145
Internat. i. ath. & Math. ci.
(1986)
Vol. ’. No.
BASIC SINGULARITIES IN THE THEORY OF INTERNAL
WAVES WITH SURFACE TENSION
M. A. GORGUI and M. S. FALTAS
l)epartment of Mathematics
Fculty of Science
Moha -rem Bay
Alexandria, EGYPT
(leceived September 6, 1984)
T],
ABSTRACT.
study f linearzed interface wave problems for two superposed fluids
often lnw) Iv,s the co)siderti,)n of dil-ferent types of singularities in one of the two
paper the line and poLnt singularities are investigated for the case
when each fluid is of finit constant depth. The effect of surface tension at the
fluids.
In
[his
surface of separation is i)luded.
KEY
W(W{DS AN{
PHRASES.
Innnal waves, surface tension, basic singularities.
1980 MS SUJ/’CT CLASSIFICACfON CODE. 76 B15, 76CI0.
1.
INTRODUCI’ION.
The study of internal waves in two fluids problems has attracted many authors in
It is found useful to permit singularities of one type or another to
occur as an idealisation of, or an approximation to certain physical situations. Prorecent years.
blems dealing with the generation of waves at the interface of two non-mixing fluids
involve the (’onsideration of singularities of different types in the fluids.
In the
case when bodies are present, waves may either be generated by the movement of the body
or reflected from it.
The two cases are essentially the same and the resulting motion
can be described by the use of these singularities in a suitable way.
For example,
Gorgui [1] has investigated into these waves using a distribution of sources on the
surface of the body.
The different types of singularities that can be used in two fluids problems have
been presented by Gougui and Kassem [21 and Kassem [3]
in both the effects of surface
the authors considered the cases when the lower fluid
is of finite constant depth and the upper fluid is of infinite height. While in [3]
tension are neglected.
In [12]
the author considered the cases when the two superposed fluids are both of finite
thickness and obtained the potentials for motions resulting from mu|tipoles submerged
in one of the two fluids.
In this paper we give a complete survey for the basic line and point singularities
when the superposed fluids are as in [3] of finite constant thickness confined between
two rigid horizontal planes but we here take surface tension into consideration.
M.A. C)RGUI and M. S. FALTAS
146
In tle two
and
dimensiona|
motion, the line singularities considered are wave
multiloles singulariti,s.
sourcL, s
Restriction is made to symmetric (or vertical) multi-
pol,s, bu the corresponding antis>mmetric (or horizontnl) multipoles can be found
For axisymm.tric motion, the point singularities considered are multipole
These tme hnrmoni singularities are described by harmonic ptential
similarly.
sin.ul,lril ies.
functins which satisfy tw
inearised conditions at the surface of separation, and
uniquenes. is ensured bv rcq,iring that there are only outgoing waves in the far field.
ThL, methoI used is an extension of that used in [2] or 131
The results obtained by
Rhles-RI,inson
2.
STA[’E,IENT OF
[4], Gru
Ka.sem [21 and Kassem [31 can be deduced as special
,nd
THE PROBLEM.
Wc, a,e concerned wth tl,e irrotational, incompressible and inviscid motin of the
two,
supcrlosed non-mix,.,
liquids
under the action of gravity and surface tension.
Each
Takin the ori.;{n 0 at the mean level of the
intc, rf,ce and the axis Ov pointing vertically downwards into the lower fluid, let the
fl[d is f infinite horizanal extent.
two,
planes y
fluid, be confined betwe,n rigid horizontal
The motion is
it is due to an os-
o
In two dmensional motion we consider
the, two fluids.
cillatin singularity in one of
-h’
h, y
simple ha-monic with a small amplitude and angular frequency
the singularity is eithc, r a line wave source or multipole and in axisymmetric motion
In each case, the velocity potentials of the lower and upper
it is a p,int multipole.
2
luids are simple harmonic with period
plex valued potentials
arc,
e
i o t
’
the real parts.
e
and it is more convenient to use the corn-
-i o t
of which the actual velocity potentials
These potentials satisfy a boundary value problem in which
-
0, V
V
’
(2.1)
0
in the reqions occupied by the fluids, except at the singularity;
D--
0
on
y
h,
(2.2)
- -’- -
0
on
y
-h’,
(2.3)
-Ty
0
Dy
by
and the linearized boundary conditions
DY
8Y
K +
K
where
(s
I)
o
g
M
T
on y
s(K’ +
y
+ M
g is the gravity,
T
is the surface tension, 0
are the densities of the lower and upper fluids respectively.
are applied for each singularity considered.
limiting conditions that
or
’
(2.4)
0
and
so
These conditions
They are supplemented by the two general
behaves like a typical singular harmonic function
near the singularity and the radiation condition that both functions represent outgoing
waves in the far field.
3.
SUBMIIRGED LINE SIN(’ULARIT[ES.
Without loss of generality, the line singularity is place at the point (o, _+
We consider only singularities symmetric in x
namely, a wave source and vertical
147
BASIC SINGUI.ARIT[ES IN THE THEORY OF INTERNAL WAVES
(R, 0)
We define polar coordinates
multipoles.
in the xy
plane based at the singu-
larity positi()n by the equations
y +_
R sin 0
x
R cos 0
according as the singularity is in the ;ower or upper fluid, so that
R
denotes the
distance from the singularity.
Wave source singularities
(i)
’
and
lere
arc solutions of the boundary -value problem stated above
having a logarithmic singularity at the source.
with
If the source is at
(o, )
then
l,g R
]1/2-
+ (y- n)
[x
R
as
(3.1)
Let
log R + c log
’
’=
[ + (y + )2
,’
obvious that
B(k), A’(k)
on
y
+ y) + B’
sinh
ky} cos kx]dk,
o, -)
is the distance from the image point
We choose
as given above are harmonic.
B’(k)
c
’
f(k)
It is
A(k)
such that the two integrals converge and the boundary conditions
-h’
h, y
y) + B sinh ky} cos kx]dk,
cosh k(h
log R + f If(k) + {A’ cosh k(h’
o
R’
where
R’ + f’[sf(k)
+ {A
o
and
y
are satisfied.
o
Under suitable conditions concerning differentiation under the integral sign and
using the relat-ions
ay (l(g R)
the conditions (2.2)
e
B
-t
cos 0
R
of
e
_foo
o
e
-k(y- n) cos
kx dk
k(y
y
n) cos kx
dk, y
n
(2.3) are satisfied if
-k(h
+ e -k(h + n)
n)
’
B’
k cosh kh
e
-k(h’ + n)
k cosh kh’
and the interface conditions are sa[isfied if
I+(-
c
A sinh kh + A’ sinh kh’
cA cosh kh
-A’ [sc
+
’
e
k
cosh kh’
s(
O,
+ (B- B’)
k(l + B k 2) sinh kh’]
(i
+ B
-kn
k 2)(’ e
k B’)
from which we have
AA
AA’
(I + B k 2) e -k
’(l +
sinh
kh’ sech kh +
k 2) e -k sinh kh tanh kh’
+
[s__
cosh kh’
F(
(I + B k 2) sinh kh’] )--cosh kh’
F(k)
c
k
where
F(k)
( +
I) e -kn cosh kh
’
O.
k(k)
e
-k(h
n)
e
-k(h + n)
-k(h’ + n) cosh
kh sech kh’
c(cosh kh sinh kh’ + s cosh kh’ sinh kh)
K
l-s’
k(1 + B k 2) sinh kh sinh kh’ ,(3.2)
M
l-s
M.A. GORGUI and M. S. FALTAS
148
F(o) =-2, therefore, the integrals involved in the assumed expressions
2c + 0(k 2) in the
A
A’ converge if we choose f(k) such that k A f(k)
dF(o)
o and if
neighbourhood of k
d-- o,
Since
for
h(l + a) + h’ a’
iee.,
o
Hence
t
B
-2
--]
CL
O
-2 cosh k(h- )
F(k)
-.
e
-kh
k
sinh k
cosh kh
sc
2[(I + 8 k 2) sinh kh’
A
A’
-2c
cosh k(h
kA
f(k)
The condition imposed on
B’
o
cosh k(h- )
cosh kh’]
cosh kh
)
does not specify it completely.
This introduces
no difficulty since it is the velocities in which we are really interested.
f(k)
found convenient to take
A(k)
Now,
k
has a simple zero at
duces simple poles for the integrals in
,’
m, say, on the real axis of
tion of the contours of the integrations.
A
A’
B’
B
It is
2c
k
this intro-
Below this pole we make an indenta-
Substituting in the above assumed forms for
we get
+
log
+ [{k(l + 8 k2)sinh kh’
2
cosh k(h
) cosh k(h- y)
k cosh kh
’
I-A-[I
2c
e
sc cosh kh’}
-kh
sinh k
sinh ky] cos kx]dk,
k cosh kh
n) cosh k(h’ + y) cos kx] dk.
cosh k(h
Two other alternative forms which will be useful in the subsequent work are
log R
I) log R’ + 2
+ (2s
-k(y + )
6cosh k(h
2
0’
2 log R
cosh K(h’
e
fo
6 -I
-kh
6
(I
2
) co__sh
+ S k2 )slnh2kh
6
kh____’cosh
cosh kh
ksinhcoshkkhsinh ky
+ 8 k2
+ y)cos kx dk + 2
cosh k(h
where
6
(I
y) cos kx dk + 2s f
cosh k(h
e
o
k(h
e
)
cosh k(k
-kh’
+
y)] cos kx
dk
(3.4)
cos kx dk
sinh kh sinh kh
) x
cosh k(h
-kh’ + [e-k(n
6
f=o [el
y)
n) cosh k(h’ + y)] cos kx ]dk
(3.5)
+ s cosh kh’ sinh kh
In the above expressions we neglected the constants
cosh kh sinh kh’
6
2s log h’ + 2s
2 log h’
+
2
(I
6
(1
+
+ 8 k2 )sinh kh
k2 )sinh
sinh kh dk
kh sinh kh’ dk in
(3.6)
and
,’
respectively.
BASIC SINGU;ARITIES IN THE THEORY OF INTERNAL WAVES
By putting 2cos kx
0’
integrals in
+
e
k
these integral tend to
m
cosh m(h
y)
m sinh mh
C(o’h h’)
.ic m
,ana rotating the contours in the indented
e
into contours in the first and fourth quadrants so that we must
include the residue term at
C(n;h,h’)
149
e
imlx
cosh m(h’ + y)
e
m sinh mh’
C(o;h,h’)
imlx
[e -m(h- )
n
n
+ (-I) n em(h- ) ]sinh mh’ sinh mh
mP)sinh2mh sinh2mh + c(h sinh2mh + h’
(I + 3
(3.7)
s
sinh2mh)
When the source is at (o, -) in the upper field, we have
’
Assume the forms
log R
+
log R
’
+ (y +
[x
R
as
)2]1/2
[s f(k) + {A cosh k(h
fo
log R + a’ log
R’ +
f
(3.8)
o
y) + B sinh ky} cos kx] dk,
If(k) + {A’ cosh k(h’ + y) + B’
sinh
ky}"
cos kx] dk,
where
R’
as above to
Ks the distance from the image point
-
(o, n)
and proceed
et the potentials
2cs
0’
)2] L
+ (y
[x
cosi k(h’
-[i
tog
n) cosh k(h
+ [(k(1 + B
)sinh kh
k
+ y) + e -kh’ ksinh
cosh kh’
cosh k(h’
y) cos kx] dk
c cosh
kh)
sinh ky3 cos kx
k A cosh kh’
Jdk
and the other two alternatives forms are
2s log R
(1 + B k2)sinh kh sinh kh’ cosh k(h’
2s
y)cos kxdk + 2s
cosh k(h
+ [e -k(y + )
’
tog RR’ + 2s
cosh k(h’
cosh k(h’
o
e
k2)sinh2kh
cosh kh
n)S----
e
-kh’
+
)cosh k(h- y)] cos kx
+ y)cos kx dk + 2 f
cosh k(h’
+ 2 f
o
(I + 8
(6
cosh k(h’
[(
e
cosh k(h’
-kh’ snh kn sinh ky
k cosh kh’
-kh’
+ y)]
cos kx dk
n)
dk
(3.9)
n)cosh k(h’ + y)
+ [e -k(n-y)
cos k
dk +
(3.10)
These potentials have the outgoing waves
-C(o;h’,h)
C(o’h’,h)
where C(n’h,h’)
Ixl
(ii) Multipoles singularities
as
Here
singularity.
’
s cosh m(h
m sinh mh
s cosh m(h’
m sinh mh’
is given by
Y)
+ y)
e
e
i
mlx]
(3.7).
are harmonic in the regions occupied by the two fluids
except at the
In the neighbourhood of this point
M.A. GORGUI and M. S. FALTAS
150
cos(n + 1)0
0
o, 1, 2
n
Rn+
(3.11)
We consider first the case when the singularity is at
(o, n)
in the iower fluid.
try as solutions
cos(n + I)._
+
0
f=o
o
Rn+
,
fo
o
A’
+ y)
cosh k(h’
y) + B sinh ky] cos kx dk,
[A cosh k(h
cos kx dk,
and use the representations
n+
(-1)
f
k
n
e
n)cos
k(y
kx dk
y
n
cos(n + 1) 0
Rn+
!
Conditions (2.2)
fo
k
n
e
n)
-k(y
cos kx dk,
y
(2.4) are satisfied if
kne-k(h-
B
o)
cosh kh
n
n+
A’
A sinh kh +
-cA cosh kh
+
_co__s(n + I)0
n +
R
-
?;
O
k
where
P(n)
(-I)
n
n
sinh kh’
sc cosh
These determine A, B,
0
+
.
A’,
A
xl
k
n
e
(-I)
k2)sinh kh’ ]A’
n+l
n!
ck
n
e-kO
which when substituted in the above assumed forms, give
f
o
n
k
e
-k(h
O) sinh
k
cosh kh
cos kx dk
n
+o-k P(n)
[e -k(h- n) + (-1 )n +
+
l[c(s
cosh kh’
,
cos kx dk
e
k(h
sinh kh’)
(3.12)
rl)
cosh k(h’
+ y)
cos kx dk,
(3.13)
k(l + 8k2)sinh kh’] e"kO
n)
-k(h
e
k(l + gk2)sinh kh’]
cosh kh
is given by equation
As
+(-I) n!
B
k(l + B
kh’
+ [cs cosh kh’
and
n
(3.14)
(3.2).
we have the outgoing waves
(n + I) C(n +
h, h’)
cosh m(h
m sinh mh
y)
cosh
0 ~-(n+l) C(n+l; h h’) m sinhm(h’+y)
mh’
where C(n; h, h’) is given by equation (3.7).
e
imlx
eimlxl
If the singularity is at (o, -D) in the upper fluid we try as solutions
where
m
0
f A cosh k(h-y) cos kx dk
,=
cos (n+l) 0
[x 2
+ (y+
o
Rn-I
+
q)2].
fo
[A’ cosh k(h’+y)+B’ sinh ky] cos kx dk;
Proceeding as above leads to
sc
kn
7 --[e
k(h’
D) + (-i) n+l
e-k(h’
)]
cos
k(h-y)cos kx dk
(3.15)
BASIC SINGULARITIES IN THE THEORY OF INTERNAI WAVES
o
(-1) n+l
cos(n+l)e
Rn+l
n
+
e-k( h’-
kn
151
)
sinh ky cos kx dk
cosh kh’
Q(n) cosh k(h’+y) cos kx dk,
(3.16)
where
[c(cosh kh
Q(n)
+
(-l)n+l[c
s sinh kh)
cosh kh
k2)sinh kh] e -kn
-k(h’
)
h2)sinh kh] ecosh kh’
k(l+ B
k(l+ 8
(3.17)
The contours of the integrals in (3.15), (3.16) are indented below the simple pole
at k
m to give the outgoing waves
cosh m(h-y) im
e
n sinh mh
cosh m(h’ +y)
-s(n+l) C(n+l; h’, h)
n sinh mh’
s(n+l) C(n+l), h’, h)
@
Ixl
as
eim[x[
+.
SUBMERGED POINT SINGULARITIES.
4.
Ix
We now define cylindrical polar coordinates (r,
,
y) with the origin 0 at the
surface separating the two fluids and the y-axls pointing vertically downwards.
@iso
(o,
We
define spherical polar coordinates (R, 0, ) based at the singularity taken at
+/-
) by the equations
r
R sin 0
R cos
y
.
We consider only point singularities for which Oy is an axis of symmetry so that the
,’
velocity potentials
are independent of
When the singularity is at (o, ) in the lower fluid the boundary value problem
for
,’
is given by (2.1)
en (cs
)
Rn+l
as R
(2.4) supplemented by the limiting condition
[r 2 + (y-n) 2]1/2
o, n=o ,1,2
If we try as solutions
Pn(COS
O)
Rn+l
’
A’
+ f [A cosh k(h-y) + B sinh ky]
o
cosh k(h’+y)
Jo
Jo(kX)dk,
(kr) dk,
and using the representations
(-1)
n!
n
o
knek(Y-n)
P n (cos O)
Rn+l
conditions
I ’I of
e-k(Y-n)J (kr)dk,
y > n,
(2.2), (2.4) are satisfied if
B=
kne-k
y-n
n!cosh kh
A sinh kh + A’ sinh kh’
-c
kn
B +
A cosh kh + [sc cosh kh’
(-I)
n!
k
n
e -kn
k(l + Bk2)sinh kh’]A’
(-l)nn.
Solving these equations and substituting, we obtain the expressions
(4.1)
152
M.A. GORGUI and M. S. FALTAS
(cos 0
+
n+l
where A
-
k
k
n
e
-k(h-)
sinh ky
s’
J (kr) dk,
k(h-y Jo(kr)dk,
is given by
(3.2), P(n)
(1)(kr)
2 Jo(kr)
Ho
(4 3)
(3.14) and as before the contour of inte-
is given by
k
m
of
A
that the radiation conditions are satisfied.
en=ureq
(4.2)
[e -h(h-) +(-) nek(h-n)] cosh k(h’+y)J (kr)dk
gration is indented below the simple root
which
kh
n
--P(n-l)cosh
!
kn
’:,:- o
f
n
o
on the positive real k-axis,
For, by putting
(2)(kr),
+ Ho
rotating the contours in each integral into contours in the first and fourth quadrants
(where
H(1)(kr),, Ho(2)(kr)
k
we obtain the diverging waves
m
C(n; h
are respectively small) and including the residue term at
cosh m(h-y__)
sinh mh
h’)
Ho(1)(mr)
m(h’+y) H!I) (mr),
m’-
cosh
-C(n; h, h
Sinh
where C(n; h, h’ is given by equation (3.7)
In a similar manner we calculate the velocity Dotential when the multiDole singularity is at (o, -) in the upper liquid. These are
as
r
.sc
,
n
[ek(h
P (cos 8)
n
R
n+l
+
where
-
fk
R
[r
2
nl-.T
+
(-I)
)
n
--F-.
n
k
o
+
(-l)ne-k(h’
fOOo kne-kh
osh kh’
)]cosh k(h-y)Jo(kr)dk
(4.4)
n
.inb ky
Jo (kr)dk
n
--Q(n-])
(V+)] 1/2
and
+ y)rosb k(’ + y)J:(kr)dko
coh k(h’
O(n}
is
iven by (3.17).
(4.5)
This motion has the diverging
cylindrical waves
C(-; h’
as
r+.
’
C(n; h’
rosb
h) _s
m(h_-,__)_
(1)
Ho
inh mh
h)
+ y)
s cosh m(h’
s inh mh
(mr),
HI)
(mr)
5.
SUBMERGED SINGULARITIES; THE LOWER FLUID IS OF FINITE CONSTANT DEPTH AND THE UPPER
IS UNBOUNDED.
A statement of the boundary-value problems for the velocity potentials
the different types of singularities can be easily written down.
,’
for
These of the present
case are similar to the corresponding ones treated in sections 3 and 4 with condition
(2.3) replaced by
V’+
o as
y
and the radiation condition taking the simplet forms
C cosh m(h
my
-C
’~
singularities, and
e
as
Ixl
for line
e
im
im
y) e
Ix[
the forms
Ixl
BASIC SINGUIARITIES IN THE THEORY OF INTERNAI WAVES
(I)
(mr),
y) H
C cosh m(h
’
my
H
A
simple root of the equation
A
0
The determination of
4#’
4#,
(mr),
is a constant multiplier and
where now
(5.1)
for each singularity can be carried out independently.
This was done by the second author [5], where he assumed
_
in the above sections.
(a)
4#,
h’
They may also be determined by letting
priate forms.
is a
m
k(l + Bk 2) sinh kh.
(cosh kh + s sinh kh)
c
(I)
C
for point singularities, where
r
as
-C e
153
’
to have the approin the formulae obtained
The velocity potentials for the different cases are as follows:
Line singularities
(o, n)
For a wave source at
(i)
log R +(2s
R’ + 2
i) log
(I + Bk 2) cosh k(h- n) cosh k(h- y)
A
o
2. e -k(y + n)
-Is
k
+ f
cos kx dk
=
in the lower fluid,
o
s6 cosh k(h
e
Y)I
n)cosh k(h-
2
where now
o to
=,
2
7o
+
A
o
e-kn
6-I
(I +B k 2
(5.2)
cos kx dk,
cosh kh
log R
-kh sinh kn sinh ky
cosh kh
)
sinh kh cosh k(h
ekYcos
kx dk
6cosh k(h- q)] e ky cos kx dk,
cosh kh
+
(5.3)
The path of integration is along
s sinh kh
k
indented below the simple pole at
Im(k)
o
m.
These potentials have the outgoing waves
m sinh mh
Ixl
as
eimlxl,
y)
cosh m(h
-C(o)
emy
C(o)
m
eimlxl,
where
ic
C(n)
m
[e
n
-m(h
hc
n)
n)
+ (-i) n em(h
+ (i + 3Bm z) sinh 2 mh
(5.4)
sinh mh
The above velocity potentials can be written in slightly different forms suitable
for use in the next section.
log R-
l-s
R’ +
-+ log
These are
(i +
2
8k2)cosh k(h- )cosh k(h- y)cos
kx dk
-kh
+ ]-s fo --k- Is2- 6(sinh
log R
* Is
6
o
2
6(sinh kn
+
2s
1o R
[e -ky
2s
o
+
s cosh
kn)(sinh ky + s cosh ky) cos kx]dk (5.5)
(|+Bk2) sinh kh cosh
s cosh
When the wave source is at
4#
kq
kn)e
(0, -)
e
ky
k(h-)ekYcos
kx dk
2
+ ls
e
-kh
k
cos kx]dk.
(5.6)
in the upper fluid,
-kn (1+Bk2)sinh
cosh k(h-y) ]cos kx dk,
kh cosh k(h-y)cos kx dk
+ 2s f
o
e
-kh
k
(5.7)
M.A. GORGUI and M. S. FALTAS
154
R’ +
log R
A e
-k(n -y)
(l+Bk 2)
2s
sinh 2 kh e
-k(n -y)
;
+ 2s
cos kx de
sinh kh m
(5.8)
cos kx dk.
or as in the previous case,
2s
4
I---
h,g R
Is
6e-k(sinh
’
2s
e
L
kv +
6ek’Y-n’cos
/
2s
o
7+
__
x
k
o
(5.9)
(l+Bk2) ek(y-n) sinh
A
-kh
e
ky) cos kx] dk,
s cosh
1-s
+ 7+7
log R’ + 2s
l,g R
a (l
-kn (l+Bk2)sinh kh cosh k(h-y)cos kx dk
-kh
kh cos kx dk
2s
e
+ 7+7
f
o -k
(5.10)
kx) dk
These potentials have the outgoing waves
m(h-y)
-C’(o) _cosh
sinh
m
m
Ix
as
my
-m-- e imlx
e
C (o)
where
-2
C’ (n)
(ii)
e imlx
sc m
n
e
n!
-m sinh2mh
(5.1)
+ (l+3Bm 2)sinh2mh
hc
Mul tipoles singularities
(0,n)
When the singularity is at
_cos(n+l)R
+
n+l
Bk2)]
[K + k(l +
,
i
foo
o
n
n
e-k(h-)
sinh ky_
sinh kh
k
cos kx dk
n
A--
n.
[(-I)
n+l a
-k (h-n)
e-kn
n.o -[ e-k(h-rl)
k
c
k
in the lower fluid,
[sc_k(l+Bk2)
+ (-1) n+l
e
k(h-.)
e__]dosh
kh
jekYcos
cosh k(h-y) cos kx dk,
(5.12)
(5.13)
kx dk,
which have the outgoing waves
cosh re(h-y)
(n+l) C(n+l)
as
e
m sinh mh
Ix
im
Ixl
(n+l) C(n+l)
e
my
m
e
im
Ix
C(n) is given by (5.4), and when the singularity is at (o,
where
in the upper fluid,
2
,=
k
sc
n
cos(n+l)0
Rn+l
e
ky
e
+
-
-k cosh k(h-y) cos kx dk
_. o
k
cos kx dk,
and have the outgoing waves
cosh m(h-y)
(n+l) C (n+l)
as
(b)
,
Ixl
m s inh mh
where
C’(n)
n
e
(5 14)
k(l+Bk 2) sinh kh]
-kn[c(cosh kh- s sinh kh)
(5.15)
eimlxl
(n+l) C (n+1)
is given by
emY e imlxl
m
(5.11).
Foint singularities
For a singularity in the lower fluid,
P (cos
n
Rn+l
+
fo kn
e
-k(h
n) sinh ky
cosh kh
Jo(kr)
dk
o
-
k
n
[(-l)n[K
+ k(l+Bk2)]
BASIC SIN(,ULARITIES IN THE THEORY OF INTERNAL WAVES
.
e
k(l+Bk 2)]
[sc
k
o
c
n
-[e
155
-k(h-)
osh
-k(h-n)
cosh k(h-y)
kh
+ (-I) n
ek(h-)
(5.16)
Jo(kr) dk
ekY
(5.17)
Jo(kr)dk,
with the outgoing waves
cosh m(h-y)
C(n)
(mr),
- -
C(n)
r
as
(I)
H
m sinh mh
-C(n)
I)
--emY
H
m
(mr)
being given by (5.4).
When the singularity is in the upper fluid,
2 sc
fo
n!
,
n
-kn cosh k(h-y)
+
Rn+1
ky
.
k
n
e
Jo(kr)
-kq
(5.18)
dk,
[c(cosh kh- s sinh kh)
k(l+k 2) sinh kh]"
(5.19)
J,(kr) dk,
,
r
’
have the outgoing waves
m(h-y)H(1)
cosh
m s inh mh
C’(n)
C’(n)
e
P (cos 0)
e
and as
n
k
emy
-C(n)
(mr)
H
m
(I)
(mr)
being given by (5.11).
SUBMERGED SINGULARITIES. BOTH FLUIDS INFINITE.
6.
Here also the boundary value problem for the velocity potentials
to the corresponding ones
i’n
’
is similar
sections 3,4 except that conditions (2.2) and (2.3) are
V o as y
,
replaced by
’
o
as
y
respectively, and the radiation condition takes the forms
ce-mY e imlx
xl
as
my
e
iml xl
for line singularities, and the forms
Ce
as
-Ce
-my
H! I)
’~-Ce my
(mr),
H! I)
o, for point singularities, where
r
(mr)
C
is a constant multiplier and
m
is now
the simple zero of the equation
k(1 + Bk 2)
’
The evaluation of
(see [5]).
c(l + s)
o
for each singularity can be carried out independently
They may also be evaluated by letting
section tend formally to infinity.
h
in the results of the previous
The velocity potentials for the different singu-
larities are as follows:
(a)
Line singularities.
(i)
Wave source.
The ve]ocity potentials are
lo R
’=
2
-l+-. ]
1-s
$ log R’
2
R + -[-$-
c0
.(l
o
2
l+s
+ Bk2)e -k(y + n)
k(1
+ 8k 2)
Bk2)e k(y- n)
+ ,k 2)
c(1+s)
(I +
k(l
for a ave source in the lower fluid and
cos kx dk,
(6 I)
c(l+s)
cos kx dk,
(6.2)
-
156
,
o
2s
2s
]-7 log R +
’=
M.A. GORGUI and M. S. FALTAS
(i + Bk 2)e
-k(y + n)
k(l + gk 2)
c(l+s)
I--S
]-+ log R’
log R +
k(y
(I + p, k2)e
,o
o
_2_s__
l+s
(6.3)
cos kx dk,
k(l + Bk 2)
n)
(6.4)
cos kx dk,
c(l+s)
for a wave source in the upper fluid.
(ii)
Hultipoles singularities.
The velocity potentials are
Rn+I
,
(_ln+l,
+
2(-1)nc
n
k
n!
o
e
[k(l + Bk 2) + K]k
b
o
n!
k(l + Bk 2)
n
e
-k(y + n)
(6.5)
cos kx dk,
c(l+s)
k(v- n)
(6.6)
cos kx dk,
k(1 + .k 2) -c(l+s)
if the singularity is in the lower fluid and
-2sc
F
=---i
,=
cos(n+1)0
R
n+l
k
n
e
-k(y + n)
cos kx
k(1 + Bk 2)
.I b kn[k(1 +
+
(6.7)
dk,
c(1+s)
f
k(1
+ Bk 2)
Bk 2)
K]
e
k(y- n)
cos kx dk,
(6.8)
Jo(kr)dk,
(6.9)
c(1+s)
if the singularity is in the upper fluid.
(b)
Point singularities
If
the singularity is in the lower fluid
(cos 0)
qb’
Rn+l
2c(-1)n+1
n!
n
+ Bk 2) +
+---.F- #[k(l
o
k(1 + Bk 2)
(-I)
F
k
n
e
k(y
K]k
n
e
-k(y + n)
c(l+s)
n)
k(1 + Bk 2)
Jo (kr)dk
(6.10)
c(l+s)
and if it is in the upper fluid,
-2 sc
n!
n
o
(cos 0
Rn+1
-k(y + q)
k
oo k(l
+
+
e
.
(6.11)
Jo(kr) dk,
Bk 2)
c(l+s)
k
o
n
[k(l + B k2
k(1 + k 2)
K]
e
k(y
n)
(6.12)
Jo (kr)dk.
c(l+s)
7. SINGULARITIES AT THE SURFACE OF SEPARATION.
Clearly the results of the previous sections are not valid for
coordinates based on the singularity at the urgin.
Here we use
o.
Then it may be shoen that the poten-
tials are as follows
(a)
(i)
Lie singularities
-
Wave source
Both fluids of finite depth
,
where
A
2cs + (k(1 + 8k2)sinh kh’
[2c + (k(l +
is given by
k2)sinh
(3.2).
kh
2cs cosh kh’)cosh k(h-y)cos
2c cosh kh)cosh k(h’
kx]dk,
+ y)cos kx]dk
!
(7.1)
BASIC SINGULARITIES IN THE THEORY OF INTERNAL WAVES
|57
Lower fluid of finite depth
2s log R
+
-ky
f
o [e
+ 2s
’=
2 log R
-1
where
6cosh k(h-y)]cos kx dk,
(7.2)
(I
+ k2)(cosh kh
Io
sinh kh e
+ 2s
+
cosh kh
s sinh kh)cosh k(h-y)cos kx dk
+ Bk )(cosh kh
(i
ky
kh)sinh kh e
s sinh
ky
cos kx dk
cos kx dk
&
s sinh kh, and
(5.1).
is given by
Both fluids infinite
2s
%
T
’=
(ii)
2
-+-
(i + Bk
2
log R-
9o
l-sl f(l
log R
2)e-ky
k(1 + 8k 2)
+
cos kx dk,
c(1+s)
k2)kY
k(1 + Bk
(7.3)
cos kx dk,
c(l+s)
Multipo]es
Both fluid of finite de_
For multipoles corresponding to
I, 3, 5,
n
2m+l
ook
ook
k(l + Bk2)sinh kh’]cosh k(h-y)cos kx
[2sc cosh kh’
A’
(2m+l)!
and for the multipoles corresponding to
=k---sinh
(l+s)
(2m)!
where
0, 2, 4, ...(odd multipoles)
n
ook
kh’ cosh k(h-y)cos kx dk,
)
2m
(7.5)
j
O, I, 2
kh cosh k(h’+y)cos kx dk, (m
---sinh
A is given by (3.2).
Lower fluid of finite depth
O, I, 2
Similarly for even multipoles (m
k
(2m+l)!
,=
kh]cosh k(h’+y)cos kx dk,
(7.4)
2m
(l+s)
(2m)! o
-c
8k2)sinh
k(l +
[2c cosh kh
.
dk,I
2m+l
(2re+l)
c
(even multipoles)
k(l + Bk2)]cosh k(h-y)cos kx dk,
[2sc
A
k
(2m+l)
2m+l
o
(7.6)
2m+l
[2c cosh kh
k(l
gk2)sinh
/
kh]ekYcos
kx dk,
A
o
and for odd multipoles
c(l+s)
(2m)!
,=
where
-
k
f ---cosh k(h-y)cos
-c(l+s)
(2m)!
k
2m
k
2m
sinh kh e
kx dk,
(7.7)
kycos
kx dk,
(5.1).
is given by
Both fluids infinite
For multipoles
qb
q
T}n-71-j-!
"
k2m+
o
o
tk(l+
k(l+Bk 2)
k
(2m+l)!
I, 3, 5
corresponding to n
2m+l
k?
cos kx dk,
c(l+s)
[k(l+ B k
lk(l+Bk 2)
2 sc]e
(even multipoles), we have
ky
(7.8)
2c ]e
c(l+s)
my
cos kx dk,
M.A. GORGUI and M. S.
158
(;ALTAS
and for odd multipoles
-c(l+s)
k
=-G)F- /
o
e
k2m
(2m)’
-ky
’os
k(]+k 2)
c (l+s)
(b)
2m
o
e
kx dk,
c(1+s)
(7.9)
ky
(m
cos kx dk,
k(l+lk 2)
o, 1,
c(l+s)
Point singularities
Both fluids of finite depth
2m
k
-;!-
’=
2-mb-i
F --&--k
o
2m
k(l+Bk2)sinh kh’]cosh k(h-y) 3o(kr) dk,
[2sc osh kh’
(7.10)
k(l+Bk2)sinh kh]cosh k(h’+y)
[2sc cosh kh
c(1+s)
(2m+l)!
k
I, 3, 5, ...(odd multipoles)
2m+l
sinh
A
o
k2m+
+s
(2m+l)
-c
A
Jo(kr) dk,
A
and for multipoles corresponding to n
where
(even multipoles)
o, 2, 4
For m,ltipoles corresponding to n
kh’ cosh k(h-y) Jo(kr)dk,
sinh kh cosh
k(h’+y)
(7.11)
(m
Jo(kr)dk,
o, I, 2
A
o
(3.2).
is given by
-
Lower fluid of finite depth
For even multipoles" (m
(2m)!
F
k
o,i,2,...)
2m
k(1+Sk 2)]cosh k(h-y)
[2sc
2m
(2m)!
k
f--[2c
cosh kh-
k(l+Bk2)sinh
Jo(kr)dk,
khJekYcos
(7.12)
kx dk,
and for odd multipoles
+s
c
k2m+
cosh k(h-y)
k2m+
ky
(2m+l)!
’=
where
(7.13)
+s)
(2m+l)!
F
-c
A
Jo(kr)dk,
A
sinh kh e
Jo (kr) dk,
(5.1).
is given by
Both fluids infinite
For multipoles corresponding to n
2m
(2m)!
(2m)!
o
o
k
2sc]ekY
[k(l+Bk 2)
k(l+Bk 2)
k
2m
(even multipoles), we have
o, 2, 4,
J (kr) dk,
c(l+s)
2c]e -ky
[k(l+Bk 2)
k(l+Bk 2)
(7.14)
Jo (kr)
dk,
c(l+s)
and for odd multipoles
([+s)
(2m+l)
-c
c(l+s)
(2m+l)
k2m+l
o
’k2m+l
o
ekY
k(l+Sk 2)
Jo (kr) dk,
c(l+s)
e-kY
k(l+k 2)
(7.15)
Jo (kr) dk.
c(l+s)
(m
o, I, 2
).
BASIC SINGUI.ARITIES IN THE THEORY OF INTERNAl. WAVES
It should be noted here that there
is a non-uniqueness for
that any multiple of a slope potential may be added.
to a continuous interface
B
159
o
to the extent
The forms given above correspond
slope t the orgin, where the interface elevation is always
finite.
8.
CONCLUSION.
A complete survey for
a|l
the basic singularities that can be used in two fluids
Results of Gorgui and Kaseem [2] and
problems with surface tension is presented.
Kaseem [3] can be recovered by putting
B
o
in the appropriate forms and also those
of Rhodes-Robinson [4] can be obtained by putting
s
o.
REFERENCES
1.
GOR(;UI, M. A.
2.
GORGUI, M.A. and KASEEM, S.E. Basic singularities in the theory of internal waves.
Quart. J. Mech. App|. Math. 31(1978), 31-48.
KASEEM, S. E. Multipole expansions for two superposed fluids, each of finite depth.
Proc. Camb. Phil. Soc. 91(1982), 323-329.
RHODES-ROBINSON, P. F. Fundamental singularities in the theory of water waves-with
surface tension. Bull. Austral. Math. Soc. 21(1970), 317-333.
FALTAS, M. S. Internal waves in two-fluids problems. M. Sc. Thesis. University of
Alexandria, Egypt, (1977).
3.
.
4.
Wave motion due to a cylinder heaving at the surface separating two
infinite liquids. Jour. Nat. Sc. Math. XVI (1976), 1-20.