a coupled-mode technique for the transformation of ship...

A coupled-mode technique for the transformation of ship-generated

waves over variable bathymetry regions

K.A. Belibassakis*

School of Naval Architecture and Marine Engineering, Section of Ship and Marine Hydrodynamics,

National Technical University of Athens, P.O. Box 64033, Zografos, 15710 Athens, Greece

Received 28 July 2003; accepted 19 May 2004

Abstract

In the present work, a coupled-mode technique is applied to the transformation of ship’s waves over variable bathymetry regions,

characterised by parallel depth-contours, without any mild-slope assumption. This method can be used, in conjunction with ship’s near-field

wave data in deep water or in constant-depth, as obtained by the application of modern (linearised or non-linear) ship computational fluid

dynamic (CFD) codes, or experimental measurements, to support the study of wave wash generated by fast ships and its effects on the

nearshore/coastal environment.

Under the assumption that the ship’s track is straight and parallel to the depth-contours, and relatively far from the bottom irregularity,

the problem of propagation–refraction–diffraction of ship-generated waves in a coastal environment is efficiently treated in the frequency

domain, by applying the consistent coupled-mode model developed by Athanassoulis and Belibassakis [J. Fluid Mech. 1999;389] to the

calculation of the transfer function enabling the pointwise transformation of ship-wave spectra over the variable bathymetry region.

Numerical results are presented for simplified ship-wave systems, obtained by the superposition of source–sink Havelock singularities

simulating the basic features of the ship’s wave pattern. The spatial evolution of the ship-wave system is examined over a smooth but steep

shoal, resembling coastal environments, both in the subcritical and in the supercritical case. Since any ship free-wave system, either in deep

water or in finite depth, can be adequately modelled by wavecut analysis and suitable distribution of Havelock singularities e.g. as presented

by Scrags [21st Int. Conf. Offshore Mech. Arctic Eng., OMAE2002, Oslo, Norway, June 2002], the present method, in conjunction with ship

CFD codes, supports the prediction of ship wash and its impact on coastal areas, including the effects of steep sloping-bed parts.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Ship-wave wash; Variable bathymetry; Coupled-mode model

1. Introduction

Waves generated by fast ships may produce detrimental

effects on the coastal environment, by damaging shoreline

property and disturbing or destructing marine operations

settled in the nearshore and/or coastal environment.

In addition, vessel wakes can be potentially harmful to

shallow water plant and animal life, and may not be

tolerated in closed and/or shallow waters, adjacent to

populated or environmentally sensitive areas. On the other

hand, the reduction of ship speeds in order to prevent wash

impact on the coast can hamper high-speed vessel

operations in coastal waters, e.g. short-sea shipping, that

depend on vessel’s speed for successful service.

Experience has demonstrated that high-speed ship wakes

can be dangerous to both humans and the natural

environment, and vessels with unsafe wakes cannot be

accepted in sensitive areas. To prevent this failure surveys of

environmental (as well as social and ecomomic) impacts

have been published [29], and ‘no harm’ criteria have been

established [30], assisting the design of low-wash vessels.

The importance of this subject in the EU is also evidenced by

recent projects, with main objectives to formulate compu-

tational fluid dynamics (CFDs) methods and software, in

order to assess the wave and wash making characteristics of

High-Speed Crafts (see, e.g. http://www.na-me.ac.uk/

research/ship_stab/flowmart.htm, Refs. [18,33]), as well as

to develop criteria covering the effects of wash-making on

the environment and means by which they can be minimised.

In general, the size and energy of any particular vessel

wake depends on numerous factors, such as the hull length,

0141-1187/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apor.2004.05.002

Applied Ocean Research 25 (2003) 321–336

www.elsevier.com/locate/apor

* Fax: þ30-210-7721397.

E-mail address: [email protected] (K.A. Belibassakis).

http://www.na-me.ac.uk/research/ship_stab/flowmart.htm

http://www.na-me.ac.uk/research/ship_stab/flowmart.htm

http://www.elsevier.com/locate/apor

beam, draft, and shape, the ship’s speed and the vessel type

(e.g. monohull or multihull, displacement, surface piercing,

or planning). Moreover, coastal features and harbor

structures, shoreline distance, bed and shoreline

composition, water depth and bottom contours play an

important role. Many of the above factors are manageable

with today’s technology. Professional vessel design

optimization, selection of route, and effective vessel speed

management, can partially ensure the avoidance of proble-

matic wakes. Important considerations in preventing harm-

ful wakes is to ensure that the vessel design incorporates

proven (by either historically or rigorous design analysis)

low-wake features. Also, the speed schedule selected for the

ship’s route should be suited to the intended operating

region.

Although, there are various methods in use and under

development to define what is an environmentally accep-

table vessel wake, one of the simplest and most readily

understood is that shoreline waves generated by passing

vessels should not significantly exceed the size and energy

of naturally occurring waves in the area under consideration

[29]. This means that part of the study is concerned with the

study of propagation of ship waves towards the coast and the

evaluation of maximum amplitudes of the wave trains

formed by fast ships that reach the shoreline, taking

into account the effects of variable bathymetry and bed

slope [7,25,27].

One way to calculate the effects of variable bathymetry is

to include the irregular bottom surface to the boundary

integral representation of the ship-wave problem [25].

This approach, does not introduce any assumptions,

however, it leads to excessive computational cost, making

it difficult for systematic use. If the distance between the

ship’s track and the bottom irregularity is relatively large,

so that the effect of the variable bathymetry on the

generation of the ship waves can be neglected, another

approach to treat the ship-wash problem is by calculating

the propagation of all wave components in the ship-wave

spectrum, including the effects of refraction and diffraction.

This can be succeeded by coupling ship-wave codes with

coastal wave models [14,27,32 and the references therein].

A broad class of approximation techniques has been

developed for the interaction of free-surface gravity waves

with uneven bottom topography [10]. In the case when the

bed is mildly sloping in the region under consideration,

approximate mild-slope models, such as the classical

mild-slope equation [8], or the modified mild-slope equation

[9,15,19] can be used for the description of wave

propagation and diffraction, with applicability to shorter

waves, and Boussinesq-type models [4,22,24], with

applicability to longer waves.

All the above coastal wave models are suitable for

mildly sloped environments. A low-cost improvement of

mild-slope models, able to treat variable bottom

topography, without imposing assumptions concerning

bottom slope or curvature, is the consistent coupled-mode

model developed by Athanassoulis and Belibassakis [1] and

extended to three-dimensional by Belibassakis et al. [6].

In these works, a local-mode series is used for the

representation of the wave potential in the variable

bathymetry region, involving the propagating and

evanescent modes and including an additional term,

called the sloping-bottom mode, leading to a consistent

coupled-mode system of equations. This model is free of

any simplifications concerning the vertical structure of the

wave field and of any smallness assumptions concerning the

bottom slope and curvature, and it is consistent since it

enables the exact satisfaction of the sloping bottom

boundary condition. Moreover, the consistent coupled-

mode model provides a rapid convergence of the local-mode

series [2], and thus, a few terms are sufficient to accurately

calculate the velocity field up to the boundaries.

The purpose of the present work is to develop and test an

efficient coupled-mode technique for the transformation of

ship’s wave spectrum in variable bathymetry regions,

based on the ship’s near-field wave data in deep-water or

in constant-depth, as obtained by the application of modern

(linearised or non-linear) ship CFD codes, or from

experimental measurements, and permitting the study of

wave wash generated by fast ships and its effects on the

nearshore/coastal environment, without restrictions

concerning the smallness of the bottom slope and/or

curvature. The consistent coupled-mode model is applied

to the problem of propagation–refraction–diffraction of

ship-generated water waves in a coastal environment,

characterised by straight depth-contours parallel to the

ship’s track. The examined environment consists of a

transition region lying between two areas of constant but

different depth (Fig. 1). Owing to the stationarity of the

wave pattern in the ship-fixed frame of reference,

the problem can be treated in the frequency domain,

by transformation of the free-wave components contained

in the ship-wave spectrum in the deeper water subregion and

being obliquely incident to the variable bathymetry region.

A similar approach has been recently applied to random

wave transformation over variable bathymetry regions [3],

permitting the calculation of the point spectra of all physical

quantities of interest, and demonstrating significant

differences in comparison with the mild-slope

approximation.

Numerical results are presented for simplified ship-wave

systems obtained by the superposition of source–sink

Havelock singularities, simulating the basic features of the

ship’s wave pattern. The spatial evolution of the ship-wave

spectra is examined over smooth but steep shoals,

resembling coastal environments, both in the subcritical

and in the supercritical case. On the basis that any ship free-

wave system, either in deep water or in finite depth, can be

represented by wavecut analysis and suitable distribution of

Havelock singularities [11,28], the present model,

in conjunction with ship CFD codes, offers an efficient

and economic alternative for the study of wave wash

K.A. Belibassakis / Applied Ocean Research 25 (2003) 321–336322

generated by fast ships and its impact on the nearshore/

coastal environment, taking into account the effects of

variable bathymetry, including steep sloping-bed parts.

2. Formulation of the problem

We consider the coastal–marine environment shown in

Fig. 1. This environment consists of a water layer bounded

above by the free-surface and below by a rigid bottom. It is

assumed that the bottom surface exhibits an arbitrary

one-dimensional (1D) variation, i.e. the bathymetry is

characterized by parallel and straight bottom contours

lying between two regions of constant but different depth,

h ¼ h1 (deeper water region or region of incidence) and

h ¼ h3 (shallower water region or region of transmission).

A motionless Cartesian coordinate system {x; y; z} is

introduced, with its origin at some point on the mean water

level (in the variable bathymetry region), the z-axis pointing

upwards and the x-axis being normal to the depth-contours.

In this system, the liquid domain D is decomposed in to

three parts DðmÞ; m ¼ 1; 2; 3 (Fig. 1), defined as follows: Dð1Þ

is the subdomain characterized by x , a1 where the depth is

assumed to be constant and equal to h1; Dð3Þ is the

subdomain characterized by x . a2 ða1 , a2Þ where

the depth is also constant and equal to h3 , h1; and Dð2Þ is

the variable bathymetry subdomain, lying between Dð1Þ and

Dð3Þ: The depth function characterizing this environment

is then written in the following form

hðxÞ ¼

h1; x # a1

h2ðxÞ; a1 , x , a2

h3; x $ a3

8>><>>: : ð2:1Þ

The liquid is assumed homogeneous, inviscid and incom-

pressible. The wave field in the examined coastal area is

excited by the waves of a ship, moving with constant speed

U in the deeper water region Dð1Þ; with direction parallel to

the depth-contours. The ship’s track lies at a distance

bðb . la1lÞ from the origin of the earth-fixed system

{x; y; z} (Fig. 1). A ship-fixed coordinate system {x1; y1; z1}

is also introduced, which is steadily translated with respect

to the earth-fixed system {x; y; z}: The longitudinal axis of

the ship Ox1 is directed to the bow, the transverse axis is

Oy1; and the vertical axis Oz1: Thus

x ¼ 2b2 y1; y ¼ x1 þ Ut; z ¼ z1: ð2:2Þ

2.1. The ship free-wave system

Let us first consider the wave system of the ship, without

the effects of reflection/diffraction from the shoal, in the

constant-depth subregion Dð1Þ; as, e.g. obtained by the

solution of the steady ship problem in a horizontally infinite

strip of constant-depth h1: The wave system generated by

the steady forward motion of the vessel (Kelvin wave

pattern) remains stationary in the ship’s system of reference

Fig. 1. Domain decomposition and basic notation.

K.A. Belibassakis / Applied Ocean Research 25 (2003) 321–336 323

{x1; y1; z1}: Under the assumptions of irrotationality

and incompressibility, the fluid motion in the ship-fixed

frame of reference is described by the wave potential

Fðx1; y1; z1Þ; and in the absence of folding (wave breaking,

etc.), the deformed free-surface is described through the

single-valued free-surface elevation, z1 ¼ hðx1; y1Þ:

We shall first restrict ourselves to the case of linearised

steady ship waves, where the free-surface elevation is given

in terms of the wave potential on the free-surface as follows:

h ¼ hðx1; y1Þ ¼U

g

›Fðx1; y1; z1 ¼ 0Þ

›x1

: ð2:3Þ

For a particular free-wave1 component k1 of the ship-wave

system (Fig. 1), of frequency v; propagating at an angle u1

ð2p=2 , u1 , 0Þ with respect to the ship’s direction, and

thus, with direction u ¼ p=2 þ u1ð0 , u , p=2Þ with

respect to the x-axis (taken to be directed normal to the

shoal), the corresponding wave crests (or any other

constant-phase surface of this wave component) will appear

stationary with respect to the ship, only if its phase speed

appropriately matches the ship’s velocity

c ¼v

k1

¼ U cos u1; ð2:4Þ

where the wavenumber k1 ¼ lk1l is connected with the

(circular) frequency v through the linear dispersion relation.

In the region Dð1Þ of constant-depth h1 the latter has the form

v2 ¼ k1g tanhðk1h1Þ: ð2:5Þ

Combining the above equations we obtain

F2ncos2ðu1Þ ¼

tanhðk1h1Þ

k1h1

; ð2:6Þ

where Fn ¼ U=ffiffiffiffiffigh1

p¼ ðkh1Þ

21=2 denotes the bathymetric

Froude number in Dð1Þ; and k ¼ g=U2 is the characteristic

wavenumber parameter.

Eq. (2.6) imposes a specific restriction between the

frequency or the wavenumber and the direction of

each free-wave component k1 ¼ ðkx1; k

y1Þ (or ðk1; u1Þ or

ðv; u1Þ) of the Kelvin free-wave spectrum. This

particular relationship is illustrated in the wavenumber

space k1 ¼ ðkx1; k

y1Þ ¼ ðk1 cos u1; k1 sin u1Þ: defined as the

Fourier counterpart of the physical space ðx1; y1Þ correspond-

ing to the horizontal plane, by means of the dispersion curve

kx1 ¼ Kx

1ðky1Þ; where u1 ¼ tan21 k

y1=k

x1

� �: ð2:7Þ

Eq. (2.7) is parametrically dependent on Fn ¼ U=ffiffiffiffiffigh1

p:

The form of the dispersion curves (2.7) for various values

of the Froude number ranging from subcritical ðFn , 1Þ to

transcritical ðFn < 1Þ and supercritical ðFn . 1Þ; in the

k1-Fourier plane, is shown in Fig. 2. We recognize in this

figure the fact that in the supercritical case, in contrast to

the subcritical case, only free waves with directions above

(or below) the cut-off value ^cos21ð1=FnÞ are present.

Thus, the directions (in the earth-fixed frame of reference) of

Fig. 2. Dispersion curves for various (bathymetric) Froude numbers Fn ¼ 0:5; 0.95, 1.0, 1.1, and 1.25.

1 Outside the vicinity of the ship, where bound waves also exist, the free

waves constitute the dominant part of the free-surface deformation.


the free-waves generated by the steady ship motion that are

able to propagate towards the coast are

0 , u ,p

2þ uco; ð2:8aÞ

where

uco ¼2cos21ð1=FnÞ; Fn . 1

0; Fn # 1

(: ð2:8bÞ

We also observe from the above relations that in the

subcritical case ðFn , 1Þ a cut-off value

vco ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikg tanhðkh1Þ

pð2:9Þ

of the angular frequency exists, corresponding to waves

propagating parallel to the depth-contours u ¼ p=2

(or parallel to the direction of the ship, u1 ¼ 0),

with ky1 ¼ 0 and kx

1 ¼ k ¼ g=U2:

The fact that the Kelvin free-wave pattern hðx1; y1Þ is

stationary in the ship fixed frame of reference, permits us to

introduce a two-dimensional (2D) Fourier transform and

obtain a representation of the free-surface elevation

through the corresponding amplitude spectrum Aðkx1; k

y1Þ;

in the k1-wavenumber space

hðx1;y1Þ¼1

ð2pÞ2Reð1

21dk

y1

ðþ1

21dkx

1Aðkx1;k

y1Þ

�expðiky1y1Þexpðikx

1x1Þ: ð2:10Þ

In Eq. (2.10), ReðsÞ denotes the real part, and ImðsÞ the

imaginary part, of the complex quantity s¼ReðsÞþ iImðsÞ;

where i¼ffiffiffiffi21

p: However, the functional dependence (2.7)

between the wavenumber components (or between

wavenumber and propagation direction) of the free ship

waves, in the ship-fixed frame of reference, permits the

simplification of the double Fourier integral to a single one,

by setting Aðkx1;k

y1Þ¼2pAðk

y1Þdðk

x12Kx

1ðky1ÞÞ; where d

denotes the Dirac-delta function. Using this relation, in

conjunction with the symmetry of the ship-wave pattern

with respect to the longitudinal axis

hðx1;y1Þ¼hðx1;2y1Þ and Kx1ðk

y1Þ¼Kx

1ð2ky1Þ; ð2:11aÞ

the representation of the ship’s free-wave pattern reduces to

the following form

hðx1;y1Þ¼1

2pReð1

21dk

y1Aðk

y1ÞexpðiKx

1ðky1Þx1Þcosðk

y1y1Þ;

ð2:11bÞ

see also Ref. [28].

The fact that the representation of the free-wave system

of the ship can be reduced to a 1D Fourier integral,

permitting the calculation of Kelvin wave spectra from

free-surface elevation data along a longitudinal cut

(parallel to the ship’s path), or along a transverse wavecut

behind the ship, has been first extensively discussed by

Eggers et al. [11]. This approach has been used by many

authors for determining the wavemaking characteristics

and the wave resistance of a ship’s hull [20,26]. The

same property has been also recently exploited by Scragg

[28] for the spectral analysis of ship-generated waves

in finite water depth. In the latter work, a method

has been developed to calculate an equivalent distribution

of finite-depth Havelock singularities, capable of

representing the far-field ship-wave system that closely

match any given wave data set, as, e.g. obtained along a

longitudinal cut parallel to the ship’s track at some

distance, where the effects of bound waves vanish. It is

shown in Ref. [28] that this approach, represents the free

ship-wave system quite well, and, although it is linear in

principle, it can be used not only with free-surface

elevation data coming from the linearised solution, but

also with wave data coming from non-linear ship CFD

codes, as well as with experimental data measured in a

wave tank.

2.2. Transformation of ship-wave spectra

The same fact is also exploited in the present work to

define the input spectrum (or the incident wave forcing) for

the calculation of the spatial evolution of ship-generated

waves in variable bathymetry regions, based on the ship’s

far-field wave data in deep water or in constant-depth. At a

particular longitudinal cut, relatively far from ship’s track in

Dð1Þ; e.g. at x ¼ a1ðy1 ¼ 2b2 a1Þ; also shown by using a

thick dashed line in Fig. 1, the ship-generated, freely

propagating waves are considered to be known in the

earth-fixed frame of reference

hðx ¼ a1; y; tÞ ¼ hðx1 ¼ y 2 Ut; y1 ¼ 2b2 a1Þ; ð2:12Þ

as obtained by the Kelvin wave spectrum (2.11), in the

ship-fixed frame of reference and using Eq. (2.2) to change

variables. Then, the signal of the free-surface elevation at

x ¼ a1; ignoring for the moment the reflection from the

coastal topography, remains stationary

hðx ¼ a1; y; tÞ ¼ hðx ¼ a1; y ¼ 0; t 2 y=UÞ

¼ hðx ¼ a1; y þ Ut; t ¼ 0Þ: ð2:13Þ

We use hIðyÞ ¼ hðx1 ¼ y; y1 ¼ 2b2 a1Þ to denote the ship

waves (at x ¼ a1) propagating obliquely incident to the

variable bathymetry, as, e.g. available from external

calculation or from measured data. Then, if the distance

between the ship and the bottom inhomogeneity is large,

so that the effect of the variable bathymetry on the

generation of the ship waves to be negligibly small,

the ship-wave-wash problem reduces to the calculation of

propagation, of all wave components j ¼ ðjxðjyÞ; jyÞ or

ðv; uðvÞÞ in the spectrum of hIðyÞ

AðjyÞ ¼ðy¼þ1

y¼21expð2ijyyÞhIðyÞdy; ð2:14Þ

taking into account the effects of refraction, diffraction,

and reflection due to variable bathymetry.


In accordance with the previous considerations, the free

ship-wave components in the earth-fixed frame of reference

obey the dispersion relation

F2n sin2ðuÞ ¼

tanhðk1h1Þ

k1h1

; ð2:15aÞ

where

jx ¼ JxðjyÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1Þ

2 2 ðjyÞ2

q; ð2:15bÞ

and

u ¼ tan21ðjy=jxÞ [ ½0;p=2 þ uco�; ð2:15cÞ

in conformity with Eqs. (2.6) and (2.8).

Accordingly, in the framework of linear theory, the wave

field at any point in the nearshore environment ðx; yÞ [Dð2Þ < Dð3Þ can be obtained by Fourier synthesis

hðx; yÞ ¼1

2pRe

ðjy¼þ1

jy¼21AðjyÞHðxlJxðjyÞ; jyÞexpðijyyÞdjy;

ð2:16Þ

where HðxlJxðjyÞ; jyÞ ¼ Hðx; y ¼ 0lJxðjyÞ; jyÞ denotes

the transfer function associated with the free-surface

elevation in the variable bathymetry region. More

specifically, the function Hðx; ylJxðjyÞ; jyÞ describes the

transformation (with the effects of refraction, diffraction,

and reflection) of the free-surface elevation of a monochro-

matic incident plane wave of unit amplitude, characterised

by the wavenumber vector ðJxðjyÞ; jyÞ; or, equivalently, by

the angular frequency v and direction uðvÞ with respect to

the bottom contours (the oblique-incident wave).

In the present work, the transfer function H is calculated

using the consistent coupled-mode model developed by

Athanassoulis and Belibassakis [1] and extended to three

dimensions by Belibassakis et al. [6]. This model is based

on domain decomposition, in conjunction with the

representation of the wave potential in the two constant-

depth subdomains Dð1Þ and Dð3Þ by complete normal-mode

series, and in the variable bathymetry region Dð2Þ by a

rapidly convergent local-mode series. The present approach

treats the full wave problem and does not introduce

assumptions concerning the vertical structure of the wave

field. Thus, one of its important advantages, especially in

intermediate-to-shallow waters, is that it permits the

consistent transformation of the incident wave spectrum

over variable bathymetry regions and the calculation of the

spatial evolution of point spectra of all interesting wave

quantities (free-surface elevation, velocity, pressure),

at every point in the domain. Another aspect is that it can

be extended to treat weakly non-linear waves, and first

results towards this direction have been presented by

Belibassakis and Athanassoulis [5].

The application of the consistent coupled-mode model to

our problem is described in the following sections. Detailed

information concerning this model and its ability to

consistently treat wave propagation over non-mildly sloped

beds with remarkable efficiency can be found in Refs. [1,6].

The rapid convergence of the present coupled-mode

technique [2] facilitates its systematic use, and supports

the parametric study of wave wash generated by various

hulls of fast ships and its spatial evolution over sloping

bottoms, assisting the minimisation of its effects on the

nearshore/coastal environment, without imposing

assumptions concerning the magnitude of the bottom

slope and/or curvature.

3. Propagation of obliquely incident waves over variable

bathymetry

Harmonic waves of unit amplitude and angular

frequency v; the same as the frequency of each component

of the oblique-incident wave system, propagating at an

angle uðvÞ over the variable bathymetry, can be represented

by a velocity potential of the form

Fðx;y;z;tÞ¼Re 2ig

2vfðx;y;z;v;uðvÞÞexpð2ivtÞ

� ; ð3:1Þ

where g is the acceleration due to gravity. The function

f¼fðx;y;zÞ is the normalized potential in the frequency

domain. In our case, the function describing the

transformation (with the effects of refraction, diffraction,

and reflection) of the free-surface elevation at any point

ðx;yÞ on the horizontal plane is equal to the values of the

complex wave potential on the free-surface ðz¼0Þ

Hðx;ylJxðjyÞ;jyÞ¼fðx;y;z¼0Þ: ð3:2Þ

Under the assumptions of linearity, the wave potential

f¼fðx;y;zÞ is obtained as solution of the following

boundary value problem

›2

›z2þ72

!fðx;y;zÞ¼0; in D; ð3:3aÞ

›f

›z2mf¼0; m¼

v2

g.0 on z¼0; ð3:3bÞ

›

›zþ7h7

� �f¼0; on z¼2hðxÞ; ð3:3cÞ

in conjunction with the requirement fðx;y;zÞ and its

derivatives remain bounded as

R¼

ffiffiffiffiffiffiffiffix2þy2

q!1: ð3:3dÞ

In Eq. (3.3), 7¼ð›=›x;›=›yÞ is the horizontal gradient

operator and m¼v2=g is the frequency parameter. The above

problem is forced by the oblique-incident wave,

characterised by the potential

fIðx;y;zÞ¼expðiðjxxþjyyÞÞcoshðkð1Þ0 ðzþh1ÞÞ

coshðkð1Þ0 h1Þ;

ðx;y;zÞ[Dð1Þ:

ð3:4aÞ


The wavenumber vector of the oblique-incident wave is

j¼jxiþjyj¼kð1Þ0 cosðuð1ÞÞiþkð1Þ0 sinðuð1ÞÞj; ð3:4bÞ

where kð1Þ0 ¼k1 is the positive root of the dispersion relation

mh1¼kð1Þ0 h1 tanhðkð1Þ0 h1Þ in Dð1Þ; uð1Þ ¼u denotes the

direction of the incident waves at x¼a1; Eq. (2.15), and

i; j denote the unit normal vectors along the horizontal axes x

and y, respectively.

We remark here the introduction of more complicated

notation for treating the problem of wave propagation in the

variable bathymetry region by means of our coupled-mode

model. Thus, we shall use an upper index within a

parenthesis to indicate the same quantity in the three

subdomains (the two constant-depth subdomains Dð1Þ and

Dð3Þ; and the variable bathymetry subdomain Dð2Þ). Also, a

lower index is introduced to denote the mode number. This is

imposed by the form of the modal series expansion of the

wave field, which contains, except of the propagating mode

(denoted by using the lower index n ¼ 0), also the

evanescent modes (denoted by using natural numbers

1,2,3…) that are generated by the interaction of the freely

propagating ship waves with the variable bathymetry.

In addition, in the variable bathymetry region, a newly

introduced term [1, Sec. 4] is present in the modal

expansion. This extra term, called the ‘sloping-bottom

mode’ (denoted by n ¼ 21) has support only on the

sloping-bottom parts, and is the tool for the consistent

satisfaction of the bottom boundary condition there,

providing a substantial acceleration of convergence of the

local-mode series.

Since the oblique-incident wave fIðx; y; zÞ is periodic

along the y-direction, the potential fðx; y; zÞ is also

y-periodic with the same wavelength l ¼ 2p=jy; where

jy ¼ kð1Þ0 sinðuð1ÞÞ [16,19]. Thus, by introducing partial

separation of variables

fðx; y; zÞ ¼ expðijyyÞwðx; zÞ; ð3:5Þ

we obtain the following 2D problem for the reduced wave

potential wðx; zÞ

72wðx; zÞ2 j2ywðx; zÞ ¼ 0; 2hðxÞ , z , 0; ð3:6aÞ

›wðx; zÞ

›z2 mwðx; zÞ ¼ 0; z ¼ 0; ð3:6bÞ

›

›zþ

dh

dx

›

›x

� �wðx; zÞ ¼ 0; z ¼ 2hðxÞ; ð3:6cÞ

supplemented by the following conditions at infinity

wðx; zÞ!½expðikð1Þ0 cosðuð1ÞÞxÞ

þ AR expð2ikð1Þ0 cosðuð1ÞÞxÞ�coshðkð1Þ0 ðz þ h1ÞÞ

coshðkð1Þ0 h1Þ;

x !21;

ð3:6dÞ

wðx; zÞ! AT expðikð3Þ0 cosðuð3ÞÞÞcoshðkð3Þ0 ðz þ h3ÞÞ

coshðkð3Þ0 h3Þ;

x !þ1:

ð3:6eÞ

In the last equations, AR and AT are the reflection and

transmission coefficients, respectively, and the direction of

the wave uð3Þ in Dð3Þ is given by Snell’s law

uð3Þ ¼ sin21ðkð1Þ0 sinðu1Þ=kð3Þ0 Þ; ð3:6fÞ

see also Ref. [19]. The wavenumbers kðmÞ0 ; m ¼ 1; 3;

appearing in Eq. (3.6) are obtained by the corresponding

dispersion relations

mhm ¼ kðmÞ0 hm tanhðkðmÞ

0 hmÞ; m ¼ 1; 3; ð3:7Þ

formulated at the depths hm; m ¼ 1; 3; respectively.

Finally, on the basis of Eqs. (3.2) and (3.5), we see that

the transfer function required in the integrand of the

right-hand side of Eq. (2.16) for the calculation of

the free-surface elevation in the whole variable bathymetry

region is given by the values of the reduced wave potential

on the free-surface, i.e.

HðxlJxðjyÞ; jyÞ ¼ wðx; z ¼ 0Þ: ð3:8Þ

4. The coupled-mode system of equations

The problem on wðx; zÞ; Eq. (3.6), is treated by means of

the consistent coupled-mode theory, based on the following

enhanced local-mode representation of the reduced wave

potential in the variable bathymetry region Dð2Þ

wð2Þðx; zÞ ¼ w21ðxÞZð2Þ21ðz; xÞ þ w0ðxÞZ

ð2Þ0 ðz; xÞ

þX1n¼1

wnðxÞZð2Þn ðz; xÞ; ð4:1Þ

and similar normal-mode representations in the two

constant-depth strips Dð1Þ and Dð3Þ: In Eq. (4.1), the term

w0ðxÞZð2Þ0 ðz; xÞ is the propagating mode. The remaining

terms wnðxÞZð2Þn ðz; xÞ; n ¼ 1; 2;… are the evanescent modes,

and the additional term w21ðxÞZð2Þ21ðz; xÞ; which exists only in

the variable depth region Dð2Þ; is a correction term called the

sloping-bottom mode, accounting for the bottom boundary

condition on the non-horizontal parts of the bottom.

The function Zð2Þn ðz; xÞ represents the vertical structure of

the nth mode. The complex amplitude of the nth mode wnðxÞ

describes its horizontal pattern. The functions Zð2Þn ðz; xÞ;

n ¼0; 1; 2;…; appearing in Eq. (4.1) are obtained as the

eigenfunctions of local vertical Sturm–Liouville problems,


formulated at the local depth hðxÞ; and are given by

Zð2Þ0 ðz; xÞ ¼

cosh½kð2Þ0 ðz þ hÞ�

coshðkð2Þ0 hÞ;

Zð2Þn ðz; xÞ ¼

cos½kð2Þn ðz þ hÞ�

cosðkð2Þn hÞ; n ¼ 1; 2;…;

ð4:2aÞ

where the eigenvalues {ikð2Þ0 ðxÞ; kð2Þn ðxÞ} are obtained as the

roots of the (local) dispersion relation

mh ¼ 2kh tanðkhÞ; a1 # x # a2: ð4:2bÞ

A specific convenient form of the function Zð2Þ21ðz; xÞ is given

by the polynomial

Zð2Þ21ðz; xÞ ¼ h

z

h

� �3

þz

h

� �2" #

; ð4:2cÞ

however, other choices are also possible [1]. By following

exactly the same procedure as in the latter work, and

using a similar variational principle, the following

coupled-mode system is obtained with respect to the

mode amplitudes wnðxÞX1n¼21

amnðxÞw00nðxÞþbmnðxÞw

0nðxÞþðcmnðxÞ2dmnamnðxÞj

2yÞwnðxÞ

¼0; a1,x,a2; m¼21;0;1;…; ð4:3Þ

where dmn is the Kronecker’s delta, and a prime denotes

differentiation with respect to x: See also Ref. [6].

The coefficients amn; bmn; cmn of the system (4.3) are

defined in terms of, Zð2Þn ðz;xÞ; and can be found in Table 1

of Ref. [1]. The system (4.3) is supplemented by the

following boundary conditions (see also Refs. [6,19]),

ensuring the complete matching on the vertical interfaces

at x¼a1 and x¼a2 separating the three subdomains Dð1Þ;

Dð2Þ and Dð3Þ

w21ða1Þ¼w021ða1Þ¼0; w21ða2Þ¼w0

21ða2Þ¼0; ð4:4aÞ

w00ða1Þþilð1Þ0 w0ða1Þ¼2ilð1Þ0 expðilð1Þ0 a1Þ;

w0nða1Þ2lð1Þn wnða1Þ¼0; n¼1;2;…;

ð4:4bÞ

w00ða2Þ2ilð3Þ0 w0ða2Þ¼0; w0

nða2Þþlð3Þn wnða2Þ¼0;

n¼1;2;3;…;

ð4:4cÞ

where the coefficients lð1Þn ; lð3Þn ; n¼0;1;2;…; are given by

lð1Þ0 ¼kð1Þ0 cosðuð1ÞÞ; lð1Þn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkð1Þn Þ2þðkð1Þ0 sinðuð1ÞÞÞ2

q;

n¼1;2;…

ð4:5aÞ

lð3Þ0 ¼kð3Þ0 cosðuð3ÞÞ; lð3Þn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkð3Þn Þ2þðkð3Þ0 sinðuð3ÞÞÞ2

q;

n¼1;2;…;

ð4:5bÞ

In the above equations,uð3Þ is defined by Eq. (3.6f), {ikð1Þ0 ;kð1Þn }

are the eigenvalues {ikð2Þ0 ðx¼a1Þ; kð2Þn ðx¼a1Þ};which remain

the same all over the region Dð1Þ; and {ikð3Þ0 ;kð3Þn } are the

eigenvalues {ikð2Þ0 ðx¼a3Þ; kð2Þn ðx¼a3Þ}; which also remain

the same all over Dð3Þ: The reflection and transmission

coefficients ðAR;ATÞ appearing in Eqs. (3.6d) and (3.6e) are

calculated from the solution of the coupled-mode system by

AR¼ðw0ða1Þ2expðilð1Þ0 a1ÞÞexpðilð1Þ0 a1Þ;

AT¼w0ða2Þexpð2ilð3Þ0 a2Þ:

ð4:6Þ

An important feature of the solution of the problem (3.6) by

means of the local-mode representation (4.1), is that the

latter exhibits a rapid rate of decay of the order Oðn24Þ

concerning the modal amplitudes lwnðxÞl: Thus, only a few

modes suffice to obtain a numerically convergent solution to

wðx;zÞ; for bottom slopes of the order of 1:1, and higher

[1,6].

The numerical solution of the coupled-mode system,

Eqs. (4.3) and (4.4), is based on the truncation of the system

(4.3) and the local-mode series (4.1), keeping the

propagating mode ðn ¼ 0Þ; the sloping-bottom mode

ðn ¼ 21Þ and a few evanescent modes ðn . 0Þ; and using

a second-order finite difference scheme to discretise the

system of differential equations (4.3) and the boundary

conditions (4.4). As an example of application of the present

coupled-mode model, we consider the case of a smooth

bathymetry, characterised by the following depth function

hðxÞ¼

h1¼4:9m; x,a1¼250m;

h1þh3

22

h12h3

2tanh

2:5px

a22a1

� �a1,x,a2;

h3¼0:5m; x.a2¼50m;

8>>><>>>:

ð4:7Þ

which models a smooth underwater shoal, with maximum

bottom slope smax¼8:6% and mean bottom slope

smean¼4:4%: A sketch of the bottom profile is shown in

Figs. 3(c) and 4 below.

In Fig. 3(a) and (c), the modulus and the phase of the

transfer function

Hmod ¼ lwðx; z ¼ 0Þl;

Hphase ¼ 2ilnðwðx; z ¼ 0Þ=HmodÞ;

ð4:8Þ

in the variable bathymetry region, 250 , x , 50 m; is

plotted by using contour lines, as calculated by the present

coupled-mode model. The case shown corresponds to a

subcritical Fn ¼ 0:743; corresponding to ship speed

U ¼ 10kn ¼ 5:15 m=s at the depth h1 ¼ 4:9 m: The

presentation is restricted to the range of frequencies from

the cut-off valuevco ¼ 1:85 rad=s; in this case (see Eq. (2.9)),

to an upper limit v ¼ 5:8 rad=s; after which the

free-wave components in the ship’s spectrum are practically

deep-water waves propagating nearly normally incident

ðu , 208Þ to the shoal. The corresponding dispersion curve

in the form v ¼ vðuÞ; is plotted in Fig. 3(b). For v . 5:8

rad=s; the contents of the free-wave spectrum are


high-frequency waves propagating without interaction with

the seabed, and the modulus of the transfer function is

practically equal to one (see also Fig. 3a).

For the same environment, the calculated wave potential

fðx; y; zÞ; as obtained by the present method by truncating the

local-mode series (4.1) keeping only the first three modes

ðn ¼ 21; 0; 1Þ; which was found enough for numerical

accuracy, is illustrated by using equipotential lines in Fig. 4.

Both the distributions of the wave potential on the horizontal

and on the vertical planes are shown. The extension of

these lines on the vertical plane below the bottom boundary

ðz ¼ 2hðxÞÞ is maintained, in order to better visualise the

fulfilment of the bottom boundary condition, which is

equivalent to the fact that the equipotential lines intersect

the bottom profile perpendicularly. The effects of wave

refraction and shoaling are very well represented by the

present method, even by using only three terms, and can be

clearly observed in the figure.

5. Numerical Fourier inversion

The last step of the present calculation procedure deals

with the evaluation of the Fourier integral (2.16), providing

the free-surface elevation and the wave field over the

whole variable bathymetry region. In the present work,

this integration is efficiently performed by applying a

Fast Fourier Transform (FFT) technique [23].

The implementation of FFT is based on the discretisation

of the interval jy [ ½2J;J�; where J is appropriately

large, into an even number (2N) of equal-length segments,

with endpoints

jl ¼ 2Jþ ðl 2 1ÞDj;

l ¼ 1;…; 2N þ 1; where Dj ¼J

N:

ð5:1Þ

Also, the interval y [ ½2Y ; Y�; in the physical space, is

subdivided into the same number equal-length segments Dy;

as follows

yj ¼ 2Y þ ðj 2 1ÞDy; j ¼ 1;…; 2N þ 1;

where Dy ¼p

DjNand Y ¼ NDy ¼ p=Dj: ð5:2Þ

On the basis of the above, the integration in the right-hand

side of Eq. (2.16) over the finite interval j [ ½2J;J� is

written in the following discrete form

hðx;yjÞ<Dj

2pRe

X2N

‘¼1

A‘ exp ipð‘21Þðj21Þ

N

� �( );

j¼ 1;…;2N:

ð5:3Þ

Fig. 3. Calculated (a) modulus and (d) phase (in multiples of p) of the transfer function H ¼ wðx; y ¼ 0Þ; concerning ship waves at Fn ¼ 0:743 propagating

over the smooth shoal with depth profile given by Eq. (4.7), and shown in (c). In this case, the corresponding dispersion curve, in the form v ¼ vðuÞ; is shown

in the upper-right subplot (b).


In the above equation, A‘ ¼ Aðj‘ÞHðxlJxðj‘Þ; j‘Þ;

where Aðj‘Þ are the values of the amplitude spectrum AðjyÞ

at the points jy ¼ j‘; as obtained from the incident ship-wave

data along the longitudinal cut x ¼ a1 by means of Eq. (2.14).

The summation in Eq. (5.3) can be very efficiently

performed, simultaneously for all ranges y ¼ yj; by

applying FFT to the array {A‘; ‘ ¼ 1;…; 2N}; if N is

selected to be a power of 2. The problem with the application

of FFT (and of DFT, in general) is that undersampling in the

Fourier j-domain causes aliasing in the physical y-domain,

due to the periodicity assumed by the discrete Fourier

transform. Actually, the evaluation of the right-hand side

of Eq. (5.3) does not yield the values of the functionhðx; yjÞ; at

the points y ¼ yj; but ratherP1

n¼21 hðx; yj þ 2nYÞ [12].

Taking this fact into account, we obtain from Eq. (5.4)

the following result

hðx; yjÞ <Dj

2pRe

X2N

‘¼1

A‘ exp i2pð‘2 1Þðj 2 1Þ

2N

� �( )

2X1

n¼21n–0

hðx; yj þ 2nYÞ;

j ¼ 1;…; 2N:

ð5:4Þ

Thus, by selecting the sampling interval Dj to be fine

enough, so that Y also to be large, the aliasing effect from

(physical) ranges lyl . Y included in the second sum of

the right-hand side of Eq. (5.4) is made small, as a

consequence of the geometrical attenuation of the free-

surface elevation (and of the wave energy) at large y-ranges

on the horizontal plane. This, however, imposes a restriction

to the physical length of the input data used for specifying

the incident wave spectrum, that, usually, have to extend

many shiplengths behind the ship.

One way to further minimise the aliasing effect is offered

by shifting the contour of integration of Eq. (2.16) to the

complex jy-plane [21]. This kind of therapy has been used for

treating similar wave propagation problems, as, e.g. with

wavenumber integration techniques in underwater acoustics

problems [13, Ch. 4]. Such a treatment necessitates the

extension of our formulation, Eq. (3.6), to complex jy; which

is possible, and an analytic representation of the incident

wave hIðyÞ and of the associated amplitude spectrum AðjyÞ;

defined by Eq. (2.14). The latter becomes possible by

implementing spectral-analysis techniques based on singu-

larity distributions with an analytical structure, as [28], for

the representation of ship-wave system in the far-field, and

part of future work is planned towards this direction.

6. Numerical examples and discussion

In this section, we shall present numerical results

illustrating the spatial evolution of ship waves over

variable bathymetry regions and the associated

Fig. 4. Calculated wave potential over the shoal with bathymetry defined by Eq. (4.7), as obtained by the present method using three modes. The case shown corres-

ponds to harmonic waves of angular frequency v ¼ 2:7 rad=s; obliquely incident to the shoal with direction u ¼ p=4; also indicated by using an arrow. This pair

ðv; uÞ matches the dispersion curve of ship waves in the subcritical case Fn ¼ 0:743; corresponding to ship speed U ¼ 10kn ¼ 5:15 m=s at the depth h1 ¼ 4:9 m:


horizontal ship-wave patterns, as obtained by the present

coupled-mode method. The basic input (forcing) is the

free-surface elevation hIðyÞ ¼ hðx1 ¼ y; y1 ¼ 2b2 a1Þ of

the ship-wave system along a longitudinal cut (x ¼ a1 or

y1 ¼ 2b2 a1) in the constant-depth subdomain Dð1Þ;

relatively far from the ship (at a distance of the order of

one characteristic wavelength), where the bound waves

are fading out and only free waves constitute the

associated amplitude spectrum AðjyÞ:

6.1. Simulation of ship-wave data

To test the present method artificial ship-wave data are

used for the specification of the incident wave, which are

easily obtained by the superposition of simple source–sink

Havelock singularities, simulating the basic features of the

ship’s free-wave system. The general expression of the

finite-depth Green’s function for a Havelock singularity,

located at ðx01; y01; z

01Þ in the ship’s frame of reference is

given by Refs. [31, Eq. (13.37),17, Eq. (2.177)]. It consists

of the free-space singularity and its mirror below the rigid

bottom (located at depth h1 in the constant-depth

subdomain Dð1Þ), a near-field contribution expressed by a

double integral and the wavelike far-field contribution

expressed by a single integral. Since we are interested in

the far-field pattern, we use only the far-field part of this

singularity, which for a source of strength s can be

expressed as follows

GFARðx1;y1;z1lx01;y01;z01Þ

¼s

pHðx012x1Þ

ðu1¼p=2

u1¼0du1 coshðk1ðz1þh1ÞÞ

�coshðk1ðz01þh1ÞÞsinðkx

1ðx12x01ÞÞcosðky1ðy12y01ÞÞ

ð6:1Þ

where H denotes the Heaviside unit step function and

k1¼ðkx1;k

y1Þ¼ðk1 cosu1;k1 sinu1Þ the wavenumber vector

obeying the dispersion relation (2.6) (see Section 2).

For source and field points on the undisturbed free-surface

ðz¼z01¼0Þ and the source at the centerline of the ship ðy01¼0Þ

the above expression simplifies considerably, and the far-

field becomes symmetric with respect to the Ox1 axis. In

this case, the associated free-surface elevation, as obtained

by Eq. (2.3) is

hFARðx1;y1lx01Þ¼sU

pgHðx012x1Þ

ðu1¼p=2

u1¼0du1Fðu1Þcosh2ðkh1Þ

�kx1cosðkx

1ðx12x01ÞÞcosðky1y1Þ; ð6:2aÞ

where

Fðu1Þ¼kþk1 cos2ðu1Þ

12kh1 sec2ðu1Þsech2ðu1Þsec2ðu1Þ

�sechðk1h1Þexpð2k1h1Þ; ð6:2bÞ

k¼g=U2:

As it is demonstrated by Scragg [28], the above expression

can be put in a more convenient form from the point of view

of numerical calculations, by changing the variable of

integration from u1 to ky1 using the relation k

y1¼k1 sinu1 and

Eq. (2.6). Accordingly, we obtain the following expression

hFARðx1;y1lx01Þ¼sUk

pgHðx012x1Þ

�ðk

y

1¼1

ky

1¼21

dky1

kx1

cos2ðu1Þ2kh1 sech2ðu1Þ

›u1

›ky1

�Re½expðikx1ðx12x01Þþ ik

y1y1Þ�; ð6:3aÞ

where

›u

›ky1

¼ k1 cosðu1Þ 1þð2k1=kÞ

2sin2ðu1Þcosh2ðk1h1Þ

sinhðk1h1Þcoshðk1h1Þ2k1h1

! !21

:

ð6:3bÞ

In the numerical results presented below, we use the above

formulae to simulate the forcing of the problem, obtained by

the superposition of a source and a sink with strengths ^s

located at a distance L apart each other, where L stands for the

characteristic ship’s length, i.e.

hIðyÞ¼hFARðx1;y1¼2b2a1lx01¼LÞ

2hFARðx1;y1¼2b2a1lx01¼0Þ: ð6:4Þ

However, this is not a restriction, since general data can be

used for the specification of the incident wave hIðyÞ; as, e.g.

are available from the output of ship CFD codes or from

experimental measurements. This enables the convenient

coupling of the present method with ship CFD codes and/or

experimental wave data.

6.2. Presentation of numerical results and discussion

The calculated ship-wave patterns and their

transformation over the variable bathymetry region for

two cases examined are presented in Fig. 5. We consider

a ship of length L ¼ 38 m (corresponding to a high-speed

small ferry), moving at speeds U ¼ 10kn ¼ 5:15 m=s and

U ¼ 17kn ¼ 8:75 m=s; respectively, in the shoaling

environment characterised by the depth function defined

by Eq. (4.7). The bathymetry profile is also plotted in the

vertical sections of Fig. 5, keeping the same scale with

the horizontal axes. The free-surface elevation is illustrated

by means of colorplots and the associated colorbars.

(However, since the source strength s has been arbitrarily

selected for simulating ship-wave data by means of

Eq. (6.3), the calculated free-surface elevation has only

relative value). In both cases, the ship’s path is taken to be

located at x ¼ 2100 m; in the constant-depth subregion Dð1Þ

ðx , 250 mÞ; where the depth is h1 ¼ 4:9 m: The first

case is a characteristic subcritical one, corresponding to

bathymetric Froude number Fn ¼ 0:743 and Froude number

based on ship’s length FnL ¼ U=ffiffiffiffigL

p¼ 0:267: The second


case is a characteristic supercritical one, corresponding to

Froude numbers Fn ¼ 1:263 and FnL ¼ U=ffiffiffiffigL

p¼ 0:453;

respectively.

In both cases, the incident free-surface elevation hIðyÞ is

simulated using Eqs. (6.3) and (6.4), along the longitudinal

cuts at x ¼ 250 m and x ¼ 2150 m; i.e. at a distance

(lateral offset) of 50 m from each side of the ship’s track.

These wavecuts are shown using dashed lines in Fig. 5.

Then, the incident wave information at x ¼ 250 m is

propagated and transformed by means of the present method

towards the shoal, in the region 250 , x , 50 m; where the

depth changes continuously from 4.9 to 0.5 m. Also, in

Fig. 5. Ship-wave patterns calculated by the present method (a) in the subcritical case Fn ¼ 0:743; corresponding to ship speed U ¼ 10kn at the depth

h1 ¼ 4:9 m (ships’ length L ¼ 38 m and FnL ¼ U=ffiffiffiffigL

p¼ 0:267), and (b) in the supercritical case Fn ¼ 1:263; corresponding to ship speed U ¼ 17kn at the

same depth ðFnL ¼ U=ffiffiffiffigL

p¼ 0:453Þ; with the effects of the bottom topography with depth function defined by Eq. (4.7). The dashed lines indicate the positions

of the longitudinal cuts where the input data hIðyÞ are specified, simulated using Eqs. (6.3) and (6.4).


order to check the accuracy of our method, the same

wave data on the other side of the track of the ship

(at x ¼ 2150 m) are also numerically propagated by

applying the present method to the constant-depth subregion

2250 , x , 2150m; where the depth is constant and equal

to 4.9 m. Finally, for illustrating the overall compatibility of

the transformed data with the input data, in the same

colorplots of Fig. 5(a) and (b) the free-surface elevation in

the strip 2150 , x , 250 m (along the ship’s track) is also

overplotted, as calculated by integration using Eqs. (6.3)

and (6.4).

We can clearly observe in these plots the effects of

refraction and shoaling on the horizontal far-field patterns.

The main effect of refraction is the asymmetry of the

ship-wave pattern produced by the bending of

the wavefronts (or constant-phase surfaces) in the variable

bathymetry region. This is more observable with the

diverging waves (the only ones in the supercritical case),

which as propagate to more shallow waters become nearly

parallel to the depth-contours, as expected. On the other

hand, the main effect of shoaling is the increase or decrease

of the local wave amplitudes, depending on the specific

incident wave conditions and the bottom surface geometry.

Turning into more specific details, in Figs. 6 and 9,

the amplitude spectra AðkyÞ and the free-surface elevations

hIðyÞ associated with the incident wave are shown, as

calculated by Eqs. (6.3) and (6.4) along the longitudinal cuts

at x ¼ 250 m and x ¼ 2150 m (at lateral offsets 50 m from

the ship’s track), and corresponding to the subcritical case of

Fig. 5(a) and to the supercritical case of Fig. 5(b),

respectively. Also, for the same two cases, the ship-wave

(or wash) profiles, as calculated by the present method, at

symmetrically located longitudinal cuts on each side of the

ship are plotted in Figs. 7 and 8, respectively. The locations

of the wavecuts with respect to the ship’s track are at: (i)

x ¼ 250 m and x ¼ 2150 m (h ¼ 4:9 m), (ii) x ¼ 0 m

ðh ¼ 2:7 mÞ and x ¼ 2200 m ðh ¼ 4:9 mÞ; and (iii) x ¼ 50

m ðh ¼ 0:5 mÞ and x ¼ 2250 m ðh ¼ 4:9 mÞ:

We observe in Fig. 7 that, in the subcritical case, the

maximum wave amplitudes over the shoal, in comparison

with the ones in the constant-depth region ðh ¼ 4:9 mÞ

are slightly decreased, and their longitudinal position is

upwave shifted (towards the ship). The de-amplification of

the ship wash in the subcritical case examined is justified by

the fact that, in the frequencies involved in the incident

wave spectrum (shown in Fig. 6a), the transmission

coefficient, defined by Eq. (4.6), is very small, and the

transfer function, defined by Eq. (3.8) and plotted in Fig. 3,

is smaller than unity. On the contrary, in the supercritical

case shown in Fig. 8, the maximum wave amplitudes over

the shoal, in comparison with the ones in the constant-depth

region ðh ¼ 4:9 mÞ are slightly increased, and their

longitudinal position is downwave shifted (towards the

ship’s wake). The amplification of the ship wake wash at

Fig. 6. (a) Ship-wave amplitude spectrum AðkyÞ and (b) free-surface elevation hIðyÞ along the longitudinal cut at x ¼ 250 m (at a distance 50 m from ship’s

track), in the subcritical case of Fig. 5(a).


lower depths, in the supercritical case examined, is due to

the higher values of the transfer function and of the

transmission coefficient, associated with the lower

frequency components involved in the corresponding

amplitude spectrum (Fig. 9(a)).

In concluding this section, we remark that, in accordance

with the present development, the calculation of the transfer

function, which serves as the key for obtaining the spatial

evolution of the ship-wave spectra over variable bathymetry

regions, is based on the solution of the reduced 2D problem

Fig. 7. Ship-wave profiles at symmetrically located longitudinal cuts with respect to the ship’s track, for the subcritical case of Fig. 5(a).

Fig. 8. Ship-wave profiles at symmetrically located longitudinal wavecuts with respect to the ship’s track, in the supercritical case of Fig. 5(b).


on the vertical plane, Eq. (3.6), for a number of frequencies.

This is accomplished by the solution of the coupled-mode

system of ordinary differential equations, Eq. (4.3), over the

horizontal x-axis. The reduction of dimensionality of the

problem, in conjunction with: (i) the rapid convergence of

the modal series (4.1), permitting the truncation of

the coupled-mode system into a small subset of equations;

and (ii) the application of the FFT scheme, Eq. (5.3), for the

numerical Fourier inversion, makes the present approach

very fast and economic from the computational point of

view, without introducing mild-slope assumptions.

For example, the calculations in the cases shown are

performed using Matlab in a Pentium IV, 2.4 GHz machine,

requiring computational time of the order of minutes.

This fact enables the systematic use of the present

method for the parametric study of wake wash in large

variable bathymetry domains, including steep bottom

features, as well as its application to hull-form optimisation

of high-speed vessels, taking into account the impact of

wake wash on the coastal environment.

7. Conclusions

In the present work, a cost-effective method is presented

for the transformation of ship’s waves over variable

bathymetry regions, characterised by parallel depth-

contours, supporting the study of wave wash generated by

fast ships and its impact on the nearshore/coastal

environment. The present method can be used in conjunc-

tion with ship’s near-field wave data in deep water or in

constant-depth, as obtained by the application of modern

(linearised or non-linear) ship CFD codes, or experimental

measurements in a wave tank.

Under the assumption that the ship’s track is parallel to the

depth-contours and relatively far from the bottom irregula-

rity, the problem of propagation–refraction–diffraction of

ship-generated waves in a coastal environment can be

conveniently treated in the frequency domain, by applying

the consistent coupled-mode model [1,6] to the calculation of

the transfer function enabling the transformation of ship-

wave spectra over variable bathymetry regions.

Numerical results are presented for simplified ship-

wave systems, obtained by the superposition of source–

sink Havelock singularities, simulating the basic features

of the ship’s wave pattern, over smooth but steep shoals

resembling coastal environments. Since any ship free-

wave system, either in deep water or in finite-depth, can

be adequately modelled by wavecut analysis and suitable

distribution of Havelock singularities [28], the present

model, in conjunction with ship CFD codes, supports the

prediction of ship wash and its impact on coastal areas,

including the effects of steep sloping-bed parts.

Future work is directed towards: (i) the prediction of run-

up of ship-waves in sloping beaches, including the effects of

weak non-linearity and dispersion; and (ii) the systematic

Fig. 9. (a) Ship-wave amplitude spectrum AðkyÞ and (b) free-surface elevation hIðyÞ along the longitudinal cut at x ¼ 250 m (at a distance 50 m from ship’s

track), in the supercritical case of Fig. 5(b).


use of the present method for the evaluation and hull-form

optimisation of high-speed vessels operating in coastal

waters.

Acknowledgements

The present work has been partially supported by the

Section of Ship and Marine Hydrodynamics of National

Technical University of Athens, in the framework of the

project ‘Wave Phenomena in the Sea and in the Coastal

Environment’.

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