On the solution of the problem of scattering
of surface-water waves by the
edge of an ice cover
By A. Ch ak ra ba rt iy
Department of Mathematics, Indian Institute of Science,
Bangalore-560012, India
Received 19 February 1999; revised 26 July 1999; accepted 1 September 1999
The mixed boundary-value problem arising in the study of scattering of two-dimensional time-harmonic surface-water waves by a discontinuity on the surface boundary
conditions, separating the clean surface and an ice-covered surface, is solved completely in the case of an in­ nite depth of water. The main problem is reduced to
that of solving a singular integral equation, of the Carleman type, over a semi-­ nite
range and the explicit solution of the original problem is determined. Neat and computable expressions are derived for the two most important quantities, known as the
re®ection and transmission coe¯ cients, occurring in such scattering problems and
tables of numerical values of these quantities are presented for speci­ c choices of a
parameter modelling the ice cover. The absolute values of the re®ection and transmission coe¯ cients are presented graphically. The present method of solution of the
boundary-value problem produces simple expressions for the principal unknowns of
the problem at hand and thus provides an easily understandable alternative to the
rather complicated Wiener{Hopf method used previously.
Keyword s: scattering; surface-water waves; ice cover; Fourier analysis;
Carleman-typ e singular integral equations; Riemann{H ilbert problem
1. Introduction
Various important and interesting methods of handling mixed boundary-value problems, associated with Laplace’s equation, arising in the study of scattering of surfacewater waves, have been developed and used by a large number of workers (see Ursell
1947; Stoker 1957; Peters 1950; Weitz & Keller 1950; Newman 1965; Evans 1985;
Evans & Linton 1994; Gabov et al . 1989; Gol’dshtein & Marchenko 1989, and others).
The problems of scattering of two-dimensional surface-water waves, by a discontinuity in the surface boundary conditions, constitute a special class, and methods involving the powerful Wiener{Hopf technique have been used by Weitz & Keller (1950),
Gabov et al . (1989) and Gol’dshtein & Marchenko (1989) to solve such boundaryvalue problems.
In the present paper we have demonstrated the use of Fourier analysis (see Ursell
1947), in a straightforward manner, to examine the problem of scattering of twodimensional surface-water waves by the edge of an ice cover. We have considered
y Present address: Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA.
Proc. R. Soc. Lond. A (2000) 456, 1087{1099
® c 2000 The Royal Society
1087
A. Chakrabarti
1088
here the case of an in­ nite depth of water whose surface is half-covered by a thin
sheet of ice, the other half being clean. Then, by following the model of Gol’dshtein
& Marchenko (1989) and assuming that the ice sheet behaves like a thin elastic plate,
the problem under consideration is reduced to that of determining the solution of the
two-dimensional Laplace equation in a half-plane, on whose boundary two di¬erent
boundary conditions are to be satis­ ed. One of these two conditions is the standard
free-surface condition involving the unknown potential and its normal derivative,
while the other condition (on the ice-covered boundary) involves the potential and
its normal derivatives of both ­ rst and ­ fth orders (see Gol’dshtein & Marchenko
1989). Other than these two boundary conditions on the free surface, the solution
will have to satisfy speci­ c types of conditions at in­ nity which are representative of
surface waves, and also, for the purpose of uniqueness, special edge conditions must
be satis­ ed by the solution at the turning point (the edge) on the surface separating
the clean and the ice-covered parts. In x 2, we describe a brief formulation of the
mathematical problem under consideration, and in x 3, the problem is reduced to a
singular integral equation of the Carleman type (see Spence 1961). The solution of
this Carleman-type equation is then obtained in x 4 and the complete solution of the
original problem is also determined. Tables of numerical values of the re®ection and
transmission coe¯ cients, associated with the problem, are presented in x 5, after a
brief explanation of the rather simple computational aspect of the work, and the
absolute values of these important physical quantities are also presented graphically.
It is emphasized that the present method of solution, of the rather complicated
boundary-value problem under consideration, is much easier to understand and
implement than the more di¯ cult ideas and calculations involved in the Wiener{
Hopf method, used for the same problem by Gol’dshtein & Marchenko (1989).
2. Mathematical formulation
Using two-dimensional rectangular Cartesian coordinates (x; y), we assume that the
region y > 0 is occupied by water of constant density and half of the surface (x > 0)
is covered by a thin sheet of ice of a di¬erent constant density, whereas the other
half of the surface (x < 0) is clean. A plane surface wave arrives from the clean side
and is scattered by the edge x = 0, separating the clean and the ice-covered surface.
The problem under consideration is to determine the complete scattered ­ eld and,
in particular, the re®ection and the transmission coe¯ cients.
Following the symbols used by Gol’dshtein & Marchenko (1989), the mathematical
problem under consideration is that of the determination of the velocity potential
Ref¿ (x; y)e¡it g (i2 = ¡ 1), where ¿ , x, y, t are all dimensionless variables, satisfying
the following equation and conditions
’xx + ’yy = 0;
’ + ’y = 0;
’ + ’y + D’yxxxx = 0;
¡
1 < x < 1;
1 < x < 0;
¡
0 < x < 1; y > 0
’ ! 0; y ! 1;
’ ! eix¡y + Re¡ix¡y ;
¡¶ y+ i¶ x
’ ! Te
Proc. R. Soc. Lond. A (2000)
;
y > 0;
y = 0;
(D > 0; const:);
(2.1)
(2.2)
(2.3)
(2.4)
x ! ¡ 1;
(2.5)
x ! +1;
(2.6)
Scattering of surface-water waves by the edge of an ice cover
’yxx ! 0;
’yxxx ! 0;
x ! +0;
x ! +0;
y = 0;
y = 0;
1089
(2.7)
(2.8)
where R and T are unknown complex constants, called the re®ection and transmission
coe¯ cients, respectively, the function eix¡y represents the incident wave term and
the functions e¡ix¡y and e¡¶ y+ i¶ x represent the re®ected and the transmitted wave
terms, respectively, in which ¶ is the positive root of the equation in ¬ , as given by
4
¬ (1 + D¬
)¡
1 = 0;
(2.9)
with the constant D (> 0) representing the ice-cover parameter.
The exact expression for the parameter D, as used by Gol’dshtein & Marchenko
(1989), is given by
D=
Eh30
12(1 ¡ ¸ 2 )» g¶
4
;
(2.10)
where E is the Young’s modulus and ¸ the Poisson’s ratio of the material of the
isotropic and elastic ice sheet, » is the density of the water on which it ®oats, g
is the acceleration due to gravity, h0 is the very small thickness of the ®oating ice
sheet of which a still smaller part is immersed in water, and ¶ represents 2º times the
wavelength of the incident wave (the variables x and y have been made dimensionless
with the help of ¶ ).
It must be emphasized that the above model for the ®oating ice sheet is meaningful
only for very small values of the parameter D, which we have called here the ice-cover
parameter.
The same model has also been used by Fox & Squire (1994) while studying the
problem of oblique re®ection and transmission of plane waves at shore fast sea ice,
and, as has been mentioned there, the present model of a ®oating ice sheet is more
appropriate to the physical situation than the ones used by Peters (1950) and Weitz
& Keller (1950), whose limitations have been pointed out by Evans (1985).
3. Reduction to a singular integral equation
In this section, we shall reduce the mixed boundary-value problem as posed by the
relations (2.1){(2.8) into a singular integral equation, over a semi-in­ nite range,
which is of the Carleman type. In order to arrive at such an integral equation in
the course of the solution of our boundary-value problem, we shall use Havelock’s
expansion theorem (see Ursell 1947), which can be stated as follows.
Theorem 3.1 (Havelock’s expansion theorem). If
Z 1
f (t)(¹ cos ¹ t ¡ k sin ¹ t) dt = F (¹ ); ¹ > 0;
k > 0;
0
then
f (t) =
with
º
2
Z
1
0
(¹
F (¹ )
(¹ cos ¹ t ¡
+ k2)
2
C0 = 2k
Z
k sin ¹ t) d¹ + C0 e¡kt ;
t > 0;
1
f (t)e¡kt dt;
0
where f (t) and its derivatives are continuous and integrable in the range (0; 1).
Proc. R. Soc. Lond. A (2000)
A. Chakrabarti
1090
We shall also require the following `generalized’ identities,
9
Z 1
>
lim
e¡"y cos(uy) cos(¹ y) dy = 12 º [¯ (¹ ¡ u) + ¯ (¹ + u)]; >
>
>
"! 0 0
>
>
>
Z 1
=
¡"y
1
lim
e
sin(uy) sin(¹ y) dy = 2 º [¯ (¹ ¡ u) ¡ ¯ (¹ + u)];
"! 0 0
>
>
>
Z 1
>
>
u
>
¡"y
>
lim
e
sin(uy) cos(¹ y) dy = 2
;
;
"! 0 0
(u ¡ ¹ 2 )
(3.1)
where ¹ , u > 0 and ¯ (x) represents the well-known Dirac delta function.
We start by setting ¿ = @ 2 Á=@x2 and assuming the following two important
representations of the function Á in the two regions (x < 0; y > 0) and (x > 0; y > 0)
Z
2 1 A(¹ )L(¹ ; y) ¹ x
Á = Á1 = ¡ eix¡y ¡ Re¡ix¡y +
e d¹
(3.2)
º 0
(¹ 2 + 1)
(for x < 0, y > 0), with
L(¹ ; y) = ¹ cos ¹ y ¡
sin ¹ y
(3.3)
and
Á = Á2 = ¡
µ
¶
T
2
¶
e¡¶
1 y+
i¶ x
+ A1 e¡¶
1 y+
+ A2 e¡¶
i¶
1x
· 1 y¡i¶ · 1 x
+
2
º
Z
1
0
B(¹ )M (¹ ; y) ¡¹
e
P (¹ )
x
d¹
(3.4)
(for x > 0, y > 0), with
M (¹ ; y) = ¹ (1 + D¹ 4 ) cos ¹ y ¡
sin ¹ y;
(3.5)
and
P (¹ ) = ¹ 2 (1 + D¹ 4 )2 + 1;
(3.6)
where R, T , A1 and A2 are four unknown constants, A(¹ ) and B(¹ ) are two unknown
functions to be determined, and ¶ is the only positive root of equation (2.9), whose
other roots are complex conjugate pairs (¶ 1 ; ¶ · 1 ) and (¶ 2 ; ¶ · 2 ), with Re(¶ 1 ) > 0,
Im(¶ 1 ) > 0, Re(¶ 2 ) < 0 and Im(¶ 2 ) > 0 (where bars denote complex conjugates).
We observe that the above forms (3.2) and (3.4) of the function Á automatically satisfy the partial di¬erential equation (2.1) and the boundary conditions (2.2)
and (2.3), as well as the in­ nity conditions (2.5) and (2.6), for an appropriate choice
of the two functions A(¹ ) and B(¹ ), which will be determined in the sequel.
We now use the fact that Á and @Á=@x must be continuous across the line x = 0,
and obtain the following two relations, respectively:
Z
2 1 A(¹ )L(¹ ; y)
T ¡¶ y
d¹ = (1 + R)e¡y ¡
e
+ A1 e¡¶ 1 y
2
º 0
(¹ + 1)
¶ 2
Z
2 1 B(¹ )M (¹ ; y)
·
+ A 2 e ¡¶ 1 y +
d¹
(for y > 0) (3.7)
º 0
P (¹ )
Proc. R. Soc. Lond. A (2000)
Scattering of surface-water waves by the edge of an ice cover
and
Z
2 1
º 0
A(¹ )¹ L(¹ ; y)
d¹ = i(1 ¡
(¹ 2 + 1)
¡
i
R)e¡y ¡
T e¡¶
¶
2
·
i¶ · 1 A 2 e ¡ ¶ 1 y ¡
º
y
Z
+ i¶
1
0
1 A1 e
¡¶
1091
1y
B(¹ )¹ M (¹ ; y)
d¹
P (¹ )
(for y > 0):
(3.8)
If we assume, for the time being, that B(¹ ) is known, then we can use Havelock’s
expansion theorem 3.1 and ­ nd that we must have, from relations (3.7) and (3.8),
Z
A1
A2
T
2D 1 ¹ 5 B(¹ )
1
+
+ (1 + R) ¡
+
d¹ = 0
(1 + ¶ 1 ) (1 + ¶· 1 ) 2
¶ 2 (1 + ¶ )
º
Q(¹ )
0
(3.9)
and
i¶ · 1
A2 + 12 i(1 ¡
(1 + ¶ · 1 )
i¶ 1
A1 ¡
(1 + ¶ 1 )
R) ¡
i
T¡
¶ (1 + ¶ )
2D
º
Z
1
0
¹ 6 B(¹ )
d¹ = 0;
Q(¹ )
(3.10)
with
Q(¹ ) = (¹
2
+ 1)P (¹ ):
(3.11)
Then relations (3.7) and (3.8) can be inverted to give the following two equations,
respectively,
·
¸
¡ T (¶ ¡ 1)
A1 (¶ 1 ¡ 1) A2 (¶ · 1 ¡ 1)
A(¹ ) = ¹
+
+
¶ 2 (¹ 2 + ¶ 2 )
(¹ 2 + ¶ 21 )
(¹ 2 + ¶ · 21 )
Z 1
[¹ 2 (1 + D¹ 4 ) + 1]
2D¹
u5 B(u)
+
B(¹ ) +
du;
(3.12)
P (¹ )
º
P (u)(u2 ¡ ¹ 2 )
0
and
·
¸
A2 ¶ · 1 (¶· 1 ¡ 1)
i
(¹ 2 + ¶· 21 )
Z
2D 1
u6 B(u)
du;
º
P (u)(u2 ¡ ¹ 2 )
0
¡ T (¶ ¡ 1)
A1 ¶ 1 (¶ 1 ¡ 1)
A(¹ ) =
+
¡
2
2
2
¶ (¹ + ¶ )
(¹ 2 + ¶ 21 )
¡
¹ 2 (1 + D¹ 4 ) + 1
B(¹ ) ¡
P (¹ )
(3.13)
where the singular integrals occurring above and even in the later parts of our work
are to be understood as their Cauchy principal values, in the usual manner.
Eliminating A(¹ ) between the above two relations (3.12) and (3.13), we easily
derive the following integral equation
Z
1 1
¶ (u)
C(¹ )¶ (¹ ) +
du
º 0 (u ¡ ¹ )
T (¶ ¡ 1)
A1 (¶ 1 ¡ 1) A2 (¶ · 1 ¡ 1)
=
¡
¡
(for ¹ > 0); (3.14)
2
2(¹ + i¶ )¶
2(¹ + i¶ 1 )
2(¹ ¡ i¶ · 1 )
Proc. R. Soc. Lond. A (2000)
A. Chakrabarti
1092
where
¶ (¹ ) =
D¹ 5 B(¹ )
P (¹ )
(3.15)
C(¹ ) =
[¹ 2 (1 + D¹ 4 ) + 1]
:
D¹ 5
(3.16)
and
Equation (3.14) is the desired singular integral equation of the Carleman type, whose
solution will determine B(¹ ). Then A(¹ ) can be determined, by using either one of
the relations (3.12) and (3.13).
The most important point that must be noted in equation (3.14) is that the forcing
term, i.e. the inhomogeneous term, of this equation contains the unknown constants
T , A1 and A2 . However, we shall go ahead solving equation (3.14) as though the
constants T , A1 and A2 are known. All the four unknown constants R, T , A1 and
A2 will be ­ nally determined by using the two relations (3.9) and (3.10), as well as
two more relations, obtainable from the edge conditions (2.7) and (2.8), along with
the representation (3.4) of the function Á, which are given by
Z
2 1 ¹ 5 B(¹ )
5
5
5
·
¶ T + ¶ 1 A1 + ¶ 1 A2 +
d¹ = 0;
(3.17)
º 0
P (¹ )
Z 1
2
¹ 6 B(¹ )
i¶ 6 T + i¶ · 61 A1 ¡ i¶ 61 A2 ¡
d¹ = 0:
(3.18)
º 0
P (¹ )
In the next section we shall determine the complete solution of the problem.
4. The complete solution
The singular integral equation (3.14) can be solved easily, by converting it into a
Riemann{Hilbert problem (see Muskhelishvili 1953; Gakhov 1966) and, for that purpose, we set
Z 1
1
¶ (u)
¤ (± ) =
du; ± = ¹ + i² 2
= (0; 1):
(4.1)
2º i 0 (u ¡ ± )
The consistency of equation (3.14) demands that ¶ (¹ ) º O(1=¹ 2 ), i.e. B(¹ ) º O(¹ 3 )
as ¹ ! 1. Therefore, we have that the sectionally analytic function ¤ (± ) is O(1=± )
as j± j ! 1, in the complex ± -plane, cut along the positive real axis.
Then, by using Plemelj’s formulae
Z 1
1
¶ (u)
§
1
¤ (¹ ) = § 2 ¶ (¹ ) +
du;
(4.2)
2º i 0 (u ¡ ¹ )
with ¤ § (¹ ) denoting the limiting values of ¤ (± ) as ± ! ¹ § i0, equation (3.14) gets
converted into the problem of solving the functional relation
[C(¹ ) + i]¤
+
(¹ ) ¡
[C(¹ ) ¡
Proc. R. Soc. Lond. A (2000)
i]¤
¡
(¹ ) =
¡ T (1 ¡ ¶ )
A1 (1 ¡ ¶ 1 ) A2 (1 ¡ ¶ · 1 )
+
+
(4.3)
2
2(¹ + i¶ )¶
2(¹ + i¶ 1 )
2(¹ ¡ i¶ · 1 )
Scattering of surface-water waves by the edge of an ice cover
1093
for ¹ > 0, which represents a Riemann{Hilbert problem for the determination of the
function ¤ (± ).
The solution of the problem (4.3) can be written using standard techniques, and
we ­ nd that
Z 1
¤ (± )
1
du
=
¤ 0 (± )
4º i 0 ¤ +0 (u)[C(u) + i](u ¡ ± )
½
¾
¡ T (1 ¡ ¶ ) A1 (1 ¡ ¶ 1 ) A2 (1 ¡ ¶ · 1 )
£
+
+
; (4.4)
¶ 2 (u + i¶ )
(u + i¶ 1 )
(u + i¶ · 1 )
¤
where
0 (± ) = exp
·
1
2º i
Z
1
0
½µ
log
·
¸
C(t) ¡ i
¡
C(t) + i
lim log
1
t!
·
C(t) ¡ i
C(t) + i
¸¶Á
(t ¡
¾ ¸
± ) dt ;
± 2
= (0; 1);
(4.5)
which is a solution of the homogeneous problem
[C(¹ ) + i]¤
+
0
(¹ ) ¡
[C(¹ ) ¡
i]¤
¡
0 (¹
)=0
(4.6)
(see Varley & Walker 1989).
The solution ¶ (¹ ) of the integral equation (3.14) can now be determined, using
Plemelj’s formulae once again, in the form
¶ (¹ ) = ¤
+
(¹ ) ¡
¤
¡
(¹ );
(4.7)
which produces B(¹ ) by using the relation (3.15), and we ­ nd that
B(¹ ) =
where
·
P (¹ )
R 0 (¹ ) =
D¹ 5
¡ T (1 ¡ ¶ )
R0 (¹ ) + A1 (1 ¡
¶ 2
¸½
¶
1 )R 1 (¹
) + A2 (1 ¡
¶ · 1 )R2 (¹ );
C(¹ )
2
2[C (¹ ) + 1](¹ + i¶ )
Z 1
¤ +0 (¹ )
du
¡
+
2º [C(¹ ) ¡ i] 0 ¤ 0 (u)[C(u) + i](u + i¶ )(u ¡
R 1 (¹ ) = fR0 (¹ ); with ¶ replaced by ¶ 1 g;
R 2 (¹ ) = fR0 (¹ ); with ¶ replaced by ¡ ¶ · 1 g:
(4.8)
9
>
>
>
>
>
>
¾ >
>
=
;
(4.9)
¹ ) >
>
>
>
>
>
>
>
;
Thus the complete solution of the problem at hand can be determined once the
unknown constants T , A1 and A2 (also R) are fully determined, and for that purpose,
we shall now use conditions (3.9), (3.10), (3.17) and (3.18), giving rise to four linear
equations involving these unknown constants. We can easily solve these equations
and ­ nd that
µ
¶
a2 t2 ¡ b2 t1 T
A1 = ¡
;
(4.10)
a 1 b 2 ¡ a2 b 1 ¶ 2
µ
¶
b1 t1 ¡ a1 t2 T
A2 = ¡
;
(4.11)
a 1 b 2 ¡ a2 b 1 ¶ 2
Proc. R. Soc. Lond. A (2000)
A. Chakrabarti
1094
T =
[U0 (a1 b2 ¡
R= ¡ 1+2
[(a1 b2 ¡ a2 b1 )¶ 2 ]
a2 b1 ) + U1 (a2 t2 ¡ b2 t1 ) + U2 (b1 t1 ¡
(4.12)
a1 t2 )]
[V0 (a1 b2 ¡ a2 b1 ) + V1 (a2 t2 ¡ b2 t1 ) + V2 (b1 t1 ¡ a1 t2 )]
;
[U0 (a1 b2 ¡ a2 b1 ) + U1 (a2 t2 ¡ b2 t1 ) + U2 (b1 t1 ¡ a1 t2 )]
(4.13)
where
Z 1
9
2
(¹ ¡ i¶ · 1 )¹ 5 R1 (¹ )
>
>
)
¡
(1
¡
¶
)
d¹
;
1
1
>
>
º
P
(¹
)
>
0
>
>
Z 1
>
5
·
>
2
(¹
¡
i
¶
)¹
R
(¹
)
>
1
2
·
>
=¡
(1 ¡ ¶ 1 )
d¹ ;
>
>
º
P (¹ )
>
0
>
>
Z 1
>
>
2
(¹ + i¶ 1 )¹ 5 R1 (¹ )
>
>
>
= (1 ¡ ¶ 1 )
d¹ ;
>
>
º
P
(¹
)
>
0
>
Z 1
>
>
5
>
2
(¹
+
i¶
)¹
R
(¹
)
1
2
5
·
·
·
>
= i¶ 1 (¶ 1 + ¶ 1 ) + (1 ¡ ¶ 1 )
d¹ ; >
>
>
º
P
(¹
)
>
0
>
>
Z 1
>
5
·
>
2
(¹
¡
i
¶
)¹
R
(¹
)
>
1
0
5 ·
·
= i¶ 1 (¶ 1 + ¶ ) ¡
(1 ¡ ¶ 1 )
d¹ ; >
>
>
>
º
P (¹ )
0
>
>
Z 1
>
5
>
2
(¹
+
i¶
)¹
R
(¹
)
>
1
0
5
>
= i¶ 1 (¶ 1 ¡ ¶ · ) + (1 ¡ ¶ )
d¹ ; >
=
º
P
(¹
)
0
Z 1
>
2iD(1 ¡ ¶ )
¹ 5 R0 (¹ ) d¹
>
>
=1+
;
>
>
º
(¹
+
i)P
(¹
)
>
0
>
>
Z 1
>
5
>
2iD
¹ R1 (¹ )
>
>
=1+
(1 ¡ ¶ 1 )
d¹ ;
>
>
>
º
(¹ + i)P (¹ )
0
>
>
µ
¶
Z
>
1
5
>
·
>
1¡ ¶ 1
2iD
¹
R
(¹
)
2
>
>
=
+
(1 ¡ ¶· 1 )
d¹ ;
>
>
1+¶ 1
º
(¹
+
i)P
(¹
)
>
0
>
>
µ
¶
Z 1
>
5
>
1
2D
¹ R0 (¹ )
>
>
=
+
(1 ¡ ¶ )
d¹ ;
>
>
1+¶
º
Q(¹ )
>
0
>
>
µ
¶
Z 1
>
5
>
1
2D
¹ R1 (¹ )
>
>
>
=
+
(1 ¡ ¶ 1 )
d¹ ;
>
>
1+¶ 1
º
Q(¹
)
>
0
>
µ
¶
Z 1
>
>
5
>
1
2D
¹
R
(¹
)
2
>
·1)
;
=
+
(1
¡
¶
d¹
:
º
Q(¹
)
1 + ¶·1
0
a1 = i¶
a2
b1
b2
t1
t2
U0
U1
U2
V0
V1
V2
5 ·
1 (¶ 1
+¶
(4.14)
The whole matter now reduces to the evaluation of the three functions R 0 (¹ ),
R1 (¹ ) and R2 (¹ ), as given by the relations (4.9). This job can be completed by using
the following result
Z
¡
d½
¤ 0 (½ )(½ + i¶ )(½ ¡
2º i
=
&)
± + i¶
·
1
¡
¤ 0 (± )
¤
1
0 (¡
i¶ )
¸
;
(4.15)
where ¡ is a positively oriented (anticlockwise) closed contour, consisting of a loop
around the positive real axis and a circle of large radius, in the complex ½ -plane. A
similar result also holds good if ¶ is replaced by ¶ 1 and ¡ ¶ · 1 .
Proc. R. Soc. Lond. A (2000)
Scattering of surface-water waves by the edge of an ice cover
1095
It is to be noted that the contour integral in the above relation (4.15) can also be
expressed as equal to
Z 1
du
2i
;
(4.16)
+
[C(u) + i]¤ 0 (u)(u + i¶ )(u ¡ ± )
0
obtained by using the two limiting values ¤ §
0 (u) on the two approaches to the real
axis, which satisfy the homogeneous relation (4.6).
Using the idea as explained above, we easily determine that
R0 (¹ ) =
P (¹ )¤ +0 (¹ )
2D¹ 5 [C(¹ ) ¡ i](¹ + i¶ )¤
R1 (¹ ) =
P (¹ )¤ +0 (¹ )
2D¹ 5 [C(¹ ) ¡ i](¹ + i¶ 1 )¤
0 (¡
i¶
R2 (¹ ) =
P (¹ )¤ +0 (¹ )
2D¹ 5 [C(¹ ) ¡ i](¹ + i¶ · 1 )¤
0 (¡
i¶ · 1 )
0 (¡
i¶ )
;
1)
(4.17)
;
(4.18)
:
(4.19)
We shall now use the above values of R0 , R1 and R2 in the relations (4.14). We ­ rst
evaluate the contour integral
Z
L(± )
¤ 0 (± )
d± ;
(4.20)
M
(± )
¡
where L(± ) and M (± ) are polynomial expressions in ± , and ¡ is the contour, same
as the one used previously, in (4.15), and we ­ nd, after using relation (4.6), that
Z
1
0
½
X
¤ +0 (¹ )
L(¹ )
d¹ = ¡ º
Re s ¤
[C(¹ ) ¡ i] M (¹ )
¾
L(± )
;
0 (± )
M (± )
(4.21)
where the expression on the right-hand side represents a sum of all the residues of
¤ 0 (L=M ) at the poles lying inside ¡ .
Using relation (4.21) to evaluate all integrals occurring in relations (4.14), we
obtain the following results:
·
¸ 9
(1 ¡ ¶ 1 ) >
5
·
a1 = i(¶ 1 + ¶ 1 ) ¶ 1 ¡
;>
>
>
>
D
>
>
>
>
a2 = 0;
>
>
>
>
>
>
>
b1 = 0;
>
>
=
·
¸
·
(1
¡
¶
)
1
(4.22)
b2 = i(¶ 1 + ¶ · 1 ) ¶ · 51 ¡
;>
>
D
>
·
¸ >
>
>
>
(1
¡
¶
)
5
·
>
t1 = i(¶ 1 + ¶ ) ¶ ¡
; >
>
>
D
>
·
¸ >
>
>
>
(1
¡
¶
)
>
5
>
t2 = i(¶ 1 ¡ ¶ ) ¶ ¡
; ;
D
Proc. R. Soc. Lond. A (2000)
A. Chakrabarti
1096
U0
U1
U2
V0
V1
V2
·
¸
9
i)
>
>
=
;
>
>
¤ 0 (¡ i¶ )
>
>
>
·
¸
>
>
¤ 0 (¡ i)
>
>
=
;
>
>
>
¤ 0 (¡ i¶ 1 )
>
>
µ
¶·
¸
>
>
·
1¡ ¶ 1
¤ 0 (¡ i)
>
>
>
=
;
=
·
·
1+¶ 1
¤ 0 (i¶ 1 )
[((1 ¡ ¶ )=(1 + ¶ ))¤ 0 (i) + ¤ 0 (¡ i)] >
>
=
; >
>
>
2¤ 0 (¡ i¶ )
>
>
>
>
[((1 ¡ ¶ 1 )=(1 + ¶ 1 ))¤ 0 (i) + ¤ 0 (¡ i)] >
>
=
;>
>
>
2¤ 0 (¡ i¶ 1 )
>
>
>
>
·
·
[((1 ¡ ¶ 1 )=(1 + ¶ 1 ))¤ 0 (¡ i) + ¤ 0 (i)] >
>
;
=
:>
2¤ 0 (¡ i¶· 1 )
¤
0 (¡
(4.22cont.)
Thus all quantities are now expressed in terms of ¤ 0 (i), ¤ 0 (¡ i), ¤ 0 (¡ i¶ ), ¤ 0 (¡ i¶ 1 )
and ¤ 0 (i¶ · 1 ) and other simple combinations of ¶ , ¶ 1 and ¶ · 1 , so that the principal
unknowns of the original problem, i.e. the functions A(¹ ), B(¹ ) and the constants
A1 , A2 , R and T , can be determined easily, where ¤ 0 (± ) is given by expression (4.5).
It is interesting to observe that in the limit, when D ! 0, we deduce from the
above results that R ! 0 and T ! 1, which was reported earlier by Gol’dshtein &
Marchenko (1989), showing that the ice cover does not a¬ect the incident wave in
this limiting case, even though it is rather hard to prove this important conclusion
from the various results obtained by Gol’dshtein & Marchenko, compared with the
ones obtained in the present paper.
5. Numerical values of R and T
We select di¬erent values of the constant D and determine ­ rst the roots ¶ , ¶ 1 , ¶ · 1 ,
¶ 2 and ¶· 2 of the polynomial equation (2.9), with ¶ > 0, Re(¶ 1 ) > 0, Im(¶ 1 ) > 0,
Re(¶ 2 ) < 0 and Im(¶ 2 ) > 0.
We shall require the following relation connecting C(¹ ) and the roots ¶ , ¶ 1 , ¶ · 1 , ¶ 2
and ¶ · 2 , which is easily established using expression (3.16) for C(¹ ):
(¹ § i)
(¹ ¨ i¶ )(¹ ¨ i¶ · 1 )(¹ ¨ i¶
¹ 5
(¹ § i)
=
[D¹ 5 + ¹ ¨ i]:
D¹ 5
C(¹ ) § i =
2 )(¹
¨ i¶· 2 )
(5.1)
Then, for a particular choice of D, we evaluate the various constants appearing
in relations (4.22), after evaluating the expressions ¤ 0 (¡ i¶ ), ¤ (¡ i¶ 1 ), ¤ 0 (i¶ · 1 ), ¤ 0 (i)
and ¤ 0 (¡ i) by utilizing expression (4.5), along with relation (5.1).
Finally, the quantities R and T are determined using relations (4.12) and (4.13)
for each of the above choices of the constant D.
The numerical values of the roots ¶ and ¶ 1 of the polynomial equation (2.9) clearly
show that as D becomes smaller and smaller, ¶ becomes closer to unity and Re(¶ 1 )
becomes larger and larger than unity, con­ rming the validity of the expected nature
of the solution, as D ! 0.
Proc. R. Soc. Lond. A (2000)
Scattering of surface-water waves by the edge of an ice cover
1097
1
|T |
0.8
|R |, |T |
0.6
0.4
0.2
|R |
0
0
0.1
0.2
D
0.3
0.4
Figure 1. Variation of jRj and jT j with D = [0; 0:4].
Table 1. Numerical values of R and T for various values of D
D
R
T
jRj
jT j
0
0.01
0.02
0.03
0.04
0.05
0.1
0.2
0.3
0.4
0
0:0226 ¡ 0:0824i
0:0149 + 0:0042i
0:0152 + 0:0095i
0:0125 + 0:0162i
0:0205 ¡ 0:0198i
0:0321 ¡ 0:0200i
0:0585 ¡ 0:0047i
0:0831 + 0:0279i
0:0987 + 0:0180i
1
0:9669 ¡ 0:0969i
0:9852 ¡ 0:0032i
0:9662 + 0:0048i
0:9433 + 0:0176i
0:8943 ¡ 0:0125i
0:8318 ¡ 0:0039i
0:7897 ¡ 0:0026i
0:7925 ¡ 0:0082i
0:7434 ¡ 0:0154i
0
0.0854
0.0155
0.0179
0.0204
0.0285
0.0379
0.0587
0.0877
0.1003
1
0.9717
0.9852
0.9662
0.9434
0.8944
0.8318
0.7897
0.7925
0.7436
The numerical results of jRj and jT j are found to be less than unity, as expected.
In table 1 we present the values of R and T , and in ­ gure 1 the graphs of jRj and
jT j, for values of D < 0:5.
6. Conclusions
The problem of scattering of plane waves, incident from the clean side of the surface,
by the edge of an ice cover, lying on the other side, has been solved completely. The
method presented here can be easily adjusted to handle the two other similar problems considered by Gol’dshtein & Marchenko (1989). One of these di¬erent boundaryvalue problems is similar to the one described by equations (2.1){(2.8), except that
the in­ nity conditions (2.5) and (2.6) are replaced by the conditions (2.5*) and (2.6*),
Proc. R. Soc. Lond. A (2000)
A. Chakrabarti
1098
as given by
¿ ! e¡¶
(ix+ y)
+ R1 e¶
(ix¡y)
as x ! +1
(2.5*)
as x ! ¡ 1;
(2.6*)
and
¿ ! T1 e¡(ix+
y)
where R1 and T1 are the new re®ection and transmission coe¯ cients which are related
to the problem of scattering of the plane wave e¡¶ (ix+ y) , which arrives at the edge
of the ice cover, from the ®uid beneath the ice (x > 0).
The second problem that can also be handled by our method is that of the determination of the ®uid motion in an in­ nite depth of ®uid whose surface is half-covered
by a thin ice sheet, and where, on the edge of the ice sheet, there act known timeperiodic concentrated forces and moments. The corresponding mathematical problem is again similar to the one described by the equations and conditions (2.1){(2.8),
except that conditions (2.5), (2.6), (2.7) and (2.8) are to be replaced by the following
new conditions:
¿ ! R2 e¡¶
¿ ! T2 e¶
¿ yxx ! M2
¿ yxxx ! F2
(ix¡y)
(ix¡y)
as x ! +1;
(2.5**)
as x ! ¡ 1;
as x ! 0+; y ! 0;
as x ! 0+; y ! 0;
(2.6**)
(2.7**)
(2.8**)
where M2 and F2 are known constants.
I ¯rst thank the referees for their constructive criticisms and suggestions to improve the presentation of the paper. I then thank the Indian National Science Academy and The Royal Society,
as well as the University of Westminster, London, for supporting my three-month visit to the
UK, during which period it has been possible for me to revise the manuscript, and I thank
Dr P. K. Bhattacharyya of the Cavendish School of Computer Science for his constant encouragement and support. Finally, I thank Dr Hamsapriye of the Bangalore University, India, and
Mr Masor Mudyr of the Cavendish School of Computer Science, University of Westminster,
London, for helping me in obtaining all the numerical results.
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