737
Internat. J. Math. & Math. Sci
Vol. 6 No. 4 (1983) 737-754
SINGULARITIES IN A TWO-FLUID MEDIUM
RATHINDRA NATH CHAKRABARTI
Narikeldanga High School
50, Kavi Sukanta Sarani
Calcutta-700085, INDIA
and
BIRENDRANATH MANDAL
Department of Applied Mathematics, Calcutta University
92, Acharya Prafulla Chandra Road
Calcutta-700009, INDIA
(Received December 30, 1981 and in revised form August 30, 1982)
ABSTRACT.
We compute the irrotational motion of two fluids with a horizontal plane
surface of separation, under gravity.
The fluids are nonviscous and incompressible,
the upper one of finite depth with a free surface; they contain a line singularity or
a point singularity.
We obtain the velocity potentials for each singularity located
in the upper or the lower fluid; if the upper depth tends to infinity, known results
are recovered.
KEY WORDS AND PHRASES. Fluid dynamics, Laplace’s equation, two-fluid problem, poi
sgularity, line sgularit, oscilg sgularit, free surface.
1980 IqATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
76B15, 31A05, 31B10, 33A45.
INTRODUCTION.
The reader of this paper should be familiar with Laplace’s equation as applied to
fluids.
For the two dimensional case studied in Sections 3 and 4, elementary complex
variable theory is needed as far as the theory of residues.
case treated in Sections 5 and
assumed.
6,
an elementary
For the three dimensional
knowledge of spherical harmonies is
In brief, you need to know typical singular solutions of Laplace’s equation,
since they dominate the potential in the neighborhood of the singularities.
To moti-
vate and better understand the integral representations of the potential assumed in
738
R. N. CHAKRABARTI AND B. MANDAL
(3.1), (3.2), (4.1), (4.2), (3.i), (5.2), and (0.i), (6.2), see Thorne ([i], pp. 708,
710, 711) and Rhodes-Robinson ([2], p. 320).
For a brief history of the subject, see
Thorne, p. 715.
Many authors have investigated different types
Some recent history is as follows.
of singularities that can be used in the one-fluid problem.
Robinson
Thorne [I] and Pdodes-
[2] gave surveys of the fundamental line and point singularities submerged in
The two-fluid problem was discussed by Gorgui
a fluid of finite or infinite depth.
and Kassem
[3], landal [4], and Chakrabarti [5J--the effect of surface
included by the last two authors as well as in
mentioned in the Introduction of
[2].
tension being
References for other cases are
[3].
In this paper we shall discuss the basic line and point singularities when they
are submerged in one of two fluids.
-
The upper fluid is on finite constant depth
with a free surface (FS); the lower fluid is of infinite depth.
’h’
The time harmonic
singularities are described by harmonic potential functions with period
which satis-
fy the boundary conditions at the surface of separation (SS); in fact it is more convenient to use complex-valued potentials
part.
The potential must also satisfy
singularity:
e -it
the actual potential being the real
limitinF conditions
in the neighborhood of the
it should behave like a typical singular harmonic function, as already
mentioned; in the far field it should represent a spreading wave.
Under these require-
ments, a unique solution will be found in all cases considered.
We note that our solution can be applied to cases when bodies are present in the
fluids, whenever the two- or three-dimensional symmetry is such that the motion can be
described by a series of singularities placed within the body in suitable positions.
Whether the waves are generated by the body, or reflected by the body. does not matter.
2.
STATIENT AND FORIULATION OF THE PROBLEM.
Consider the irrotational motion of two non-viscous incompressible fluids under
the action of gravity.
Their SS is a horizontal plane, the lower fluid of infinite
depth, the upper of finite height
’h’.
Their motion is due to an oscillating singu-
larity in one of the fluids; it is assumed to be simple harmonic with period---the velocity potentials
too.
@i
and
2
so
(of the lower and upper fluids respectively) will be
SINGULARITIES IN A TWO-FLUID MEDIUM
739
We take origin 0 in the mean SS, axis 0y pointing vertically downward into the
lower fluid and chosen so it passes through he singularity which is then located at
(0,N)
or
(0,-N) according as the singularity
is in the lower or upper fluid respec-
tively.
Then, for all x,
except at the singular point.
02
0 -< y -< h
V2@2
O,
y < 0
Also
Sy + K@ 2
0
on y
V@I
0
on y
02
s
g
Ix
2
(2.2)
0
+
on y
(2.3)
0
is the acceleration due to gravity, and
g
=--01density.
the upper fluid
condition as
S (K@
1
h
on y
0y
K@ 1 +
2
when K
0,
y
-
and
V2@I
-oo.
@I
Finally,
and
@2
01
is the lower and
must satisfy the so-called radiation
This condition is that the potential function represent diverging
waves at a large distance from the singularity.
3.
UPPER FLUID OF FINITE DEPTH.
SINGULARITY;
SUBMERGED LINE
We first consider a line singularity placed at the point (O,-N) in the upper fluid
or depth
Now
@i
iF"
@2
log
o
13
A, B, C,
2.j
R
fh’
gj
0
Rj
{x 2 + (y +
log R as R
)2}1/2
0.
can be represented as
+
Rj
c. log
where d
@2
Then
and
@i
fj
@2
and
’h’.
oo dj
+
x
2
R’.j
log
log
+ (y +
gi’ cj,
dj
+
(3.I)
cos kx de
A(k)e
-0
J
j0
2jh-
)2
R’. +
[B(k) cosh k(h+y)+C(k)sinh k(h+y)]coskxdk (3.2)
and
2j
R’o
x
2
+ (y + 2jh +
are to be found from the conditions
(2.3) and also the condition that the integrals are
)2,
=0,_+1,+ 2
(2.1), (2.2), and
to be convergent.
The radiation
[3], page 34,
will be needed
condition will be dealt with in the sequel.
The following integral representation used also in
740
R. N. CHAKRABARTI AND B. MANDAL
--
in our calculations
Y
Y
-
e
log
Rj
-f
-k(y + 2jh- n)
ek(Y
+
i
a’o=
J
y >-2jh
+ N
) cos kx
dk,
y < -2jh
+
0
+ 2jh + N) cos kx dk,
e-k(y
log
2jh
cos kx dk,
e
y > -(2jh + )
k(y + 2jh + )
cos kx dk,
y < -(2jh
+ N)
so that,
Y
N)cos
kx dk,
G
log
Rjl y=-h
e
e
)
i h
k(2j
-k(2j
1 h
+
cos kx dk
)cos
kx dk,
j
1,2,3
cos kx dk,
j
0,-1,-2,-3
j
1,2,3,
cos kx dk,
j
0,-1,-2,-3
cos kx dk,
j
0,1,2,
cos kx dk,
j
-1,-2,-3,...
i h
+
+
i h
0
Y
Y
i h-
e-k(2j
J y=-h
e
+ rl)
1 h
k(2j
0
e
log
Rjl
-k(2jh- N) cos kx
dk,
0
y=0
fo
e
k(2jh- )
0
e-k(2jh
log
+ N)
0
R’.
J y=O
I
e
k(2jh
+ )
2
+
0
Condition (2.1) gives
K
2
cj log{x +
+ E d.
0
I
+ B
0
J
io g
i h- N)
(2j
x2
cos kx dk
+ (2j
i h
+
IF" c_j. log{x
N)2j1/2+
F. d
2
io g
+ (2j +
x2
+
(2j
N)2} 1/2
N)2j1/2
i
] + [e-kcj
i
0
(2j "I
h-)kx
cos
dk- Y.
0
-k(2j +I
h+rl)c
e
cos
kx dk
SINGULARITIES IN A TWO-FLUID MEDIUM
+E d
+
e
I h +
-k(2j
1
J
10
k c cos kx dk
cos kx dk- E d
0 -J
0
e
741
-k(2j
+
n) cos
kx dk
i h
(3.3)
0
from which we obtain
Cj+l +
d
O,
j
0,1,2
+
c
O,
j
0,1,2,...
(3.4)
dj+l
Since d
-j
i, we obtain
O
c
(3.5)
-i
I
Condition (2.3) gives
log{x2 + (2jh
fj
+If
1
rl)cos
e
J
q)2}1/2
+
cj
2
Z d j+l log{x + (2jh + N)
2
c. e
+ k(B
-k(2jh
1 h-
+
ri)+
1
+
sinh kh
c cosh kh)
J
e
gJ
-k(2jh
I
+
0(B
cosh kh
+
-k(2jh + rl)
+
d. e
]
A cos kx dk
0
+ rl) cos kx
0 cJ +llg{x2+(2jh-r) 2}1/2+
log{x2+(2jh-rl)2} 1/2-
0
2
Y.
0
0
SI
+
+
kx dk
q)2}1/2
log{x2 + (jh +
gJ
0
d.
J
I
kA cos kx dk
dk-
log{x2+(2jh+n)2}1/2
c sinh kh)cos kx d
+S
o y.
t])+
2
d. e
-k(2j
i
+
i h
cj e
(1 +
dl)e
-kn
(3.6)
cos kx dk.
from which we obtain by considering the coefficients of the different logarithmic terms
Cj+l)
fj,
j
1,2,3,...
dj+I)
S(d0- dl)
gj,
j
1,2,3
S(cj
S(dj
(3.7)
go’
and then (3.6) reduces to
(k
K)A + S(K cosh kh + k sinh kh)B + S(K sinh kh + k cosh kh)C
-2S[d I e
-kQ
+
1
e-2kjh(cj+1 e-kn + dj+I
e
-kr_)]
(3.8)
Condition (2.2) now gives
Y. (i- S)(c.
J
ekN
i
+
d
I(1
+ d.
J
e-kN
+ S)e -kN
e-2kjh
+ Z (i +
i
Cl(l
S)e -kB
S)(cj+1
e krl
+
-2kjh
dj+I e-kD)e
-k(A + B sinh kh + C cosh kh)o
Now for convergence of the integrals in the expressions for
I
and
2’
(3.9)
G(k) must be
742
R. N. CHAKRABARTI AND B. MANDAL
O, where G(k)
zero for k
Y. (i
1
+
S)(cj
dj)
is the expression in the left side of
+ Y(l
S)(cj+1
1
+
dj+I)
c
(3.9),
so that
S) + d I(I + S)
I(I
0
This is satisfied by choosing
(i
S)cj
+ (I +
S)cj+I
0,
j
1,2,...,,
(i
S)dj
+ (i +
S)dj+1
0,
j
1,2
and
From (3.4), c
-d
o
(i- S)c
S
1
where
I
(i + S)d
I
so that we obtain
I + S
c.
(-i)
d.
(-i)
I
j
j-i
j
1,2,...
j
J
j
1 2
(3.10)
(3.8) can be written as
(k- K)A- s(K cosh kh + k sinh kh)B + s(K sinh kh + k cosh kh)C
2S
-kN
e
-2kh
1
(ekN
+
De-kN
+ e -2kh
1
(3.11)
From (3.3), we obtain
+ kc
KB
2e-Pd(e kN
l+e
+ De -kN)
0
(3.12)
-
(3.13)
-2kh
From (3.9), we obtain
A + B s inh kh + c co sh kh
0
Solving for A,B,C from (3.11), (3.12) and (3.13), we obtain
2e-kh(e kU+ De-kN)sinh
A
m(l + De
i
B
[2e
-kh(e k
1
+
1
k---K +
C
le
-2kh
+ De
-2kh
(S
kh
sinh kh-cosh kh)
+(
-kN)
k
i
k
K
+ (S -l)sinh
)
kJ]
F
1
(3.15)
I) sinh kh |
J FI
A(k)
(3.14)
(3.16)
where
2SDe -k
F
and
2e-kh(’ekN +
1
e-kN)[SDe-kJ
+ e -2kh
1
(i
A
A(k)
2S)
{k(l- S)
+ 1K {(K-k+Sk)sinh kh +
sinh kh
cosh kh
SK}sinh kh- K cosh kh
SK cosh
kh}]
(3.17)
(3.18)
SINGULARITIES IN A TWO-FLUID MEDIUM
Now, A(k) has
k
poles k
I
> O, say on the real axis of k (there are also
0
n + iBn
kn
and complex poles
A,B,C have simple poles k
k
I
n
kn
where
kn-(n-l)
are purely imaginary.
Hence
In the line
K on the positive real axis of k.
and k
0
0,
where when S
The zeroes of the denominator of F
(cf. [2])).
as n
k
one simple pole at k
743
we make indentation below these poles which account for the be-
integrals from 0 to
haviors of the potential functions at infinity particularly as
Ix
oo.
This will be
evident later.
Substituting the above results, we have
i
S
Z
[(-I)JP j-I
K
+
1
k_
(-I)J[P
+ S
+ D j+l] log R’o
]
0
I
+
log R
e-kh(e k + e -kN
-2kh
2__
0
(-l)JP j]
+
k
+
l)sinh kh(
(S
-ky
cos kx dk
pe
sinh kh-cosh kh)F
(
K
s inh kh e
-ky
e
1
cos kx dk
cosh kh)F
sinh kh
0
2
(-l)Jp j-I
+
log R. +
(-l)Jp j
log
0
f
+?0
1
k
(-l)Jp j+l
R’ +
-J
I
log R
0
+
i
k
+ pe -kn)
pe
-2kh
k
where M
+ y)]F 1
(3.19)
R’.
+ y)
cos kx dk
cos kx dk
cosh k(h+y)]F
I
cos kx dk
(3.20)
it is possible to obtain
K
i+
2S
91
1
2
log R
s
s
+
log R0 +
S
0
2S
i+
e
S
-k (y+n)
+ i1 + SS log R0
cos hx dk
M
k
2S
1
+
e
S
0
k (Y-n)
k
M
cos kx dk
which are the results derived by Gorgui and Kassem
vestigate the behavior of the integral for large
2 cos kx
Then
cos kx dk
cosh k(h
0
Now, as h
-ky
log
i
cosh k(h
l)sinh k(h+y)
(S
(-l)Jp
+
-]
e-kh(e k4
2
[sinh k(h + y)
K
+
0
e
1
3
iklxl
xl,
we put
+e -iklxl
[3].
Now to in-
R. N. CHAKRABARTI AND B. MANDAL
744
k
1
k
-’K
sinh kh- cosh kh)F
xl
Ii eikl
dk
e
1
-ky
+
cos kx dk
I
0
e-
I
(3.21)
dk, say
Whe re,
I
i
(
1
k
FI
cosh kh)
sinh kh
e
-ky
(3.22)
For the first integral of (3.21), we consider in the complex k-plane a contour in the
first quadrant bounded by the real axis of large length X with an indentation below
I
K, an arc F of radius X1 with center at the origin and the line joining
<
Then for considering the behavior
where 0 <
the origin, with the point X e
1
the pole k
as
xl
at k
oo,
we only need to consider the behavior of the term arising from the residue
K, because the integral along the arc becomes exponentially small as X I
the integral along the line 0 to X
e
is
(0 <
1
which becomes exponentially small for large
I
1
eiklxl
<
xl.
will have a factor e
Hence making X
dk--2?ri Residue of I
e
I
iklxl
I
Ixl
sin
we find that as
1
at k
-X
and
K
For the second integral of (3.21), we consider in the complex k-plane a contour in the
fourth quadrant bounded by the real axis from 0 to X
K, an arc F’ or radius X I with center
where 0 < a <
origin with the point X 1 e
pole k
1
at the
with an indentation below the
origin and the line joining the
Since now the singularities on the
real axis are taken to be outside the contour and following a similar argument as
above, we obtain that as
Ixl
I
1
e
-iklxl
0
dk
Again,
0
A(k) (S
l)sinh kh(
k
sinh kh
cosh kh)F
12 e iklxl
1
dk
e
-ky
+
where
i
12 (S
1)sinh kh(
k
sinh kh
cosh kh)
cosh kx dk
12 e -iklxl
F1
e
-ky
dk
(3.23)
(3.24)
For the first integral of (3.24), we choose a similar contour as was chosen for the
SINGULARITIES IN A TWO-FLUID MEDIUM
intetral with I
745
that the indentation is now below k
I excepting
k
instead of K.
0
contribution from the poles of X(k) which lie inside the contour has a factor e
where
+ iB n
n
neglect it.
A(k) in the first quadrant so that for large
is a zero of
The line may cross some singularities of A(k).
Ixl
The
n
we may
To avoid this, if it
crosses a zero of A(k), we indent the line about it so that it lies outside the region
bounded by these contours and the contribution for this indentation will also contain
xl’
which becomes exponentially small for large
a factor e
12 e iklxl
2i Residue of
dk
a
Ix
Hence, we find that as
singularity of this type.
m + ibm being
12
e
ik x
at k
k
0
By a somewhat similar argument as was used in the second integral in (3.21), we obtain
12 e -iklxl
0
xl
oo,
cosh kh)
s(sinh Kh
(l-2s)sinh Kh-cosh Kh
e-K
Hence, we find that as
2i
+ 2i(s
l)sinh k
e
o
o h(--sinh
Ix
as
tends to
i
1
k
0
(eK+e -K) e-Kh+sinh
-2Kh
e-KYeiklxl
+ e
e
cosh k h)
k h
o
o
k
-k h
_ko
e
o
(e
+
+
e
o
-2k h
1
se
-k
o
(K-k 0 +sk 0 )sinh k0 h + sK cosh k
o
+e
e
where
D
2s)sinh ko h
[(i
xl-
Similarly as
oo,
2
cosh k
,-v-----v--7-. 7-.------
h][h{k o(I
----.-,
I-K
+2i(s-
l)sinh
se
o
koh[sinh ko(h+y)+ 1
sK}cosh k h + (i
o
e-Kh(eK+e )-KDe -Kh
|De
zs)sxnn
i
s)
s
hK)sinh k o h]
tends to
y)-cosh
h+y)
sinh K(hK(+
2is
o
(k-k
o
+
ik+
o
cosh
sk )snh k h
o
o
ko(h+y)][se
+ sK
cosh k h
o
snn n. cosn n
(e
e
o
leiKlx
+e
_2ko h
e
where
D
Thus
[(1
@i
and
2s)snh ko h
2
cosh k
o
h][h{ko(1
s)
sK}cosh k h + (1
o
satisfy the radiation condition at infinity.
(3.25) and (3.26) take respectively the following forms
s
hK)s[nh k o h]
Now as h tends to
infinit,
746
R. N. CHAKRABARTI AND B. MANDAL
-k (y+)
s
2i
o
e
ik
o
xl
ik
o
Ixl
e
and
where now k
I
o
s
k (Y-D)
s
-2i
o
e
_-r--.
l+s
e
M, and these agree with results of Gorgui and Kassem [3].
K
WAVE SOURCE SUBMERGED IN LOWER FLUID.
4.
In this case, there is a logarithmic type singularity at the point (O,N).
Now
01
02
and
01
0
can be represented as
cj
+
log R
J
log
j
0
+
Rj
5’
log
d
Rj’
log
iZ fj
+
A(k)e
-ky
(4.1)
cos kx dk
0
+
-17" gj
log
5’
+
JO
[B(k)cosh k(h,+ y) + C(k)sinh k(h + y)]cos kx dk
where C
i.
o
(4.2)
Condition (2.1) gives
f
o
+ g_
i
0
fl
0
(4.3)
+ g-j
fj+l
g’+13 +
f
-3
0,
j
1,2
0,
j
1,2
and
-k(2j
KB + Z f. e
1 J
i h- N)
Z f
0
-J
e
-k(2j + i + N)
+ Z g_
e
-k(2j
i h
+ N)
1
7.
i
g_j
e
-k(2j
+
i h- n)
+kC=0
(4.4)
Condition (2.3) gives
d
o
Sf
o
-i
cj
S(fj
fj+l)
j
1,2
dj
S(gj
gj+l )’
j
1,2
(4.5)
and
(k
K)A + S(K cosh kh + k sinh kh)B + S(K sinh kh + k cosh kh)C
2Sf
Condition
o
e -kn
(2.2) gives
2e-kN
-k(2jh- N)
e
2S I f
1 j+l
2S
I
I
gj+l
e
-k(2jh
+ n)
(4.6)
SINGULARITIES IN A TWO-FLUID MEDIUM
k(A + B sinh kh + C cosh kh)
+
S I
i
I
+
-k(2jh
S
IF" (fj
fj+l)e
.
+ n)
f
I
+ N) -Z
i gj
-k(2jh
e
gj+l
1
gj +i )e
(gj
-kh
-e
e
e
J
-k(2jh
747
-k(2jh- r)
-k(2jh- N)
+ N)
f
+
j+l
i
e
f
+ (Sf
o
i)
e-k
e
-k(2jh- N)
(4 7)
Now for convergence at k-- 0, we obtain
F.[(S-
I
(S +
l)fj
+ F.[(S-
l)fj+I]
(S +
l)gj
i
l)gj+l]
+ (S + l)f o
0
2
(4.8)
This is satisfied by choosing
From (4.5), d
(S-
l)fj
(S +
l)fj+I
0,
j
1,2,
(s-
1)gj
(s +
1)gj+I
0,
j
1,2
-
o
f
o
(4.9)
2
I+S
S
I+S
i
where
From (4.3), (4.5) and (4.9), we obtain
gj
2
I+S
2
f-j
1
C.
0
d
(_l)j j
S
i_
(-I)
$2
1,2,
j
1,2
j
1,2
j
1,2,
j
1,2
j
1,2
(4.10)
(-1) jj+l
4S
J
j
j j
(4.4) can be written as
KB + kC
2
i
+
S
e
-k(h + )
2e-2kh
+
i
+e
-2kh
[
e
-k(h + N)
i
e
+ S
N)
k(h
i
S
]
J
0
(4.11)
(4.6) can be written as
(k
K)A + S(K cosh kh + k sinh kh)B + S(K sinh kh + k cosh kh)C
-2e
-kN
4S
2 -k(2h
e
(i- S)(I +
+ B)
e
-2kh
(4.12)
(4.7) gives
A + B sinh kh + C cosh kh
0.
(4.13)
748
R. N. CHAKRABARTI AND B. MANDAL
Solving for A,B,C from (4.11), (4.12) and (4.13), we obtain
(
A
+ N)
-2kh
+ 2e -2kh
l+e
1+S
=- [k_
2e
K
(S
1
C=
]
i) sinh kh
A(
k- K
e
K
k(h
n)
e
]
-k(h + )
S
i
+
i
S
(4.14)
[
I
-k(h+n)
kh] GI+K(I +
(S- l)sinh
k
sinh kh
J GI
k- K
2e-k(h
B
l)sinh kh ]
A(k)
(S
kh)
s inh kh- co sh
2De-2kh
e
+
S)
K.l+e-2kh)
k(h-)
e
-k(h+)
I + S
S
i
G
]
(4.16)
whe re,
G
-2e
I
2
1
e
2S
+ I+S
-k (h+N)
e
+e
1
2 e
/
-2kh
-2kh
l+e-2kh
I
i
[(K-
-2kh
e
k (h-n)
e
+ Sk)sinh kh +
k
SK cosh
kh]
-k (h+D)
1- S
/(l-2S)sinh kh-cosh kh
+ S
1
(4.17)
(k) being given previously.
Thus using the above results, we obtain
log R
o
+
0
S+ S+I,i log Ro +
i
+
0
k
1
k- K
#Ak
2e-k(h+rl)
sinh kh
J0
2
+ s
2
I + S log
+
0
[sinh
[sinh
Ro
+
2
1
+
k(h + y)
k(h + y)
e
k
2
(-l)J j
1
e
1
-2kh
+ 2e -2k
-
(-l)J j+l
k
e
e
I+S
+
J
cos kx dk
k(h-n)
e
+
y)]
+
Rj
I
e
-ky
cos kx dk
e-kYcos
I+S
2
(-l)JJ
S
i
GI cos
y)]
G
1
-k (h+n)
i- S
log
cosh k(h
log R
cosh kh)G
l+e
S I
-ky
sinh kh
cosh k(h
I- S
Now as h
S
sinh kh- cosh kh)G
l)sinh kh(
(S
4S
i_
kx dk
cos kx dk
+
I
log
i
0
A(k)
R.’
(S
l)sinh kh
2e
+
+ y)
cosh k(h
i
o
S
1
log R
I+S
o’
-2kh
K(l+e
2
I+S
oo
0
e
-k(y + )
k-M
-2kh)
(4.19)
cos kx dk
we obtain
log R
(4.18)
kx dk
cos kx dk
SINGULARITIES IN A TWO-FLUID MEDIUM
@2
2
2
+ I+S
log R
I+S
e
749
n)
k(y
cos kx dk
k-M
which are the results derived by Gorgui and Kassem
[3].
Proceeding similarly as in the previous case we see that the above potentials
Ix
have the following forms as
e-K(h+)
+
1
e-K(l
cosh Kh)
2i(sinh Kh
+
S
e-2Kh
l+e-2kq
(
k
2i(l-S)sinh k h(
o
o
(e
)
] e-KYe
o
I+S
S
1
sinh k h-cosh k h)
o
o
+ )
)l]e-koy eikolx,I/
S)
ko(l
o
o
[
e
I
2
+
1
+
-K(h + N)
e
S
-2e -K
-2Kh
o
ri(S-1)sinh ko h{sinh ko (h+y)i
( ek(h
1
where D
k
o
)
S
+ Sk o )sinh ko h +
ek(h
+ )
I + S
[(I-2S)sinh k o h
Now, as h tends
to infinity,
o
eikIxll/
cosh k h][h
o
l+e-2ko h
-
(4.20)
o
+
+ (i-S-hK)sinh k o h]
S(sinh Kh + cosh Kh)
+ q)
S
e-2koh
(h+y)} -2e -krl(l+ 2S
1+S
2
o
e
1
D
e-K(h
! i-
SK cosh k h}
)]
+
1
cosh k
--if-
+
l+e-2koh
l+e-2Kh
S
1
-2k o h
e -2koh
e
N)
/ eK(h
+ 2e -2Kh
l+e
{(K-
2S
+ I+S
i
e
2S
I+S
SK cosh k h
and
gi{sinh K(h + y)
cosh K(h + y)}
2S)sinh Kh- cosh Kh
(i
cosh Kh)
/(2S-1)sinh Kh + cosh Kh
(I +
I+S
cosh k h][h
2S)sinh k h
[ -k o
o
I+S
[(i
+ S(sinh Kh +
-2Kh
eK(h-)
e-koh(h
I-S
where D
l+e
o
o
k(h
+ I+S
-ko(h
l+le-2koh
2e -2kh
+ )
l+e _2ko
h
(4.21)
D
ko(l-S)- SK
cosh k h
o
+ (i
S
hK)sinh k hi.
o
(4.20) and (4.21) take respectively the following forms:
-M(y
2i
e
I+S
+ N)
e
iMlx
and
2i
5.
SUBMERGED POINT SINGULARITIES
P (cos 8)
n
as R
Here,
o
,n+l
2
S
M(y
e
I+S
N) iM x
e
UPPER FLUID OF FINITE DEPTH.
r
2
+ (y + ) 2 1/2
0,
n
0,1,2
n
where r
R. N. CHAKRABARTI AND B. MANDAL
750
I
A(k) e
y- ).
We assume
(5.1)
J (kr) dk
o
0
P (cos e)
r
n
2
r
tan-l(
e
is the distance from the y axis, and
J
+
R ,n+l
o
{B(k)cosh k(h + y) + C(k)slnh k(h + y)}J o (kr)dk.
0
(5.2)
We use the following integral representations
P
n
Ro
cos,
i___
n+l
n
i
e- k (y + )
n
k
J (kr)dk,
o
0
(5.3)
(-i)
I
n
n’
n
k
e
0
k(y + ) J
(kr)dk,
o
From conditions (2.1), (2.2) and (2.3), we obtain
t._l)n+l kn( k
n!
KB + kc
)
K)_k( h
+
n
k
A + B sinh kh + C cosh kh
(5 4)
e -kB
(5.5)
and
(k- K)A+s(K cosh kh+k sinh kh)B +s(K sinh kh+k cosh kh)C
.
S
n
k
e-k (k-K)
(5.6)
Solving for A, B, C we obtain
n
n
-k + (-i) (k + K) e -k(hK
k
A
W
h IK
)sinh
k
W
+
k
I
+- (S
(-i)
B
n
kn(k +
e-k(h
K)
n’. K
)
k
E
1
g
+(
sinh kh
l)sinh kh(
+
cosh kh)
sinh kh
(S-l
cosh
kh)
(5.7)
kh-]
inh
w1
K
where
n
n- [(S-l)(k-K)e-k
k
Wl
(-1)
n
+--- {(K-k+Sk)sinh
2S)
.
and A is the same expression used in
0
+
+
and,
+
W1
k
k
K
(S
+
SK cosh
3,
(5.8)
so that we obtain
e-kYJ o (kr)
(k + K)sinh kh e
cosh kh) e
sinh kh
l)sinh kh(
kh}(k + K)e -k (h-) ]
sinh kh- cosh kh
(i
i
kh
k
sinh kh
-ky
dk
J (kr) dk
o
cosh kh) e
-ky
Jo(kr)
dk
(5.9)
SINGULARITIES IN A TWO-FLUID MEDIUM
PR
n
2
(c os
O
+
18 i
n+l
(-i)
+
,n+-
0
sinh k(h
0k-K
kn(k
n!
+ y)
to infinity,
and
i
2
+ Y
+ y)
cosh k(h
+ y)J O (kr)dk
J (kr)dk
o
cosh k(h
+ y
J (kr)dk
O
(5.1o)
take respectively the following forms:
o.0
2SM
n’.(l + S)
)
e-k(h
cosh k(h
inh k(h
l)sinh kh
Now, as h tends
+ K)
K
75i
kn
-k(y + n)
e
Jo(kr)dk
(5.11)
and
Ro
l+S
S
M
where
K i
ting
2J (kr)
O
ving
Ho(1)(kr)
H
+
,n+l
O
+ (i + S)(k
which are the
(i)(kr)
result,s
+ HO (2)(kr)
(5.12)
Jo(kr)dk
e
M)
[3].
lerived by Gorgui and Issem
Now put-
and rotating the contour in the integral invol-
Ho(2)(kr)
in the first quadrant and in the integrals involving
in the
fourth quadrant, we can reduce the integrals into suitable forms from which it is seen
oo,
that as r
2hiS
n
(-i,)
n!
+{i
+
i and 2 respectively take the
eK(q h- y) H (1)(kr)
Kn+l. (2S l)slnh Kh +o cosh Kh
ko
n
(’-.l)n
K
where D
k
(S
n-q--
l)sinh
koh(
o
following forms:
-k q
sinh
koh
koh)S
cosh
l)(k
-k (h-rl
o
I
-kYH
{(K-ko +Sko )sinh ko h + SK
cosh k
[(l-2S)sinh kO h
h][h{k O (i-S)-SK}cosh k O h + (i
cosh k
O
o
h}(ko+K)e
o
K) e
e
(1)
o
(kor)
/D (5.13)
hK)sinh ko h]
S
and
2i
(-l).n
n+l
K
n!
k
+ {i
(i
K(h+y)]e-K(h-)Ho(-l)(kr)
k
(S
o
l)sinh kO h[sinh ko(h + y)
n
+ (-I)
K’ {(K-kO +SkO )sinh kOh +
where D
cosh
2S)sinh Kh- cosh Kh
n
o
-
[sinh Kh + cosh Kh][sinh K(h+y)
SK cosh k
cosh k
[(l-2S)sinh k O h
O
O
cosh k (h
O
h}](ko+K)e-k(h
h][h{k o(l-S)
l)(k
K)e -ko
)HO (1) (kor) }/D
(5.14)
+ y)]
[(S
SK}cosh k h + (i
O
S
hK)sinh k h]o
O
Now as h tends to infinity, (5.13) and (5.14) take respectively the following forms:
2SK
(S
i)
(M
K)2
e
-M(y + n)
HO
(i)
(Mr)
752
R. N. CHAKRABARTI AND B. MANDAL
n
M
i
6.
S)(M- K)
(I
2SK n’
MULTIPOLE SUBMERGED IN LOWER FLUID.
P (cos e)
2
n
as R
Here, qb
[r
n+l
1
O
R
2
D)H (1)(Mr)
M(y
e
O
N)2]1/2
+ (y
0, n
1,2
where
0
e
tan
-i
r
y_
We as sume
R(cs
@)
+
n+l
0
I
92
0
i
A(k)e
-ky
J (kr) dk
(6.1)
o
n
[B(k)cosh k(h + y) + C(k)sinh k(h + y)]J (kr)dk.
o
(6.2)
We use
P (cos )
i
n
n--
R
n+l
r
k
n
e
-k(y
)
J (kr)dk,
y >
J (kr)dk,
y <
o
0
O
oo
.(_!)n
n
k
n J0
e
k(y
o
Proceeding much as in the previous case, we obtain
P (cos )
n
91_
R
+
n+l
oo
v
+
k
(S
l)sinh kh(
’+
[sinh k(h + y)
v (S
-ky
J (kr)dk
-
cosh kh)e
sinh kh
and
K
o
O
0
92
e -kn J (kr)dk
cosh kh)e
sinh kh
K
k
n
k
n
0
0
+
(-i )n
k
cosh k(h
l)sinh kh[sinh k(h- y)
whe re,
+
k
-kyJ
o
(kr)dk
y)]Jo(kr)dk
cosh k(h + y)
]Jo (kr)dk
kne -kN
2 (-I)
cosh kh]
2S)sinh kh
nK
V
n![(l
(6.4)
(6.5)
(6.6)
and A(k) is the same expression used previously.
Now,
as h tends to infinity,
P (cos )
n
R
n+l
O
and,
+--(-i)
n’.
n
91
and
oo
0
2
{i +
take respectively the following forms:
2M
(i + S)(k
M)
}kne-k (y+) Jo(kr)dk
(6.7)
SINGULARITIES IN A TWO-FLUID MEDIUM
2 (-I) nM
n’.(l + S)
where M
K
I+S
S
i
ar_e
which
I
0
kn
M
k
e
k (y-N)
753
(6.8)
Jo(kr)dk
the results derived by Gorgui and Kassem [3].
Proceeding much as ir the previous case, we find that as r
ively take the following forms:
n+l
-K(y+)
cosh Kh)e
K
(sinh
2i
2 i
n’ (S
+
[(l-2S)sinh
and
2
respect-
2S)sinh Kh- cosh Kh
(i
(-l)n
i
Ho(1)(kr)
(-I)
n!
oo,
l)Kk o
n
k
o
sinh k h( --sinh k h
o
o
K
eko(Y+N) Ho (i) (kor)
koh+(l -S-hK)sinh koh]
cosh k h)
o
koh-Cosh koh][h{ko(l- S)- SK}cosh
(6.9)
and
2i
(-l)n
n!
+{2i
n+l
K
(-l)n
where D
n’
sinh
(i
K(y + y)
cosh K(h + y)
2S)sinh Kh
cosh Kh
H
(i)
o
(Kr)
k
(S-I)Kk o nsinh k h[sinh k (h+y)o
o
[(l-2S)sinh k h-cosh k
o
o
o
(i)
-cosh ko(h+y)]e -ko H o (kor)}/D
h][h{ko(l-S )- SK}cosh
(6.10)
k h +(l-S-hK)sinh k h]
o
o
Now, as h- ,(6.9) and (6.10) take respectively the following forms:
n
iKM
(-l)n
Sn
and
n
iKMn
7.
(-i)
S(I
S)n’.
CONCLUS ION.
M
-
i)
M
eM(Y
l)e
+ ) H (i)
(Mr)
o
M(y- n)
No
(i)
(Mr).
Integral representation of the potential function in a two layered fluid medium
where the upper layer is of finite depth with a free surface and lower layer is of infinite depth have been obtained.
The different results reduce to the known results of
Gorgui and Kassem [3] when the FS in the upper layer is taken to infinity (i.e., h-).
We note that these authors thought there was no difficulty except longer equations and
a bulkier result in adding the surface tension term in their problem, closely related
to ours.
ACKNOWLEDGEIENT.
We take this opportunity of thanking the editor for his valuable
suggestions in the improved version of the paper.
754
R. N. CHAKRABARTI AND B. MANDAL
REFERENCES
Multipole expansion in the theory of surface waves, Proc. Cambridge
Phil. Soc. 49 (1953), 707-716.
I.
THORNE, R.C.
2.
RHODES-ROBINSON, P.F. Fundamental singularities in the theory of water waves with
surface tension, Bull. Australian Math. Soc. 2 (1970), 317-333.
3.
GORGUI, M.A. and KASSEM, S.E. Basic singularities in the theory of internal waves,
0uart. Jour. Mech. and Appl. Math. 3__i (1978), 31.
4.
MANDAL, B.N.
5.
CHAKRABARTI, R.N. Singularities in a two-fluid medium with surface tension,
(to appear) Bull. Cal. Math. Soc. (1982).
Singularities in a two-fluid medium with surface tension at their
surface of separation, Jour. Tech. 26 (1981).