Interlat. J. Math. & Math. Sci.
Vol. 9 No.
(1986) 175-184
175
TWO-DIMENSIONAL SOURCE POTENTIALS IN A TWO-FLUID
MEDIUM FOR THE MODIFIED HELMHOLTZ’S EQUATION
B. N. MANDAL and R. N. CHAKRABARTI
Department of Applied Mathematics
University College of Science
92, A. P. C. Road
Calcutta- 700 009, India
(Received March I0, 1985)
ABSTRACT.
Velocity potentials describing the irrotational infinitesimal motion of two
superposed inviscid and incompressible fluids under gravity with a horizontal plane of
surface
mean
of
either
of
separation,
are
whose
fluids,
the
derived due
strength,
sinusiodal[y along its length.
of
extension
the
a
to
besides
vertical
being
line
harmonic
source
in
present
time,
in
varies
The technique of deriving the potentials here is an
technique used for the
of only time harmonic vertical
case
line
The present case is concerned with the two-dimensional modified Helmholtz’s
source.
equation while the previous is concerned with the two-dimenslonal Laplace’s equation.
KEY
AND
WORDS
PHRASES.
Helmholtz’s
Modified
two-fluid
equation,
medium,
source
potentials, surface of separation.
1980 AMS SUBJECT CLASSIFICATION CODES:
[.
76B15, 31A05, 35A45.
INTRODUCTION.
Velocity potentials due to the presence of different types of singularities in an
incompressible and inviscid one-fluid medium,
amplitudes,
assuming irrotational motion of small
play an important role in dealing with problems involving radiation or
scattering of surface waves by obstacles present in the medium.
These problems can be
reduced to equivalent problems of solving some singular integral equations of second
by a suitable use of Green’s integral theorem in the fluid medium
kind in general,
the
with
Thorne
help
[I] gave
of
these
medium and Rhodes-Roblnson
the
free
obtained
either of
singular
a survey of
surface
(FS).
potentials
due
the fluids.
potentials
[2] modified
Gorgul
to
(generally
called
Green’s
function).
the fundamental singularities submerged in an one-fluid
and
it to include the effect of surface tension at
[3]
Kassem
oscillating
line
considered
and
point
a
two-fluid
singularities
medium and
submerged
in
The upper fluid of the two-fluid medium considered in [3] is
extended infinitely upwards and the lower fluid is of either infinite or finite depth
below the mean surface of separation (SS).
Later the model is modified to include a
number of generalizations, e.g. presence of interracial surface tension in the SS (cf.
Rhodes-Robinson [4], Mandal [5]) upper fluid of finite depth with a free surface with
or without surface tension (cf. Chakrabarti and Mandal
[6], Chakrabarti [7], Kassem [8]).
In problems dea[i,g with the scattering of obliquely incident surface waves in an
one-fl,lid
medium by
horizontal
plane barriers (cf.
Heins
[9], Green and Helns [I0]
B.N. MANDAL and R. N. CHAKRABARTI
176
etc.) or vertical plane barriers (el. Mandal and Goswaml
[13]), half-
[II], [12]
immersed or fully submerged infinitely long circular cylinder (cf. Mandal and Goswami
[14], Levine [15], by exploiting the geometry of the obstacles, the velocity potential
can be as:;umed to have a harmonic variation in the lateral (z) direction, same as the
Thus the potential function satisfies a two-dimensional reduced
incident wave field.
Helmholtz’s
problems are essentially boundary value problems
the
Hence
equation.
(BVP) involving the He[mholtz’s equation, and the construction of a two-dimenslonal
source potential (the Green’s function) is necessary to reduce the BVP’s to equivalent
Both for infinite and finite constant depth of fluid, this source
integral ,equations.
(in x)(cf. Heins
potential can be constructed by the method of Fourier transform
[9], Levlne [15], Miles [16] etc.) or by the method of separation of variables (cf.
Rhodes-Roblnson [17] where the effect of surface tension of FS is included), thereby
obtaining a linear combination of potentials due to the source in an unbounded fluid
together with an ’image’ potential in the FS boundary condition.
the
In
paper
present
due
potential
a
to
we
two-fluid
a
consider
source
llne
vertical
medium
and
either
of
in
present
derive
the
velocity
whose
fluids
strength varies harmonically with time and also with the co-ordlnate measured along
This is the same as deriving the source potentials in a two-fluld medium
its length.
for the reduced two dimensional Helmholtz’s equation.
The corresponding problem for
the two-dimensional Laplace’s equation was considered in
[4].
When the strength of
the llne source is made independent of the co-ordlnate along its length, known results
When the density of the upper fluid is made
for a two-fluid medium are recovered.
zero, the results derived here reduce to corresponding known results for an on-fluld
medium.
2.
STATEMENT OF THE PROBLEM.
We
consider
two-fluid
a
The mean SS is
inviscid.
medium,
both
the
fluids
taken as the
horizontal and
being
incompressible
xz-plane
and
y-axis pointing
A line source is assumed to be present in either of the fluids
and the y-axis is chosen to pass through the singular point so that the point of
(O,q) or (0,-) ( > 0). The strength of the
singularity is situated either at
vertically downwards.
llne source is
assumed
be the densities of the lower and upper fluids respectively so that
The motion
is
assumed
described,
by
velocity
the
fluid
irrotational
be
’s
and
is
of
small
Pl
>
P2-
amplitude,
Re {.(x,y,z) exp(-it} (j-l,2),
and
can
be
where
is
3
in
Laplace’s
equation
three-dimensional
the
satisfy
potentials
frequency.
circular
respective
to
Let Pl, P2
z.
to vary sinusoidally with time as well as with
regions except at the point of singularity where it exists.
The
linearized SS conditions are
K
>I
+-y
(K +
3--),
on
y
3y
s
y
0,
y= 0,,
2
where
K
depth
’h’
/g,
g
being the gravity and
below the mean SS, then
s
P2/Pl < I.
If the lower fluid is of
TWO-DIMENSIONAL SOURCE POTENTIALS IN A TWO-FLUID MEDIUM
0
y
h,
y
on
177
otherw[se,
iI
#21
grad
Also
grad
Further,
I,
z
0
as
y
=.
0
as
y
=.
.
satisfy the radiation condition that both represent outgoing waves
in the far fiels as
Ixl
Assuming the z-variation of the strength of the line source as
possible to extract the z-variation completely from the functions
exp(iz),
.(x,y,z).
it is
Thus we
can write
(x,y,z)
j
where now
j
(x,y) exp (iz)
1,2
j
satisfy the two-dimensional modified Helmholtz’s equation
j’s
2
2
(V
0
j
DI,D 2
except at a point of singularity, where
2
V
pied by lower and upper fluids and
(2.1)
in D.
3
denote respectively the regions occu-
is the two-dimensional Laplacian operator.
Near a point of singularity the potential behave as K
-
lar solution of Helmholtz’s equation,
R
second kind and
Kgl +
(z)
K
being the distance from the poln.t.
(K2 + ,-17-)
s
y
y
y
0
2
(UR)
which is a typical singu-
being the modified Bessel function of
The boundary conditions are
Y
0;
(2.2)
y
0;
(2.3)
y
h
(2.4a)
when the lower fluid is of finite depth, otherwise,
IVII
(2.4b)
0 as y
when the lower fluid is of infinite depth; also
IVY21
0 as y
+-=;
(2.5)
and finally,
’I’ 2
Thus
satisfy a boundary value problem (BVP) described by (2.1) to (2.6).
I, 92
satisfy the radiation condition in the far field as
section 3 we will decompose this BVP into two
In
BVP’s by defining two sets of component
potentials where the first set accounts for the singularity in the medium but die out
in the far field while the second set is non-singular but accounts for the radiation
condition
to these
in the far field as
Ix
=.
BVP’s assuming the lower fluid
In sections 4 and 5 we will obtain solutions
to be of infinite and finite depth respect-
ively, thereby deriving the source potentials in the two fluids completely.
B.N. MANDAL and R. N. CHAKRABARTI
178
3.
DECOMPOSITION INTO TWO BOUNDARY VALUE PROBLEMS.
We define potentials
@j
where
(x,y), A_. (x,y)(j--1.2)
9j
D.
j-- 1,2
j
satisfy
@j
2
in
+
j
such that
a
except at
2
(v
-v
point
of
(3.2)
--0
9j
and near a singularity the appropriate
singularity,
conditions are
Wj
Ko(R)
91
s
3y
as R
92,
y--O,
(3.4)
0,
(3.5)
Y
3y
(3.3)
0.
2
91’ 2
{n
DI, D2
Thus
respectively.
(hereinafter Pl).
Then
(V2- 2) Xj
g
(I +-y (I
k2
i)X
---y,
}---
y
+
in
hl)
(3.6)
1,2),
(j
Dj
{Kx 2
s
(3.7)
+-y (W 2 + X2)}
y
(3.8)
0;
(3.9)
0
I
8X’I
y
2)I/2
satisfy the BVP described by (3.2) to (3.6)
9Z
@I,
+ y
satisfy the BVP (hereinafter P2) described by
Xl, A2
0
(x
0 as
y
3y
(3.10a)
h
if there is a bottom to the lower fluid, otherwise,
IYXII
0
as
y
IVX21
0
as
y
and finally,
Xl, X2
=,
(3.10b)
(3.11)
",
satisfy the radiation condition in the far field as
In the conditions (3.8) and (3.10a),
and
91
2
are assumed to be known (solution of
pl).
4.
LOR FLUID OF
(i)
INF[NIT
DEPTH.
Wave Source in the Lower Fluid.
described by (2.1) to (2.6) where
@i
In this case we seek a solution to the BVP
K0(r)
{x2 +
as r
(y-B)2}I/2/
0.
Thus in PI
the precise form of (3.2) and the condition (3.3) are
2
(V
2
(V
-
2
91
0, y
>
0
2
0, y
<
0,91
2
except at (O,Q),
K0(r)
as r
0
(4.1)
(4.2)
TWO-DIMENSIONAL SOURCE POTENTIALS IN A TWO-FLUID MEDIUM
Let
K0(vr) +
i
z
{x
r*
where
2) }I/2
+ (y +
Clearly
unknom constants.
conditions (4.2) and (3.6).
K0(r*) @2
cl
c2
179
K0(r)
is the distance from the image point and
cl, c 2
are
as given above satisy the equations (4.1) and the
@I, 2
We choose
cl and c2
such that the conditions (3.4) and
(3.5) are satisfied.
The following integral representations will be needed in our calculation
f
Y
K0(vr)
Y
K0(r*)
-I
k
x
cos
exp {+ k(y-n)} dk,
x
cos
{ k(y-n)} dk,
exp
and the upper (lower) sign is for
(k
where
f
$
-1
k
y >(<)n.
Thus
K0(r)
Y
K0(r,
=;f
k4
f
k
Conditions
cos
x
exp(-kh) dk,
y= 0
K0(r)
Y
-1
K0 (r,)
-I cos
4 exp
{-k(hSn)}
dk.
y= h
(3.4) and
(3.5) give after making use of appropriate integral representa-
tions given above
+ Cl
s c2,
l-cl
from which we obtain
c
(l-s) (l+s) -I
1
K0(r)
c2
2(l+s) -I
Hence
I--S
K0(r*)
(4.3)
2
2
K0(r)
Again, let
K2
/
A 4 -I cos 4x exp(-ky)dk, y
f
B
-I
exp(ky)
cos
dk, y
>
(4.4)
0,
<
0,
where A,B are unknown functions of
(3.10b) and (3.11).
the
radiation
sequel.
k.
Clearly I, 2 given above satisfy (3.7),
The contour in the integrals is to be chosen in such a way that
condition
is
satisfied
automatically.
The conditions (3.8) and (3.9) lead to
(K-k) A- s(K+k) B
A+B
2(1-s)(l+s)
O.
-I
exp(-k0),
This
will
be
shown
in
the
180
B.N. MANDAL and R. N. CHAKRABARTI
Thus
A, B
2(l+s)
+/-
-I
-!
k(k-M)
exp (-kq)
whe re
(l+s) (l-s)
M
-I K.
(4.5)
-
Hnce
I-s
2
ls K0(vr*) + ls
Pl
K0(r)
2
T K(r)
2
k
2
k---
k
cosx
exp
where the contour is indented below the pole at
(i
To
infinity.
tion at
[xl)+
(4.6)
(k(y-q)
(4.7)
M
k
formed into a line from
(where
X
to
X
just above the cut from
v
<
in the integrals by exp
exp (i
k
number) on the real
M, the quarter of a circle of radius
iX
0
to
and a line from
.
z)}, y <
In this case there will be a contribution from the pole at k
{- My + i (M cos x + M sin
from the
contribution
integral involving
circular arc will
exp
M sin
0 then
(-i[x[)
As X
M.
X
in the fourth quadrant, and the
M
k
lles outside the
As X
closed contour, there will be no contribution to the integral from this.
cancel
ix
the
Comb-ining
out.
(which
representations
for
@I, 2
K0(r)
2
+
I-s
+
f
above and below the cut
0 to
we
integrals
two
account
the
for
condition
in
the
the
to
will
alternative
far
field
as
as
l-s
--s
[i M N
-I
K0(r*)
ix[}
exp {-M (y+) + i N
2)I/2 .!}
2
k {k
k
from
finally obtain
will
radiation
-
The contribution
the contribution from the circular arc will be exponentially small.
from the real line from
0 to
on the real line with an indentation below the
In this case as the point
-iX to 0.
the
(R),
Similarly in the
be exponentially small.
M, the quarter of a circle of radius
imaginary axis from
exp
However see section 6).
the contour is deformed into a llne from
to X
below the cut, a line from
point k
inc
I
(In fact if we assume an incident wave field represented by
M.
0 to
It is being assumed
in the complex k-plane.
to
is de-
is a large positive
k
in the first quadrant, the imaginary axis from
tat
to ensure the radiation condi-
The contour in the integral involving
axis with an indentation below the pole at
X
dk
2 cos Cx
this, we replace
establish
Ixl).
ex p (-i
cosCx exp {-k(y+)} dk
cosk(y+)
M sin
(k 2
+
0
eXp {-(k +
k(7+)}
M2)(k
2
+
2/2
dk]
(4.8)
and
2
2
[i M N
-I
exp {M (y-) + i N
Ixl}
2
+
/
0
where
N
(M
k{k cosk (y-n)
M sin k(y-n)} exp {-k + n
2
2
k
+
M2)(k
+ 2/2
2)1/2 ix I}
dkl
(4.9)
(4.o)
TWO-DIIENSIONAL SOURCE POTENTIALS IN A TWO-FLUID MEDIUM
In this case
Wave Source in the Upper Fluid.
(ii)
2
K0(r*)
r*
as
0
18!
so that
K0(r*)
2
O.
r*
as
By writing
WI
and
K0(r*),
cl
AI, A2
2
c2
K0(r) + K0(r*)
the same integrals as in (4.4) (with different A and B) we will.slmilarly
obtain
41
2s
l--s [K0(r*)
42
K0(r*)
k
(k-) cosx
l-s
K0(r)
+s
+ 2s
l--s
exp {-k(y+rl)} dk]
k
’(k-’M)’ cosx
where the contour is indented below the pole at
tion at infinity.
k
Alternative representation for
(4.)
exp {k(y-)} de
M
91,
(4.12)
to ensure the radiation condi-
2
can be obtained following
the same method mentioned above as
5.
i
2s
l--s [K0(r*)
2
2s
l-s
K0(r*) + l--s K0(r) + l-s [the
the terms in the square bracket in (4.8)],
terms in the square bracket in
(4.9)].
LOWER FLUID OF FINITE DEPTH.
Wave Source in the Lower Fluid.
(i)
Section 4(i), while
xz
XI,
In this case
@l, 2
are the same as in
satisfy P2 with the condition (3.10a) in place of (3.10b).
Le t
cos
2
XI, 2
/
-I
C
x
cos
{A cosh k(h-y) + B sinh ky} dk, 0
x
exp (ky) dk, y
<
<
y
<
h,
O.
given above obviously satisfy (3.7) and (3.11).
The
SS
conditions
(3.8),
(3.9) and the bottom condition (3.10a) yield the following three equations for the
derivation of A, B, C.
A(Kcosh kh
ksinh kh)
+ kB-s(K+k) C
A sinh kh
B + C
B cosh kh
exp {-k(h-)} -(l-s)(l+s)
2(l-s)(l+s)
-1
k exp(-k),
0,
-I
exp {k(y+)}.
Solving for A, B, C we obtain
41
+
K0(r)
l-s
ls K0(r*)
2
l--s j [----exp(-kh) {s(K+k)-k}
(sinh
kq
+ s cosh Kq) sech kh
(l-s) exp(-kq) cosh k(h-y)
+ exp(-kh)(sinh
kq+s cosh kq) sech kh sinh
ky]
cosx
dk
(5.2)
B.N. MANDAL and R. N. CHAKRABARTI
182
2
-i- K0(r)
2
+
2
f
l--s
[(slnh ko + s cosh k0) sech kh exp {-k(h-y)}
{exp (-kh) {s(K+k)-k} (slnh
A
+ s cosh kq) sech kh
kq
cosx
(l-s)k exp(-kr)} sinh kh exp (ky)]
K cosh kh + {s(K+k)-k} sinh kh,
A(k)
where
(5.3)
dk,
(5.4)
k=k
and the contour in the integrals is indented below the pole at
g(k),
only real positive zero of
which is the
to ensure the satisfaction of the radiation cond-
tion at infinity.
ni
Note that poles do not occur at kh
the
a(k).
of
zeros
When
other is negative.
has
real zeros,
two
3
The only poles occur at
i,
one
are complex in general.
be.come purely imaginary (see
zero).
+/-
k
positive,
is
say, and the
0, magnitudes of these real zeros become the same.
s
a(k)
of
zeros
remaining
a(k)
+/----,
Rhodes-Robinson [18]
The
O, the complex zeros
When s
with surface tension put equal to
To ensure the radiation condition in the far field as
,
Ixl
the same
steps of section 4 (i) can be followed in the deformation of the contours tn the first
and fourth quadrants with the modification that the contours are indented below the
pole at
the large radius of the circular arc in the first and
and
k=k0,
quadrants is chosen in such a way that no complex zero of
&(k)
The far
is crossed.
field behaviour will come only from the contribution to the integrals at
fourth
(when
k=k
the contour is deformed in the first quadrant), other contributions from the imaginary
axis, from the poles at complex zeros lying in the first and fourth quadrants will die
out in the far field as
Ixl
above and below the cut from
=.
(h-y)
cosh k
The contribution from the real line from
v to v
k
NO
-I exp (i N
WI+ D1
sinh
Z
-I exp (i N
exp (k0h) N0
k0h
will cancel out.
Thus as
0 to
Ixl
Ixl),
(5.5)
where
DI
Nu
(kO
D1
l--s
Ixl),
(5.6)
and
2
ni
k 0} (sinh k0q + s cosh k0q) sech
[exp(-k0h) {s(k0+ K)
sinh
(l-s) k
A’(k0)
where
(ii)
d
d---
A (k)
exp(-k0)]
k0h
k0h
(5.7)
g,(k0)
k=k
Wave Source Submerged in the Upper Fluid.
same as given in Section 4 (il) while
i,
X2
In this case
I, 2
are the
may be assumed to have the same type of
representation given in (5.1) (with different A, B, C).
Then A, B, C satisfy
TWO-DIMENSIONAL SOURCE POTENTIALS IN A TWO-FLUID MEDIUM
A sinh kh- B + C
O,
2s (l+s)
B cosh kh
183
-I
exp {-k(h+n)}
Thus we will finally obtain
exp(-kh)sech kh
{’ (1-s)k+ {sK-(l+s)k}
8
2s
Ko(r*) + -1-s
2s
*
exp(-kn)cosh k(h-y) + exp {-k(h+n)} sech kh sinh ky}
K0(r*) +
2
!
As
l-s
1--s K0( r+2s f
cosx
(5.8)
dk,
[exp (-kh) sech kh
{(l_s)k+ {sK-(l+s)k} ep(-kh)sech kh}
sinh kh exp k
(y-n)]
cosx
(5.9)
dk
Ixl
cosh k 0(h-y)
D2
91
k0h
sinh
N01
exp (i
NO
Ixl),
(5.10)
(k0Y)
exp
42
-D2
D2
2s (l+s)
-I
N
exp (i N
Ixl),
where
-I
i
k0h
sinh
A’(k
and
6.
N
is given by
+ (sK-(l+s)k 0} exp(-k0h) sech k0h]
[(l-s) k
(5.1)
exp(-k0n)
(5.6).
CONCLUSION.
We have derived in the present paper source potentials for the two-dimenslonal
modified
Helmholtz’s
equation
two-fluid
a
in
medium.
The
parameter
in
the
Helmholtz’s equation has been assumed to be less than the wave parameter
infinite depth of
However, if
u
the lower fluid)
or
k
(for
M
(for finite depth of the lower fluid).
is greater than the wave parameter then the potentials will no longer
represent outgoing waves in the far field, rather they will die out in the far field
(see the corresponding one-fluid case with surface tension in the
Making
([16] with
potentials
recovered.
results.
s
0
surface
due
in
the above results,
tension
put
equal
to only time-harmonic
to
FS
[16]).
in
source potentials in an one-fluid medium
zero) can be recovered.
line source in a two-fluid medium
One can also include the effect of surface tension of the
SS
v
O,
[3] can be
Making
in these
B.N. MANDAL and R. N. CHAKRABARTI
184
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