Ship Waves in a Stratified Fluid

Abstract

The far-field asymptotics of the linear waves excited by a moving object are obtained for a fluid consisting of n density layers (n is arbitrary). The structure of the perturbations of free surface and density interfaces is analyzed as a function of the depth of the object and its velocity. The amplitudes of different types of waves are compared. The present model also confirms and generalizes the features of the ship waves that were known previously only for the cases of one-layer and two-layer fluids.

Full text

Journal of Ship Research, Vol. 45, No. 1, March 2001, pp. 1–12
Journal of
Ship Rese arch
Ship Waves in a Strati’ed Fluid
Timour Radko
Dep artm ent of Earth, Atmospheric, and Planetary Sciences, MIT, Cambridge, Massa chusetts
The far-’ eld asymptotics of the linear waves excited by a moving object are obtained for
a ‘ uid consisting of n density layers (n is arbitrary). The structure of the perturbations of
free surface and density interfaces is analyzed as a function of the depth of the object
and its velocity. The amplitudes of different typ es of waves are compared. The prese nt
model also con’ rms and generalizes the features of th e ship waves that were known
previously only for the cases of one-layer and two-layer ‘ uids.
1. Introduction
Problems of generation and evolution of waves on  ows with
a free surface constitute a large and important part of the the-
ory of waves. In addition to their general theoretical signi cance,
the results of studies on the subject have several practical appli-
cations. For instance, the wave resistance for ships and other
towed a nd self-propelled oceanic devices may be estimated. And
of broader geophysical signi cance, a theoretical analysis of the
strong interaction of large-scale  ows with bottom topography
can be performed.
Surface waves that are commonly seen spreading behind a
moving ship have attracted the attention of scientists since the
beginning of the century. The  rst results on the problem, which
is now known in hydrodynamics as the theory of ship waves, are
due to Kelvin (1905).
1
Kelvin considered the  ow due to a sin-
gular disturbance moving with uniform velocity along the surface
of an incompressible inviscid  uid of in nite depth and presented
a famous “herring bone pattern” of the steady waves formed in
a domain “between two lines draw n from the ship’s bow and
inclined to the wake on its two sides at equal angles of 19

28
0
.”
Kelvin’s solution was later rediscovere d by Havelock (1908) who
extended Kelvin’s theory to the case of a  nite depth of a  uid, H.
It follows from Havelock’s work that if the ve locity U of the pres-
sure impulse is less than th e speed of the long gravity waves, the
free surface behind the impulse is disturbed by two wave systems,
a tra nsverse an d a divergent one, also known as a “longitudinal”
Manuscript received at SNAME headquarter s November 4, 1999; revised
manuscript received July 13, 2000 .
1
An extremely interesting and controversial history of the discovery of
ship waves can be found, for example, in Eggers (1990). Here we present a
conventional view of the events that led to the formulation of the shi p wave
theory.
system, as shown in Fig. 1 (from Stoker 1957). The aforemen-
tioned studies considered only the far- eld region, the precise
determination of the form of the wave surface close to the ves-
sel being extremely dif cult and a problem not yet totally solved
even from a num erical point of view. Within such approximations
the investigation of ship waves in water of constant density was
completed by Ursell (1960), who properly treated the portion of
the ship waves along the wedge line where the method of sta-
tionary phase yields the singularity of the solution. The rigorous
mathematical analysis of the far- eld asymptotics and the discus-
sion of limitations of the classical theory of ship waves was given
by Maz ’ ya & Vainberg (1993). In addition to th e aforementioned
analytical theories, ship waves have also been studied numeri-
cally (Miyata & Nishimura 1985) and experimentally (Arabadzhi
1996).
There are several applications of the ship wave theory in which
the effects of the density strati cation are of fundamental impor-
tance. These include a so-called “dead-water” phenomenon where
slow ships lose their speed and steering capabilities when mov-
ing in the regions of strong density strati cation. This effect was
explained (Miloh et al 1993) by increased wave resistance of the
ship due to the work done in generating the internal gravity waves
(in addition to the surface waves considered above) . The devel-
opment of the theory was also motivated in part by its realization
in nature in meteorological observations: the trapped lee waves
in the atmospheric  ows over obstacles of appropriate scale are
dynamically analogous to ship waves. The reader is referred to
a review by Wurtele et al (1996) for geophysical applications of
the ship wave theory.
The inclusion of strati cation, however, prevents the possibil-
ity of introducing the (single) velocit y potential and signi cantly
complicates th e problem. First results for a strati ed problem
were obtained by Hudimac (1961) who considered a two-layer
MAR CH 2001 0022-4502/01/4501-0001$00.49/0 JOU RNAL OF SHIP R ESEARCH 1

Fig. 1 Wave crests for a straight course. Surface is perturbed by two
types of waves: transverse and divergent. From Stoker (1957)
ocean with an in nite ly deep lower layer. He showed that in this
case (in addition to the surface mode of the ship waves) there
is also an internal wave mode which has a pattern similar to the
classical Kelvin ship waves. When the speed of the source is less
than some c ritical value the internal mode consists of a system
of divergent and transverse waves (as in Fig. 1), and when the
speed is suf ciently large only the divergent waves are present.
Hudimac’s theory was extended by Crapper (1967) who sugge sted
a simpler analytical approach. Crapper pointed out that the sur-
face mode is only slightly affected by the strati cation, while th e
relative amplitude of the internal wave depends on the (upper
layer) depth and on the density diffe rence between the layers, and
can be very large.
For the purpose of examining the patte rns of ship waves the
so-called ray theories proved to be very successful. Keller &
Munk (1970) suggested a general method for obtaining the two-
dimensional patterns of the internal m odes which required only
the knowledge of the dispersion relationships for the free wave
modes . They found an explicit expression for the pattern of the
internal modes for the area near the source path, where the inter-
nal modes are concentrated for supercritical speeds of propaga-
tion. Later Yin & Zhu (1989) simpli ed the ray method and
extended consideration of the patterns to the case of the subcrit -
ical (fo r the internal mode) speeds as well; they considered the
one-layer model and the two-layer model with an in nitely deep
lower layer. The ray methods, however, provide little information
abou t the amplitude of the disturbance as a function of the various
parameters of the problem. The latter question was addressed by
Tulin & Miloh (1991) who estimated the amplitude of the inter-
nal waves in terms of the spectral amplitude function generated
by the ship for the supercritical velocities.
It should be noted that most of the theoretical studies con-
sider a highly supercritical case when th e speed of the ship is
much greater than the phase speed of the internal waves. Recently
Yeung & N guyen (1999) considered the most general case of
the  uid consisting of the two layers of  nite depth and did not
assume that the density difference is small (or, equivalently, that
the speed of the object is large compared to the velocity of the
internal waves). These authors consider in detail the behavior of
the two resulting modes (surface and internal) of ship waves and
found that for very slow motion each of the modes is con ned
to an angle —ˆ— < 19

28
0
and look s like the well-known Kelvin
wake. With an increase of the speed of propagation this angle also
increases until it reaches 90

for the certain (mode-dependent)
critical velocity. When the velocity is further increased the angle
of ship waves starts to decrease, but the pattern of the ship waves
then consists only of the divergent waves.
The present study extends and rationalizes Yeung & Nguyen’s
(1999) results in several ways. We examine the surface and inter-
nal waves caused by a moving subme rged body in a strati ed
three-dimensional  uid consisting of n layers of constant density
(n is arbitrary). The moving body is represented by a mass force
acting on a  nite volume of  uid which is located in the arbitrary
layer  < n. This model allows us to analyze how the amplitudes
of various types of waves (given by the elevation of the free sur-
face and the density interfaces) depend on the location of our
object, i.e. , on its depth and on the number 4 5 of a layer con-
taining the body. First we consider the case when the propagation
velocity U is asymptotically small (highly subcritical propa ga-
tion) and derive an explicit expression for the surface mode and
for the internal modes (or, more correctly for our case, “interfa-
cial” modes) of the ship waves. We analytically show that in this
limit all the internal modes as well as the surface mode will be
con ned to the angle —ˆ— < 19

28
0
and consist of the divergent
and trans verse systems. The surface mode in this limit is not sig-
ni cantly affected by the strati cation, and (if the object is mov-
ing at a suf ciently deep leve l  25 the perturbation of the free
surface is determined mostly by the long internal modes, rather
than by the surface mode. When the periodic oscillations of the
amplitude of forc ing are considered (Se ction 4) we observe the
propagating surface and internal waves superimposed on the ship
wave pattern. In Section 5 our asymptotic (small U ) results are
generalized to the case where the speed of the object is compara-
ble to the speed of the internal waves. The results for the  nite U
case con rm our asymptotic predictions (Section 3) and are con-
sistent with the (two layer)  ndings of Yeung & Nguyen (1999)
with regard to the structure of the surface and the internal modes
for subcritical and critical propagation. The main purpose of the
present paper is to estimate the amplitudes of the various systems
of waves on the free surface and on the density interfaces, and to
demonstrate how the previous results for one-layer (Kelvin 1905)
and for two-layer  uids (Yeung & Nguyen 1999, among others)
could be extended to the models with more sophisticated repre-
sentation of the strati cation.
The multilayer model considered here is capable of approxi-
mating a rather general density strati cation, and at the sam e time
it is much simpler, both conceptually and in terms of analytical
tractability, than models with (more realistic) continuously vary-
ing density. The problem of waves generated by a moving source
in a continuously strati ed  uid ha s attracted considerable atten-
tion in the literature, although mostly for the case of a spher e
moving in a vertically unbounde d  uid wit h uniform strati ca-
tion, and such m odels will not be discussed herein. The reader
is referred to Voisin (1994) for a review of the known results for
the continuous problem.
2. One-layer model
In order to establish notation and to develop a method for solv-
ing the general problem let us  rst re-derive the classical solution
(Havelock 1908), extending it to the case of a  nite volume dis-
turbance. The reader who is already familiar with the classical
2 MA RCH 2001 JOU RNAL OF SHIP RESEARCH

one-layer ship wave problem may proceed directly to the strati-
 ed model in Section 3.
Consider waves excited by a source moving at constant speed
U in the negative x direction in an incompressible inviscid  uid
of  nite depth H . For reasons of analytical tractability it is con-
venient to use a mass force
E
F 4F
x
1 F
y
1 F
z
5 as a disturbance (rather
than consider the inviscid over ow of the submerged body or
the “source of mass” model). In the coordinate system associated
with the moving object the linearized Euler equations reduce to
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
¡u
¡t
CU
¡u
¡x
D ƒ
1

¡p
¡x
CF
x
¡v
¡t
CU
¡v
¡x
D ƒ
1

¡p
¡y
CF
y
¡w
¡t
CU
¡w
¡x
D ƒ
1

¡p
¡z
CF
z
¡u
¡x
C
¡v
¡y
C
¡w
¡z
D 0
(1)
where 4u1 v1 w5 are the velocities in the moving coordinate sys-
tem. Equation (1) will be solved subjec t to the (linearized) bound-
ary conditions
p D g‡1 w D
¡‡
¡t
CU
¡‡
¡x
1 for z D 0
w D 0 for z D ƒH
The  uid is assumed to be at rest initially: u D v D w D 0 and
the elevation of the free surface ‡ D 0 at t D0. Consider the m ass
force
E
F which is consta nt in time for t > 0 and
E
F D 0 otherwise.
In order to  nd a resulting wave  eld 4t !Cˆ5 let us employ
the Fourier transform in x and y and the Laplace-Carson trans-
form
2
in time. When these are applied to the linear system (1)
the resulting ordinary differential equations in the new variables
k, l, z, and S can be reduced to a single equation for the pressure
image Op:
d
2
Op
dz
2
ƒŠ
2
Op D 4z5 (2)
where the horizontal wavenumber Š D
p
k
2
Cl
2
and  D4ik
b
F
x
C
il
b
F
y
C
d
dz
b
F
z
5. The boundary conditions reduce to
8
>
<
>
:
d Op
dz
C
4S CUik5
2
g
Op D0 z D 0
d Op
dz
D 0 z D ƒH
(3)
The general solution of the homogeneous version of (2) is a com-
bination of hyperbolic functions cosh4Šz5 and sinh4Šz5 and there-
fore the complete solution of (2) with the boundary conditions
(3) is obtained by a method of variation of constants resulting in
O‡ D
Op
g
zD0
D
1

R
ƒH
0
cosh Š4H Cz
0
54z
0
5dz
0
gŠ sinh4ŠH5 C4S CUik5
2
cosh4ŠH5
(4)
2
Laplace-Carson transform
e
E4S5 D S
R
Cˆ
0
E4t5 exp4ƒSt5dt is similar to
the Laplace transform but it is more convenient to use for problems of wave
generation by impulsive forcing.
The amplitudes of the ship waves and the corresponding wave
resistance signi cantly depend on the shape of the moving object
(Kostyukov 1968) and such a dependence will not be considered
here. Instead, in order to obtain the general qualitative properties
of the ship waves, we will (rather arbitrarily) choose the forcing
function such that the theory would be easily amenable to ana-
lytical treatment. Consider a compact forcing function
E
F 4t > 05
which is nonzero only within a speci ed volume 4—x—< a, —y—< b,
—z Ch
f
— < ãh5 and
F
z
D D4zCãh Ch
f
5 for z < ƒh
f
F
z
D D4ƒz Cãh ƒh
f
5 for z > ƒh
f
F
x
D F
y
D 0
(5)
For the force in equations (5) div4
E
F 5 DD if ƒh
f
ƒãh < z < ƒh
f
and div4
E
F 5 D ƒD for ƒh
f
< z < ƒh
f
Cãh, and since 4z5 is
just a Fourie r image of  div4
E
F 5, equation (4) simpli es to
O‡ D
2D

2
sin4ak5 sin4bl5
kl

sinh Š4H ƒh
f
56cosh4Šãh5 ƒ17
Š6gŠ sinh4ŠH5 C4S CUik5
2
ch4ŠH57
Now return to the physical space x1 y:
‡4x1 y1 S5 D
2Dab

2
g
Z
2
d
Z
Cˆ
0
d‹

sin a
‹
H
cos  sin b
‹
H
sin 
a
‹
H
cos  b
‹
H
sin 

sinh ‹
H ƒh
f
H
cosh ãh
‹
H
ƒ1
‹ sinh ‹ C
4SHCUi‹ cos  5
2
Hg‹
cosh ‹
 exp6i ‹r cos4ˆ ƒ57
where the new (nondimensional) variables r 1 ‹1 ˆ1  are de ned
by k D
‹
H
cos 1 l D
‹
H
cos 1 x D rH cos ˆ1 y DrH sin ˆ.
The properties of the Laplace-Carson transform are such
that for any function E4t5, eventually converging to a steady
 nite value li m
t!Cˆ
E4t5 D E
0
, its limiting form could be sim-
ply obtained from its transform
e
E4S52 lim
s!C0
e
E4S5 D E
0
0 This
allows us to obtain the structure of the ship waves for t ! Cˆ
by considering the limit S ! C0. If S D 0 there is a pole ‹
ü
in
the inner integral in the above expression; considering in nites-
imally small but positive S allows us to properly compute the
contribution from the pole in a way which would satisfy the radi-
ation conditions. (The a lternative methods for treating the poles
which appear in the literature for similar problems usually involve
inclusion of small  ctitious dissipative forces; use of the dissipa-
tive methods and the Laplace-Carson transform certainly yields
identical results.) For small positive S the pole is displaced in the
positive Im4‹5 direction on the complex ‹-plane if ƒ

2
<  <

2
and in the negative Im4‹5 direction otherwise. This implies that
the path of integration should be deformed as shown in Fig. 2.
In order to employ the residual theorem w e close the integrat-
ing contour using the curve C
1
if cos4ˆ ƒ5 > 0 and using C
2
MAR CH 2001 JOURNAL OF SHIP RESEAR CH 3

Fig. 2 Path of integration in complex ‹-plane (see text)
otherwise. Integration over these curves is asymptotically small in
the far  eld 4r ! ˆ5 compared with the part from the residual s
at the poles (Radko 1992, among others ), and the double integral
(above) is simpli ed to
‡ D
8Dab
 g
Z
ƒ

2
<<

2
cos
4ˆ
ƒ
5>
0
Id CO4r
ƒ1
51 (6)
I D
sin a
‹
ü
H
cos  sin b
‹
ü
H
sin 
a
‹
ü
H
cos  b
‹
ü
H
sin 

sinh ‹
ü
H ƒh
f
H
cosh ‹
ü
ãh
H
ƒ1
‹
ü
1
cosh ‹
ü
ƒƒ
2
cos
2
 cosh ‹
ü
 sin6‹
ü
r cos4ˆ ƒ57
where ƒ D
U
p
gH
is a Froude number, and ‹
ü
is the pole, which
satis es the equation tanh4‹5 D ƒ
2
cos
2
‹.
The expression (6) can be evaluated asymptotically for large r ,
i.e., far away from the moving object, by the rule of stationary
phase, also known as Wehausen’s procedure, an asymptotic solu-
tion for the integral of rapidly oscillating functions (described in
many sta ndard texts on perturbation methods and  uid dynam-
ics; see, for exam ple, Debnath 1994). After changing of the vari-
able  in the integral (6) to V D tan , conditions of the point
of stationary phase become: ê
0
V
D 0, where ê is the phase ê D
‹
ü
cos ˆ
1
p
1CV
2
Csin ˆ
V
p
1CV
2
, and cos ˆ CV sin ˆ > 0. These can
be simpli ed to
8
>
>
<
>
>
:
tan ˆ D
V CV
V
2
C1
ƒ
2
cosh
2
‹
ü
2V
2
C1 ƒ
V
2
C1
ƒ
2
cosh
2
‹
ü
ƒ

2
< ˆ <

2
(7)
and the equation for the pole 4‹
ü
5 tanh ‹
ü
D
ƒ
2
1CV
2
‹
ü
can have (at
most) only one positive root.
Existence of a point of stationary phase determines whether the
far- eld asymptotics behave as O
1
p
r
or as O
1
r
. The critical
value ˆ
c
, such that a point of stationa ry phase (7) can be found
only when —ˆ— < ˆ
c
, is extremely important as it determines the
angle of the domain to which the wave phenomena are restricted
(at the leading order). It can be shown (Havelock 1908) that the
structure of ship waves is fundamentally different for ƒ
2
> 1 when
only a single system of diverging waves makes up the ship wave
pattern and for ƒ
2
< 1. The discussion in this section will be
mostly limited to the latter case of rather slow motion, and the
valu e of the angl e of ship waves ˆ
c
is then obtained from (7):
tan ˆ
c
D max
V>0
(
V CV
V
2
C1
ƒ
2
cosh
2
‹
ü
2V
2
C1 ƒ
V
2
C1
ƒ
2
cosh
2
‹
ü
!
(8)
and ˆ
c
can be easily computed in the two following limiting cases.
When ƒ
2
! 0 th e expression
V
2
C1
ƒ
2
cosh
2
‹
ü
uniformly approaches
zero, so
tan ˆ
c
D max
V>0
V
2V
2
C1
D
1
2
p
2
1 V
max
D
1
p
2
4 MA RCH 2001 JOU RNAL OF SHIP RESEARCH

which corresponds to the well-known value of 19

28
0
for the
angle of ship waves in the in nitely deep  uid. If ƒ
2
is small but
nonzero the value of this angle can be easily estimated by the
small perturbations technique:
tan ˆ
c
1
2
p
2
1 C
9
ƒ
2
exp
3
ƒ
2
which implies that for ƒ
2
! 0 waves converge ver y fast to their
expression in the absence of a bottom. For ƒ
2
< 002 the  uid
can be effe ctively considered in nitely deep. For ƒ
2
! 1
ƒ
the
expression on the right-hand side of (8) could be in nitely large
and therefore ˆ
c
D 90

.
In the intermediate case 0 < ƒ
2
< 1 the angle of the ship waves
can be computed numerically from the expression
tan ˆ
c
D max
tanh
‹
ü
‹
ü
<ƒ
2
q
ƒ
2
‹
ü
tanh ‹
ü
ƒ14sinh ‹
ü
cosh ‹
ü
C‹
ü
5
2ƒ
2
‹
ü
cosh
2
‹
ü
ƒ4sinh ‹
ü
cosh ‹
ü
C‹
ü
5
which is algebraically equivalent to (8). The resulting relationship
ˆ
c
4ƒ
2
5 is presented in Fig. 3.
The leading term of the far- eld asymptotics obtained by the
method of stationary phase (not shown) is a complicated expres-
sion that is dif cult to analyze. However, it signi cantly simpli es
for small ƒ
2
, when ‹
ü
1. This corresponds to the approximation
of an in nitely deep  uid in whic h case terms sinh ‹
ü
1 cosh ‹
ü
are
asymptotically close to 005 exp4‹
ü
5 and sinh
H ƒh
f
H
‹
ü
is close to
Fig. 3 Angle of ship waves as a function of ƒ
2
D U
2
/gH. When prop-
ag ation velocity is small 4ƒ
2
! 05, wave phenomenon is restricted to
an angle —ˆ— < 19

2 8
0
a s in the case of an in nitely deep  uid (Kelvin
19 05). When ve locity is larger, th e angle of ship waves increases, and
for ƒ
2
! 1 it reaches 90

. Dynamics of ship waves are different for
ƒ
2
> 1 where only divergent waves are presen t, and supercritical region
is not presented
005 exp
H ƒh
f
H
‹
ü
(unless th e source is located near the bottom).
In most applic ations these conditions are satis ed and the result-
ing expression is:
¬
‡ D
8Dab
 g
X
jD11 2
2
r—ê
00
j
—
005
sin
h
a
‹
ü
j
H 4V
2
j
C15
0
0
5
i
a
‹
ü
j
H 4V
2
j
C15
0
0
5
 ®

sin
h
b
‹
ü
j
V
j
H 4V
2
j
C15
0
0
5
i
b
‹
ü
j
V
j
H 4V
2
j
C15
0
0
5
cosh ‹
ü
j
ãh
H
ƒ1

exp ƒ
h
j
H
‹
ü
j
1 CV
ü
j
sin rê
j
C4ƒ15
jC1

4
CO4r
ƒ1
5 (9)
where ‹
ü
j
D 4V
2
j
C15=ƒ
2
, and V
j
are the stationary phase points
determined from ê
0
D 0:
V
11 2
D
1
p
1 ƒ8 tan
2
ˆ
4 ta n ˆ
The term sin4rê
j
C4ƒ15
jC1

4
corresponds to the fast mod-
ulation (in space) of the ship waves, and the condition
rê
j
C4ƒ15
jC1

4
D

2
C2 n describes the crests of the waves,
while rê
j
C 4ƒ15
jC1

4
D ƒ

2
C 2 n describes the troughs.
Equation (9) implies that the wavelength is of order ƒ
2
, and there-
fore the condition ƒ
2
1 corresponds to the waves that are much
shorter than the depth of the  uid. Of the two systems of waves
in (9) one with j D1 is identi ed with the “divergent” waves (see
Fig. 1), whose crests are given by
r D
 41 C8n5
cos ˆ43 ƒ
p
1 ƒ8 tan
2
ˆ5
 1 C
1
16
cot
2
ˆ41 C
p
1 ƒ8 tan
2
ˆ5
ƒ005
and j D 2 corresponds to the transverse waves:
r D
 41 C8n5
cos ˆ43 C
p
1 ƒ8 tan
2
ˆ5
 1 C
1
16
cot
2
ˆ41 ƒ
p
1 ƒ8 tan
2
ˆ5
ƒ005
Terms
¬
,

, and
®
in equation (9) describe the interference
of th e waves emitted by the different points of our ( nite size)
object. When
a
H
1
b
H
1
ãh
H
ƒ
2
equation (9 ) reduces to the classi-
cal expression for th e ship waves from a point source (Havelock
1908). The rough estimate of the magnitudes of waves in this case
—‡
j
—
Dab4ãh
2
5
gH
2
ƒ
3
r
005
exp ƒ
h
f
H
V
2
j
C1
ƒ
2
indicates that if the source is suf ciently close to the surface
4h
f
U
2
=g5 the amplitude s of the transverse and divergent waves
are comparable, but otherwise the divergent waves are much
weaker.
MAR CH 2001 JOURNAL OF SHIP RESEAR CH 5

If the scale of the object signi cantly exceeds the wavelength
a
H
1
b
H
1
ãh
H
> ƒ
2
we similarly obtain
—‡
j
—
Dƒ
5
H
2
gr
005
exp ƒ
h
f
ċh
H
V
2
j
C1
ƒ
2
in which case the amplitude of the ship waves does not depend
signi cantly (fo r a  xed D) on the horizontal dimension s of the
source 4a1 b5.
3. Ship waves in a strati’ed ‘uid
Inclusion of strati cation in the problem introduces a new
nondimensional para meter, the relative variation of density ˜. As
will be shown later, the structure of the ship waves strongly
depends on how its magnitude compares to ƒ
2
D U
2
=gh, where
h is a characteristic thickness of a density layer. De pending on
a particula r situation ˜ could be either smaller, larger, or of the
same order as ƒ
2
. In this section, mostly for reasons of analyti-
cal tractability, we consider the case ˜ ƒ
2
(an extreme case of
the subcritical propagation). This limit could be re alized in the
ocean, although it is most appropriate to interpret it as an expla-
nation of a laboratory experiment w ith several layers of different
liquids. This will provide us with an insight into the intermedi-
ate case ˜ ƒ
2
(considered in Section 5), and when ˜ ƒ
2
the
solution reduces to the cla ssical problem of ship waves in a  uid
of constant density.
Let 
1
< 
2
< < 
n
be the densities of the layers located
at the depths ƒH
i
< z < ƒH
iƒ1
i D 11 : : : 1 n respectively, and
h
i
D H
i
ƒH
iƒ1
is the thickness of layer i. Assume that the mass
force is still given by (5) and it is located in the layer  6D n (the
case when  Dn ca n be considered similarly). Linearize equations
of motion in each layer as in (1), a dd the boundary conditions at
z D 0 and z D ƒH
n
(see Section 2), and introduce the matching
conditions on the density interfaces 4H
i
5:
(
w
i
D w
iC1
p
i
ƒp
iC1
D 4
iC1
ƒ
i
5g‡
i
When the Fourier transform in 4x1 y5 and the Laplace-Carson
transform in t is applied to the resulting systems, the problem
reduces to the equation for the images 4 Op
i
5 of the pressure per-
turbations in each layer (as in Section 2):
8
>
>
<
>
>
:
d
2
Op
i
dz
2
ƒŠ
2
Op
i
D 0 i 6D 
d
2
Op
i
dz
2
ƒŠ
2
Op
i
D 4z5 i D 
(10)
The boundary and matching conditions become
d Op
1
dz
C
4S CUik5
2
g
Op
1
D 0 for z D 03
d Op
n
dz
D 0 for z D ƒH
n
3
1

i
d Op
i
dz
D
1

iC1
d Op
iC1
dz
1 Op
iC1
ƒ Op
i
D ƒ4
iC1
ƒ
i
5

g
4S CUik5
2

i
d Op
i
dz
for z D ƒH
i
1 i D 11 : : : 1 4n ƒ15.
The general solution of differential equation (10) is
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
Op
i
D A
i
cosh6Š4z CH
iƒ1
57
CB
i
sinh6Š4z CH
iƒ1
57 if i 6D
Op
i
D A
i
cosh6Š4z CH
iƒ1
57
CB
i
sinh6Š4z CH
iƒ1
57
C
Z
z
ƒH

ƒ1
sinh6Š4z ƒz
0
57
Š
4z
0
5dz
0
if i D
When these solutions are substituted in the boundary and match-
ing conditions, the following relationships between A’s and B’s
appear:
A
1
4S CUik5
2
gŠ
CB
1
D 01 ƒA
n
sinh Šh
n
CB
n
cosh Šh
n
D 03
A
iC1
B
iC1
D
T
i
A
i
B
i
if i 6D 1
A
iC1
B
iC1
D
T
i
A
i
B
i
C
X
Y
if i D 
where
T
i
D
0
B
B
B
@
coshŠh
i
C
gŠ˜
i
4S CUik5
2
ƒsinhŠh
i
ƒ
gŠ˜
i
4S CUik5
2
 sinhŠh
i
 coshŠh
i
ƒ41C˜
i
5sinhŠh
i
41C˜
i
5coshŠh
i
1
C
C
C
A
X
Y
D
0
B
B
B
B
B
B
B
B
B
@
ƒ
Z
ƒH

ƒH

ƒ1
sinh6Š4H

Cz
0
57
Š
4z
0
5dz
0
ƒ
gŠ˜

4S CUik5
2
Z
ƒH

ƒH

ƒ1
cosh6Š4H

Cz
0
57
Š
4z
0
5dz
0
41C˜

5
Z
ƒH

ƒH

ƒ1
cosh6Š4H

Cz
0
57
Š
4z
0
5dz
0
1
C
C
C
C
C
C
C
C
C
A
and ˜
i
D4
iC1
ƒ
i
5=
i
is a relative density difference. In order to
distinguish between the various modes that appear below assume
that these ˜’s are different 4˜
i
6D ˜
j
5.
The general expression for the image of the free surface pe r-
turbation becomes
O‡ D ƒ
1

1
g
4ƒsinh Šh
n
1 cosh Šh
n
5
T
0
Y
X
4ƒsinh Šh
n
1 cosh Šh
n
5
T
1
ƒ
4SCU ik5
2
gŠ
(11)
and the perturbation of the density interfaces is
O‡
m
D ƒ

1

m
4ƒsinh Šh
n
1 cosh Šh
n
5
T
00
1
ƒ
4SCU ik5
2
gŠ
O‡1
m D 11 : : : 1 4n ƒ1 5 (12)
where
T
D
T
nƒ1
T
nƒ2
T
1
1
T
0
D
T
nƒ1
T
nƒ2
T

1
T
00
D
T
mƒ1
T
mƒ2
T
1
.
6 MA RCH 2001 JOU RNAL OF SHIP RESEARCH

The denom inator of (11) in the limit S ! C0 has a  nite num-
ber of zeroes, and (as in the one-layer case) it can be shown that
the far- eld asymptotics of the free surface are dominated by the
part from the residuals
‡ D ƒ
4
h
2
Z
dè res
‹
ü
6 O‡‹ sin4‹r cos4ˆ ƒ557CO4r
ƒ1
5 (13)
where integration is carried over the region where cos4ˆ ƒ5 > 0,
ƒ

2
<  <

2
. Here, again, k D
‹
h
cos 1 l D
‹
h
cos , x D rh cos ˆ,
and y D rh sin ˆ.
The equation for the poles
ƒsinh ‹
h
n
h
1 cosh ‹
h
n
h
T
SD0
1
ƒƒ
2
cos
2

D 0 (13a)
is dif cult to solve in general. However, when ƒ
2
!0, all its solu-
tions ‹
ü
i
! ˆ (Radko 1992), in which case cosh4‹4h
i
=h55 and
sinh4‹4h
i
=h55 become asymptotically close to 0.5 exp4‹4h
i
=h55
and therefore
T
i
can be simpli ed to
T
i
D 005 exp ‹
h
i
h
a
i
ƒa
i
ƒb
i
b
i
61 CO4exp4 ƒ2‹557 (14)
where a
i
D 1 C4g
‹
h
˜
i
5=4S CUi
‹
h
cos 5
2
and b
i
D 1 C˜
i
.
The further transformations are based on the following alge-
braic property of the matrices of type (14):
a
2
ƒa
2
ƒb
2
b
2
a
1
ƒa
1
ƒb
1
b
1
D
a
2
ƒa
2
ƒb
2
b
2
4a
1
Cb
1
5 (15)
This simple relationship allows us to simplify (11) and (12) for
large ‹’s (near the poles), reducing the products of matrices in the
numerators and denominators to the products of ordinary alge-
braic expressions. When this is applied to (11) and the explicit
expression for the mass force (5) is used in X and Y , the result is
O‡ Dƒ
2Dh
2
ab

2
g



1
sin4a
‹
h
cos 5
a
‹
h
cos 
sin4b
‹
h
sin 5
b
‹
h
sin 
exp4ƒ‹
h
f
h
56cosh4ãh
‹
h
5 ƒ17
‹
2
6ƒ2ƒ
2
‹ cos
2
7
ƒ1

1
6ƒ42 C˜
1
5ƒ
2
‹ cos
2
 C˜
1
7 6ƒ42 C˜
ƒ1
5ƒ
2
‹ cos
2
 C˜
ƒ1
7 6ƒƒ
2
‹ cos
2
 C17
(16)
From this expression we can see that the number of poles on the
real ‹ a xes exactly equals the number of layers 45 where the
object is located and therefore there are only  corresponding
systems of waves!
3
Thus, the poles in (13) are given by
‹
ü
i
D
˜
i
42 C˜
i
5ƒ
2
cos
2

i D 11 : : : 1 4 ƒ151
‹
ü
0
D
1
ƒ
2
cos
2

(17)
3
If the terms exp4ƒ2‹4 h
i
=h55 were not neglected in (11) , there (generally)
would have been n systems of waves. However, the amplitudes of 4n ƒ5 of
those become asymptotically small compared with the other  systems when
ƒ
2
˜.
The free surface perturbation (13) reduces to sum of  compo-
nents c orresponding to the poles in (17)
‡ D ‡
405
C‡
415
C C‡
4ƒ15
CO4r
ƒ1
5 (18)
(Eqs. (16), (17) are written with the understanding that the
neglected terms are O6exp4ƒ2‹
ü
i
h
i
h
57 which are small since we
assumed that ƒ
2
˜05
Let us now evaluate each component ‡
4i5
in (18) in the far-
 eld region. Substitute (16) in (13), use (17) and then apply the
method of stationary phas e. The resulting expression for i D 0
becomes:
‡
405
D
8Dab
 g



1
X
jD112
s
2
r—ê
00
j
—
sin a
‹
ü
0
j
h
p
V
2
j
C1
a
‹
ü
0
j
h
p
V
2
j
C1

sin b
‹
ü
0
j
V
j
h
p
V
2
j
C1
b
‹
ü
0
j
V
j
h
p
V
2
j
C1
cosh ‹
ü
0j
ãh
h
ƒ1

exp ƒ
h
f
h
‹
ü
0j
1CV
2
j
sin rê
j
C4ƒ15
jC1

4
CO4r
ƒ1
5 (19)
where tan 
11 2
D V
11 2
are the stationary phase points determined
from ê
0
D 0:
V
11 2
D ƒ
1
p
1 ƒ8 tan
2
ˆ
4 ta n ˆ
Remarkably, the component ‡
405
looks nearly like the total per-
turbation of the free surface in a one-layer problem (see eq 9 in
Section 2) and it will be identi ed with a surface mode.
The phase
ê D ‹
ü
0
cos ˆ
1
p
1 CV
2
Csin ˆ
V
p
1 CV
2
is exactly the same and the amplitude differs fro m (9) only by
a c oef cient



1
. Clearly, these waves exist only within the angle
—ˆ— < arctan 1=42
p
25 D 19

28
0
, and all other conclusions from
Section 2 can be applied to ‡
405
.
The fundamental difference between the strati ed and nonstrat-
i ed problems results from the internal mode s ‡
4i5
where i 6D 0,
MAR CH 2001 JOURNAL OF SHIP RESEAR CH 7

for which we similarly obtain:
‡
4i5
D
8Dab
 g



1
˜
ƒ2
i
4˜
i
ƒ˜
1
5 4˜
i
ƒ˜
ƒ1
5

X
jD11 2
s
2
r—ê
00
j
—
sin a
‹
ü
ij
h
p
V
2
j
C1
a
‹
ü
ij
h
p
V
2
j
C1

sin b
‹
ü
ij
V
j
h
p
V
2
j
C1
b
‹
ü
ij
V
j
h
p
V
2
j
C1
cosh ‹
ü
ij
ãh
h
ƒ1

exp ƒ
h
f
h
‹
ü
ij
1 CV
2
j
sin rê
j
C4ƒ15
jC1

4
CO4r
ƒ1
5 (20)
where ‹
ü
ij
D ˜
i
=42 C˜
i
54V
2
j
C15=ƒ
2
and V
j
are the points of sta-
tionary phase computed from ê
0
D 01 ê D ‹
ü
i
cos ˆ41=
p
1 CV
2
5 C
sin ˆ4V=
p
1 CV
2
5 to be (again) V
11 2
D ƒ41
p
1 ƒ8 tan
2
ˆ5=
4 ta n ˆ.
Let us analyze equation (20). The phase ê in (20) differs
from that in (19) by the constant factor ˜
i
=42 C˜
i
5 and therefore
the new modes 4i 6D 05 are signi cantly longer than the i D 0
mode, but their structure is quite similar to that of ‡
405
. They are
certainly con ned to the 19

28
0
angle and each ‡
4i5
consists of
the longitudinal 4j D 15 and transverse 4j D 25 subsystems as in
Fig. 1.
A rough estimate of the amplitudes of these waves indicates
that when
a
h
1
b
h
1
ãh
h
442 C˜5=˜5ƒ
2
our object can be considered
to be a point source since there is no interference of the waves
from the different points of the object and
‡
4i5
j
Dab4ãh5
2

k
 gh
2
ƒ
3
r
005

1
˜
i
2C˜
i
105
exp ƒ
h
f
h
˜
i
2C˜
i
V
2
j
C1
ƒ
2
(21)
If the scale of the object exceeds the wavelength
a
h
1
b
h
1
ãh
h
>
442 C˜5=˜5ƒ
2
we obtain
‡
4i5
j
Dh
2
ƒ
5

k
 gh
2
r
005

1
2 C˜
i
˜
i
205
exp ƒ
h
f
h
˜
i
2 C˜
i
V
2
j
C1
ƒ
2
(22)
When these are compared with the amplitude of the i D0 mode,
we see that “‡
405
” “‡
4i5
” i D 11 : : : 1 4 ƒ15. Thus , unless the
object is moving in the upper la yer, the amplitude of the free
surface perturbation is dominated by the i 6D 0 modes.
Now let us similarly analyze the structure of the density inter-
faces. Consider a density interface m located above the moving
object 4m < 5. Apply the procedure use d above for the pertur-
bation of the free surface to compute the density interfaces (11).
The formula (not shown) analogous to equa tion (16) indicates
that on the density interface number m there will be only 4ƒm5
different waves:
‡
m
D ‡
4m5
m
C‡
4mC15
m
C C‡
4ƒ15
m
CO4r
ƒ1
5
where
‡
4i5
m
D
8Dab
 g



m
˜
ƒ2ƒm
m
4˜
i
ƒ˜
m
5 4˜
i
ƒ˜
ƒ1
5

X
jD112
s
2
r—ê
00
j
—
sin a
‹
ü
ij
h
p
V
2
j
C1
a
‹
ü
ij
h
p
V
2
j
C1
sin b
‹
ü
ij
V
j
h
p
V
2
j
C1
b
‹
ü
ij
V
j
h
p
V
2
j
C1

h
cosh ‹
ü
ij
ãh
h
ƒ1
i
exp ƒ
h
f
h
‹
ü
ij
1 CV
2
j
 sin rê
j
C4ƒ15
jC1

4
CO4r
ƒ1
5 (23)
and the poles ‹
ü
ij
, stationary phase points V
j
, and the phase ê
j
are exactly the same as for the i 6D 0 modes of the surface waves.
Thus, on the density interface m there exist 4 ƒm5 systems of
waves, each of which ha s a structure very similar to that of the
classical ship waves (i.e., these waves are con ned to an angle
of 19

28
0
and consist of transverse and divergent subsystems).
Note that all of these interfacial waves are longer than the waves
in Section 2. Using e quation (23) one can easily estimate the
amplitudes of ‡
4i5
m
, as was made in (21), (22) for equation (20).
The above analysis assumed that the object is located in the
interior of a density layer. Since the number of the various wave
modes on the free surface (and on the interfaces m <  above
the object) increases when the moving body descends from, say,
layer  to layer 4 C15, it becomes important to analyze how
the additional wave system appears during the crossing of an
interface. Such a transition occurs in a thin layer 4H

ƒh
f
442 C˜5=˜5U
2
=g5 just above the density interface as indicated in
the schematic diagram in Fig. 4 (Radko 1992).
Fig. 4 Schematic diagram for amplitudes of different ship wave modes
at a free surface as a function of location of a moving object. Amplitudes
of (19) and (20) are estimated without exponents exp ƒ4h
f
/h5‹
ü
ij
. An
ob ject loca ted in the interior of layer  effectively excites only  systems
of waves. This changes when the object is approaching a thin layer
ab ove the density interface (close parallel dashe d lines), where a new
m ode appears
8 MA RCH 2001 JOU RNAL OF SHIP RESEARCH

When we conside r the density interfaces below the moving
object 4 m5 it becomes apparent that there are no pole s on the
real ‹ axes, and for this reason the leading term of the asymptotics
vanishe s (within the speci ed approximations). This makes it dif-
 cult to rigorously analyze those interfaces using our method .
The main conclusion for m  is that the waves located on th e
interfaces below the object are much smaller tha n the waves at
the same vertical distances above it. Such an “asymmetry” appar-
ently results from the boundary condition of no vertical motion
at the bottom z D ƒH, in contrast to the free surface conditions
at z D 0.
4. Moving oscil lator
The problem of an oscillator moving in a nonstrati ed  uid is
well known (e.g., Kha skind 1973). The periodic oscillations of
the amplitude of the forcing results in appearance of another type
of motion (in addition to the ship waves considered previously)
which will be referred to herein as propa gating gravity waves.
Below we will demonstrate how the method in Section 3 can be
modi ed to solve the strati ed problem with an oscillator and
formulate conditions under which th e ship waves and the gravity
waves could be considered independently.
Let — denote the frequency of oscillations in the amplitude
of mass force in (5) D D D
0
exp4i—t5, and introduce the corre-
sponding nondimensional parameter ‚ D 4—U 5=g . As t ! Cˆ
waves gradually evolve to a regime which could be described
by
¡
¡t
D i—, that corresponds to S ! i— C0 (S is a Laplace-
Carson transform variable). Since the derivation of the expres-
sion for the Fourier image of the free surface and interfaces
is very similar to that in Section 3 we omit them here. The
result shows that the perturbations in the oscillating problem are
related to that used in equations (11) and (12) in Section 3 by
O‡
osc
D exp4i—t5 lim
S!i—C0
O‡. Our technique, such as employed in
Section 3, requires the poles of Fourier image to be large 4‹
ü
15
which occurs when
ƒ ‚ ˜ (24)
The condition ‚ ˜ implies that the oscillations are slow enough
for the structure of the ship wave component not to be modi ed
signi cantly, as will be shown below.
When the small terms exp4ƒ2‹ 5 are neglected in the vicinity
of the poles and the algebraic property (15) is applie d to the
result, the equivalent of the expression (16) for the oscillating
case becom es
O‡ D ƒ
2D
0
ab exp4i—t5h
2
g
2



1
sin4a
‹
h
cos 5 sin4b
‹
h
sin 5
a
‹
h
cos b
‹
h
sin 

exp4ƒ h
f
‹
h
56cosh4ãh
‹
h
5 ƒ17
‹
2

6ƒ2G7
ƒ1
6ƒ42 C˜
1
5G C˜
1
7 6ƒ42 C˜
ƒ1
5G C˜
ƒ1
7 6ƒG C17
(25)
where G D
1
‹
44‚=ƒ5 Cƒ‹ cos 5
2
. It can be shown again that the
far- eld asymptotics 4r ! ˆ5 are dominated by a contribution
from the poles, an d since each of the terms 6ƒ42 C˜
i
5G C˜
i
7 in
the denominator of (25) has two roots if equation (24) is satis ed,
the free surface perturbation in physical variables becomes:
‡ D ƒ
4
h
2
R
d
cos4ˆƒ5>01ƒ

2
<<

2
è res
‹
ü
si
6 O‡‹ sin4‹r cos4ˆ ƒ557
ƒ
2
i
h
2
R
d
cos4ˆƒ5>0
è res
‹
ü
si
6 O‡‹ exp4i‹r cos4ˆ ƒ557 (26)
The poles computed from (25) are:
‹
ü
si
D 
i
1 ƒ2
‚

i
cos  C
q
1 ƒ4
‚

i
cos 
2ƒ
2
cos
2


i
ƒ
2
cos
2

1
(27)
and
‹
ü
gi
D 
i
1 ƒ2
‚

i
cos  ƒ
q
1 ƒ4
‚

i
cos 
2ƒ
2
cos
2

‚
2

i
ƒ
2
1 (28)
where 
i
D ˜
i
=42 C˜
i
51 
0
D 11 i D 01 11 : : : 1 4 ƒ15.
When (25) and (27) are used to simplify (26), the “ship wave”
terms in (26), i.e., those corresponding to the poles in (27),
reduce exactly to their expression in the absence of oscillations
{see equations (19) and (20)} multiplied by ex p4i—t5, and the
same occurs for the density interfaces. This implies that in the
parametric regime (24) there is almost no in uenc e of the oscil-
lations on the str ucture of ship waves.
The free surfac e perturbation by the gravity waves obtained
from equations (25), (26), (28) by a method of stationary phase
consists of  systems:
‡
405
g
D
4D
0
ab
 g



1
r
2
r’
2
sin a
’
2
h
cos ˆ
a
’
2
h
cos ˆ
sin b
’
2
h
sin ˆ
b
’
2
h
sin ˆ

h
cosh ’
2
ãh
h
ƒ1
i
exp ƒ
h
f
h
’
2
 exp
h
i —t ƒ’
2
r ƒ

4
i
CO4r
ƒ1
5 (29)
where ’ D
‚
ƒ
D —
q
h
g
,
‡
4i5
g
D
4D
0
ab
 g



1
˜
ƒ2
i
4˜
i
ƒ˜
1
5 4˜
i
ƒ˜
ƒ1
5
r
2 
i
r’
2

sin a
’
2
h
i
cos ˆ
a
’
2
h
i
cos ˆ
sin b
’
2
h
i
sin ˆ
b
’
2
h
i
sin ˆ

h
cosh ’
2
ãh
h
ƒ1
i
exp ƒ
h
f
h
i
’
2
 exp
h
i —t ƒ
’
2

i
r ƒ

4
i
CO4r
ƒ1
5 (30)
where i D 11 : : : 1 4 ƒ 15. {When the density interfaces were
treated similarly we found that on the interface m there effec-
tively exist 4ƒm5 systems of waves.} The wave component ‡
405
g
differs from the expression of the gravit y waves in a nonstrati ed
case only by a factor



1
, and their phase speed c
f
D
g
—
certainly
equals that of the free waves in an in nitely deep  uid. From
the estimate of the leading order propagating gravity waves in
MAR CH 2001 JOURNAL OF SHIP RESEAR CH 9

equations (29) and (30) it follows that these do not depend on U .
Thus, when (24) holds, propagation and oscillation are not cou-
pled (at the leading order) and their effects could be considered
separately.
Let us compare the a mplitude of the various waves on a free
surface. Consider, for example, the case when the size of a source
is suf ciently small so that there is no interference between the
waves from different points. This occurs when a=h1 b=h1 ãh=h
˜=’
2
in which case
‡
405
g
D
0
ab4ãh5
2

k
’
3
gh
2
r
005

1
exp ƒ
h
f
h
’
2
and
‡
4i5
g
D
0
ab4ãh5
2

k
’
3
gh
2
r
005

105
i

1
exp ƒ
h
f
h
’
2

i
This estimate shows that the propagating gravity waves (unlike
the ship waves!) are always dom inated by the surface wave com-
ponent ‡
405
g
in the considered parametric regim e. When the ampli-
tudes of the gravity waves are compared to the corresponding
ship wave modes, i.e., to the components of the  rst sum in (26),
denoted as ‡
4i5
s
below, we  nd that ‡
405
g
‡
4i5
g
‡
4i5
s
‡
405
s
i D1 1 : : : 1 4ƒ15 for pa rameters satisfying (24), unless the object
is moving in the upper layer in which case ‡
405
g
and ‡
405
s
can be
of the same order.
Thes e results, indicating that there is no in uence of the prop-
agation on the (completely isotropic) gravity waves and no in u-
ence of the oscillations on the structure of the ship waves, have
been obtained using only the leading term in the Taylor expansion
over
‚

i
in (27) and (28). Taking into ac count the two terms results
in a sligh t modi cation of such a conjecture. The ship waves
will be con ned to larger angles of arctan41=2
p
2 C
p
3‚=4
i
5
and the gravity waves will become slightl y anisotropic, since the
term41 ƒ2
‚

i
cos ˆ5 will appear in their phase.
5. Ship waves for objects moving with
’n ite velocity
What happens when the object is moving with an arbitrary
velocity U , such that the asymptotic theory in Section 3 is not
necessarily valid? The general properties of kinematical patterns
of the linear ship waves in the far  eld are fairly well understood
(e.g., Keller & Munk 1970). The density strati cation determines
a system of normal modes with distinct dispersion relationships
for the free gravity waves. (The number of such modes could
be  nite, as in our multilayer case, or in nite for a continuously
varying strati cation). Th e wake produced by a moving object
also consists of a s ystem of modes each of which directly cor-
responds to one of those free wave modes, and the patte rn of a
particular ship wave mode then could be computed from the cor-
responding free wave dispersion relationship c D c
i
4k5 (Keller &
Munk 1970). If speed of the object U is less than the maximum
phase speed of the particular free wave mode, which occurs for
long waves 4k ! 05, th e corresponding ship wave mode consist s
of divergent and transverse waves as in Fig. 1. In the case when
U > c
i
405 the transvers e waves cannot be steady relative to the
moving object and therefore only the divergent waves are present
in the wake.
The phase spee ds of the different modes of long gravity waves
6c
i
4057 are related by a simple relationship derived originally by
Yanowitch (1962):
X
c
2
i
405 D gH
where H is a total depth of  uid an d summation is made over the
all possible free wave modes. This property has a direc t coun-
terpart in the ship wave problem. In our n-layer model the total
number of different modes is n. Thus, the critica l Froude numbers
F
r
i
, based on the critical mode-dependent speeds U
i
associated
with fundamental changes in the structure of the corresponding
ship wave modes, should satisfy
n
X
iD1
Fr
2
i
D 1 (31)
where Fr
i
D U
i
=4gH5
005
D c
i
405=4gH5
005
, i D 11 : : : 1 n. Equation
(31) is an important feature of the strati ed s hip waves which will
be conveniently used for checking and explaining the following
theory.
In order to give a more quantitative prediction of the ship wave
structure for  nite values of the propagation velocity U we con-
sider a three-layer  ow and start from equations (11–13). In this
section it is more convenient to nondimensionaliz e the wave num-
bers in equations (11) and (12) by the total depth of the  uid
H2 k D
‹
H
cos 1 l D
‹
H
cos , x D rH cos ˆ, y D r H sin ˆ, and the
equation for the poles ‹
ü
i
(eq 13a) for a three-layer case reduces
to:
ƒ
2
‹
ü
cos
2
 D L
i
4‹
ü
5 (32a)
where ƒ D U =4gH5
005
is the Froude number and L
i
4‹51 i D11 21 3
are the roots of the third-order equation
A4‹5L
3
CB4‹5L
2
CC4‹5L CD4‹5 D 0 432b5
Coef cients in equation (32b) are functions of ‹ only:
A D s
1
s
2
c
3
Cs
1
c
2
s
3
Cc
1
s
2
s
3
Cc
1
c
2
c
3
C˜
1
4c
1
s
2
s
3
Cc
1
c
2
c
3
5
C˜
2
4s
1
s
2
c
3
Cc
1
c
2
c
3
5 C˜
1
˜
2
c
1
c
2
c
3
1
B D ƒ41 C˜
1
C˜
2
54c
1
c
2
s
3
Cc
1
s
2
c
3
Cs
1
c
2
c
3
Cs
1
s
2
s
3
5
ƒ˜
1
˜
2
4c
1
c
2
s
3
Cc
1
s
2
c
3
Cs
1
c
2
c
3
51
C D ˜
1
4s
1
c
2
s
3
Cs
1
s
2
c
3
5 C˜
2
4s
1
c
2
s
3
Cc
1
s
2
s
3
5
C˜
1
˜
2
4s
1
c
2
s
3
Cs
1
c
2
s
3
Cc
1
s
2
s
3
51
D D ƒ˜
1
˜
2
s
1
s
2
s
3
where s
i
D sinh44h
i
=H5‹5, c
i
D cosh44h
i
=H5‹5.
In order to dete rmine the angle of a ship wake in the far-
 eld we once again use the rule of stationary phase. This method
implies that the asymptotics 4r !ˆ5 of the integral in (13 ) would
have a suf ciently strong expression 4 r
ƒ005
5 only when ˆ is
such that a point of stationary phase  could be found:
¡
¡
6‹
ü
cos4ˆ ƒ57 D 0 (33)
Now we apply this condition separately to each of the three s hip
wave components in (13) by substituting ‹
ü
D ‹
ü
i
, i D 11 21 3
10 MARCH 2001 JOU RNAL OF SHIP RESEARCH

in (33). Since the poles ‹
ü
i
are the roots of the transcendental
equatio n (32a) we now eliminate  in (33) using (32a). This sim-
pli es th e condition for existence of a point of stationary phase
(for a wave s ystem number i ) to:
tan ˆ D
s
ƒ
2
‹
ü
i
L
i
4‹
ü
i
5
ƒ1
L
0
i
4‹
ü
i
5‹
ü
i
CL
i
4‹
ü
i
5
2ƒ
2
‹
ü
i
ƒ6L
0
i
4‹
ü
i
5‹
ü
i
CL
i
4‹
ü
i
57
(34)
Thus, the signi cant 4 r
ƒ005
5 ship waves are con ned to the
region where ˆ is such that (34) has a real solution for ‹
ü
i
.
For each of three systems of waves there generally exists a
critical tan ˆ
ci
equal to the maximum of the right-hand side of
equatio n (34) which corresponds to the observed angle of ship
waves.
In order to determine how these angles ˆ
ci
depend on the
Froude number ƒ, we numerically computed the maximum of the
right-hand side of (34) for each i over the range of ƒ; the func-
tions L
i
were obtained using the known expression fo r roots of
the third-order polynomial (32b). For
h
1
D 0035H1 h
2
D 0035H1 h
3
D 003H1
and ˜
1
D 0051 ˜
2
D 008 (35)
the angles of the ship waves a re plotted in Fig. 5 as functions
of ƒ
2
. This  gure shows that the angles of all the modes of
ship waves reduce to 19

28
0
when the propagation velocity is
small 4ƒ
2
! 05 as was anticipated by the asymptoti c theory in
Section 3. When the propagation velocity is larger, the angle of
each mode increases and  nally for critical ƒ D Fr
i
it reaches 90

and when ƒ > Fr
i
this angle starts to decrease to zero (see the
dotted curves in Fig . 5). It can easily be shown that in the super-
critical region (ƒ > Fr
i
5 the dependence of ˆ
ci
on ƒ
2
is simply
given by the so-called Mach angle ˆ
ci
D arcsin4Fr
i
=ƒ5.
Now let us analyze the critical velocities U
i
i D 11 21 3 that
determine the existence of a transverse system of ship waves.
The expansion of the coef cients of the cubic equation (32b) in
powers of ‹ shows that its roots L
i
are of order ‹ for ‹ ! 0, and
therefore p
i
lim
‹!0
L
i
=‹ is  nite and L
i
D p
i
‹ Co4‹5 . When
ƒ
2
D p
i
and ‹ ! 0, the right-hand side of (34) tends to in nity,
max4tan ˆ5 D ˆ, and therefore ˆ
ci
D 90

. Thus, p
i
D Fr
2
i
and the
values of p
i
are easy to obtain as functions of ˜
i
1 h
i
since they
are the roots of the third-order equation:
A
1
p
3
CB
1
p
2
CC
1
p CD
1
D 0
where
A
1
D lim
‹!0
A4‹5 D 1 C˜
1
C˜
2
C˜
1
˜
2
1
B
1
D lim
‹!0
B4‹5
‹
D ƒ41 C˜
1
C˜
2
C˜
1
˜
2
51
C
1
D lim
‹!0
C4‹5
‹
2
D ˜
1
h
1
h
3
Ch
2
h
3
H
2
C˜
2
h
1
h
3
Ch
2
h
3
H
2
C˜
1
˜
2
h
1
h
2
Ch
1
h
3
Ch
2
h
3
H
2
1
D
1
D lim
‹!0
D4‹5
‹
3
D ƒ˜
1
˜
2
h
1
h
2
h
3
H
3
Fig. 5 Same as in Fig. 3 but for a three-layer  uid. The three modes
of ship waves behave qualitatively similar. All angles ˆ
ci
are equal
19

2 8
0
in the limit ƒ
2
! 0. When velocity increases, ship wave angles
al so increase until they reach 90

for U D U
i
. These critical velocities
are mode-dependent. Dotted curves correspond to supercritical regions
wh ere only di vergent waves are present
According to Viet’s theorem the sum of the roots of a polynomial
is equal in magnitude and opposite in sign to the ratio of the
second and the  rst coef cie nts of the polynomial, and therefore
p
1
Cp
2
Cp
3
D ƒ
B
1
A
1
D 1, or
U
2
1
CU
2
2
CU
2
3
D gH (36)
The fact that the sum of squares of the critical velocities equals
the squared speed of long gravity waves certainly agrees with the
previously derived general relationship ( 31), providing an impor-
tant check on the theory in this section. For the numerically com-
puted c ritical velocities with the parameters given by (35) (see
Fig. 5) we have Fr
2
1
D 00821 Fr
2
2
D 0013, and Fr
2
3
D 0005 which
con rms (31) and (36).
Note that the analysis in thi s section concerns the angles of the
ship waves but not their amplitudes. It should be understood that
some of the three modes considered herein could be very small in
amplitude, according to the results of Section 3, depending on the
speed of the object, its location, and on the depth of a particular
interface. This explains why in Section 3 w e were dealing only
with 4 ƒm5 modes and here with all n D 3 of them.
6. Conclusions
We have studied the far- eld asymptotics of the waves caused
by a moving object in a strati ed three-dimensional  uid consist-
ing of n layers of constant density. The classi cation of the pat-
terns of ship waves in the multiple-laye r model as well as their
genera l properties are consistent with those for a two-layer  uid
MAR CH 2001 JOURNAL OF SHIP RESEARCH 11

(see Yeung & Nguyen 1999). The wave systems and the density
interfaces ar e perturbed by n wave mode s, one surface and 4nƒ15
interfacial, which are fairly similar to classical ship waves (Kelvin
1905). For each mode both divergent and transverse subsystems
are presen t if the speed of the object is less than some critical
(mode-dependent) velocity U
i
, and only the divergent component
is sustained in the supercritical regime 4U > U
i
5. We show that
in the multilayer model these critical velocities satisfy a simple
relationship
P
n
iD1
U
2
i
D gH.
When the speed of an object is suf ciently small (Section 3)
all the waves on the free surface and on the density interfaces
are con ned to an angle of arctan41=2
p
25 D 19

28
0
. For such
velocities 4U ! 05 it is poss ible to derive an explicit asymptotic
expression for the elevation of the free surface in the far  eld.
Among n different wave modes only  modes ( is the num-
ber of a layer where the object is located) have amplitudes large
enough to appear in the leading term of an asymptotic expan-
sion for the free surface perturbation. The results in Section 3 are
generalized in Section 4 to the case of an oscillating an d moving
source.
The case when the propagation velocit y is not assumed to be
small U
p
4ã=5gH is considered in Section 5 using a three-
layer model, which shows how the angles of each of the three
ship wave modes change as a function of speed. When the object
is moving slowly, the results are consistent with the asymptotic
(small U ) theory in Section 3 as the angles of all modes are equal
19

28
0
. As we increase the pr opagation speed U , the angle of
each wave mode 4ˆ
ci
5 increases until it reaches the critical value
U DU
i
when ˆ
ci
D90

. The further increase of velocity results in
a decrease of the shi p wave angle and in the signi cant changes in
the structure of the ship waves as only th e divergent components
remain in the supercritical regime.
Acknowledgments
I want to take this opportunity to thank Prof. B. Sebekin for
directing my attention to these problems and Prof. J. Winchester
for helpfu l editorial comments. The  nancial support by the
National Science Foundation is gratefully acknowledged.
References
Arabadzhi, V. 1996 On the active cancellation of ship waves. Exp. Flu-
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