

i

ij

ξξααξ

ξα

αρσα

µ

(38)

−

−

2

3

,

2

3 ji

are taken equal to zero for

1== ji

.

( )

( ) ( )( )

( )

( ) ( )

[ ]

,6,2,6,2

,,

,

32

02000

2

2

2

2

4

2

0

4

2

00

==

−+−×

×−−×

×

Ψ

−=

∫ ∫

− −

−−

ji

dxdxJxJP

xPx

q

j

L

L

L

L

j

i

i

ij

ξξαξααξ

ξα

αρσα

µ

(39)

where

2=q

if

ji =

, otherwise

1=q

,

( )

Hth

Hch

H

0

0

2

0

0

1

α

α

α

α

+

=Ψ

, functions

( )

0

,

α

xP

i

and

( )

0

,

α

xQ

i

are given by (32) – (34),

0

J

and

2

J

are

Bessel functions.

For calculation of inertia forces acting on an os-

cillating vessel potential functions in the near field

must be used. To avoid the difficulties of integration

the source-like functions in the vicinity of ship hull

an alternative method is used. The method is based

on the fact proved in (Landau, Lifshits, 1964), dis-

cussed and used in (Kotic, Mangulis, 1962).

It was demonstrated that added masses and damp-

ing coefficients are proportional to integral sine and

cosine transformations of identical functions. It is

enough to find a couple of transformations

( ) ( )

( ) ( )

dx

x

x

ijij

ijij

∫

∞

−

∞−

+∞=

0

22

2

σ

µµ

π

λσλ

, (40)

( ) ( )

( ) ( )

[ ]

dxx

x

x

ijij

ijij

∫

∞

−

∞−

−∞=

0

22

2

σ

λλ

π

σ

µσµ

, (41)

where

( )

σλ

ij

,

( )

∞

ij

λ

,

( )

σµ

ij

,

( )

∞

ij

µ

= added mass

and damping coefficients for frequency

σ

and infi-

nite frequency consequently.

Integrals in (40), (41) are introduced us principle

value integrals. It is known that mostly

( )

0≡∞

ij

µ

.

The value of

( )

∞

ij

λ

for a ship can easily be calculat-

ed using strip method and solving standard integral

equation in the layer

Hz ≤≤0

.

The hydrodynamic characteristics of 200000

DWT tanker (Oortmerssen, 1976) for motions in

shallow water conditions

2.1=

T

H

are demonstrated

in Fig. 1-5. The values calculated using the results of

paper are given by solid lines, while the results of

experiments conducted in towing tank Odessa Na-

tional Maritime University are presented by dots.

The coincidence of theoretical and experimental re-

sults is satisfactory for practical uses.

Coefficients of added mass, exciting forces and

damping are plotted against undimensional frequen-

cy

g

L

σν

=

.

420

Figure 1. Longitudinal exciting forces

Figure 2. Transverse exciting forc-

es

Figure 3. Longitudinal damping

Figure 4. Sway added mass

Figure 5. Transverse damping

6 CONCLUSION

The results derived on the base of MAEM were used

for systematic calculations of hydrodynamic charac-

teristics for a ship floating in regular waves under

shallow water conditions.

The calculated values demonstrated good agree-

ment with the results of model experiments conduct-

ed in towing tank of Odessa National Maritime Uni-

versity.

REFERENCES

Vorobyov, Y.L., 2002. Ship hydrodynamics in restricted waters.

St.P., Shipbuilding, 224 p. (in Russian)

Haskind, M.D., 1973. Hydrodynamic theory of ship motions,

M.Science, 327 p. (in Russian)

Newman, J.N., 1959. The damping and wave resistance of

pitching and heaving ships// Journal of Ship Research.

Vol.3, N1, p.p. 1-19.

Landau, L.D . & Lifshits , E.M., 1964. Theoretical physics, v.5,

Statistical physics, M., Science, 567 p. (in Russian)

Kotic, G. & Mangulis, V. 1962. On the Kramers-Kronig rela-

tions for ship motions// International Shipbuilding Pro-

gress. Vol.9. N97, pp.361-368.

Oortmerssen, G.M, 1976. The motions of a ship in shallow wa-

ter// Ocean Engineering. Vol.3. N4, pp.221-255.

.