Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 741-748
741
TRANSMISSION OF OBLIQUELY INCIDENT SURFACE
WAVES THROUGH A NARROW GAP
B.N. MANDAL and P.K. KUNDU
Department of Applied Mathematics
University of Calcutta
92 A.P.C. Road
700009
Calcutta
(Received January 22, 1987)
ABSTRACT.
This note is concerned with the transmission of a train of surface water
waves obliquely incident on a thin plane vertical barrier with a narrow gap.
Within
the framework of the linearized theory of water waves, the problem is reduced to the
solution of an integral equation which is solved approximately.
The transmission and
reflection co-efficients are also obtained approximately and represented graphically
against the different angles of incidence for fixed wave numbers.
Water waves, linearized theory, narrow gap, oblique incidence,
reflection and transmission coefficients.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
76B.
INTRODUCTION.
When a train of surface water waves is incident on a plane vertical barrier with
a gap, it is partially reflected and partially transmitted.
incidence Tuck
For the case of normal
[I] used the method of matched asymptotic expansion
to solve the
problem for a gap which is assumed to be narrow in the sense that the depth of sub-
mergence of the midsection of the gap is large compared to its breadth.
Packham and
Williams [2] also considered the same problem and used an integral equation (IE)
formulation wherein IE is on the horizontal component of velocity across the gap.
The IE is then solved analytically by an approximate method.
Porter [3] used a
reduction procedure and also an IE formulation (different from the approach of Packham
and Williams [2]).
Both the methods considered by Porter [3] lead to finding the
solution of the same Riemann-Hilbert problem. In the IE formulation considered by
Porter [3], the IE is on the difference of potential across the barrier above and
below the gap. While Tuck [1] and Porter [3] considered the problem only for deep
water, Packham and Williams [2] extended it to water of finite constant depth.
The transmission and reflection coefficients obtained by Porter [3] are in
principle exact, but that obtained by Tuck [I] or Packham and Williams [2] are
approximate under the assumption that the depth of submergence of the midsection of
the gap is assumed to be large compared to its breadth.
In the present note we generalize the problem in deep water to the case of a
742
B. N. MANDAL AND P. K. KUNDU
An IE formulation is used to reduce the
wave train impinging the barrier obliquely.
problem to the solution of an IE involving the unknown horizontal component of
IE’s by
This IE is reduced to the solution of a system of
velocity across the gap.
an approximate procedure which is successfully used in some related obliquely
incident water wave scattering problems involving obstacles in the form of plane
vertical barriers (cf. Mandal and Goswami [4], [5],
[6]) and
horizontal circular cylinder (cf. Mandal and Goswami [7]).
2.
a half immersed
STATEMENT AND FORMULATION OF THE PROBLEM.
We consider the motion in an inviscid and incompressible fluid occupying the
region y
0, the mean free surface (FS) coinciding with the plane y
O,
Ixl
O.
A plane vertical barrier of infinite horizontal extent with a gap is present in the
0, the gap being situated in the region
fluid, the barrier being represented by x
a < y < b.
Assuming the motion to be irrotational and simple harmonic in time with
be described by
and of small amplitude, a velocity potential exists which may
o
cylinder frequency
Re{x(x,y,z) exp (-lot)}.
Under the linearized theory
X
satisfies
the Laplace’s equation in the fluid region, the FS condition
3y
+ KX
0
on
O,
y
the barrier condition
0
3x
where
K
0 < y < a
0,
x
on
and
b < y <
,
02
g being the gravity.
g
A train of surface water waves represented by X
is assumed to be incident at an angle
a
exp{-Ky+iKcos x+iKsinz}
with the normal to the barrier.
In view
of the geometry of the barrier we can assume,
(x,y) exp (iz)
X(x,y,z)
Then (x,y) satisfies
K sin a.
where
32
3y + K
0
where.(x,.y)
cross x
2)
+__
3y 2
3x 2
0
on
on
x
0
y
0,
(2.1)
in the fluid region,
(2.2)
0,
0 < y < a
.
and
b < y <
,
(2.3)
and its partial derivatives are continuous everywhere except possible
0, 0 < y < a and b < y <
We also require that
and
V
are bounded
everywhere away from the two sharp edges of the wall at the gap, and near these edges
iVI
O(r
})
where
"r
denotes the distance from the edges.
Let us write
(x,y)
where
o(X,y)=
o(X,y) +
(x,y)
(2.4)
exp(-ky+iKcosux), then (x,y) satisfies (2.1), (2.2) and the edge
743
TRANSMISSION OF OBLIQUELY INCIDENT SURFACE WAVES THROUGH A GAP
As the incident wave train experiences reflection by the
condition stated above.
barrier and transmission through the gap, the asymptotic behavior of #(x,y) as
xl
is given by
T
where
#(x,y)
T
#(x,y)
o(X’Y)
R
and
#o(X’Y)
as
+ R
,
x
o(-X’Y)
(2.5)
-,
x
as
(2.6)
(complex) transmission and reflection co-efficients
are the
respectively.
By an appropriate use of Green’s integral theorem in the region
and y > 0, x < 0
we obtain following Mandal and Goswami
b
/ G(o,y;,n) f(y) dy
(,n)
y > 0, x > 0
[6]
(2.7)
> 0,
for
a
b
f G(o,y;,) f(y) dy +
a
o(’)
(o,y) dy
-/" G(o,y;,n)
for
(2.8)
< O,
o
x
f(Y)
where
G(x,y;,)
and
(o,y), a
is the
Ko(0)
G(x,y;,n)
y
(2.9)
b,
Green’s
function given by Levine [8] as
Ko(0
(kZ- 2) cos{ (k2- 2) (y+)
2
K 2 + k2
+ 2
exp(-klx-
)dk + 2i
K sin{ (k2- 2)
exp{-K(y+) +
(K2- 2
(y+)
i(K2-2)1/21x-l}.
(2.10)
Considering the continuity of (x,y) across the gap we obtain the IE for f(y) as
b
G(o,y;o,n) f(y) dy
a
3.
0(y2-a2) -1/2
f(y)
where
as
y
(2.11)
-o(’)
a
and
0(b2-y2) -1/2
f(y)
as
y
b.
SOLUTION OF THE INTEGRAL EQUATION.
To solve the IE (2.11) we first expand its Kernel in terms of
so that 0
g
< I.
0
where
sina
corresponds to the case of a normally incident wave for
which the solution is supposed to be known.
This technique is used earlier
successfully by Mandel and Goswami [4], [5], [6], [7] in connection with the scattering of an obliquely incident wave train by obstacles of some particular geometrical
shapes.
Now following Mandal and Goswami [6]
G(o,y;o,)
G (o,y;o,)
o
+ 2
+ 2 n
G2(o,y;o,
Gl(O,y;o,)
+ 0( n
,
)
(3.1)
B. N. MANDAL AND P. K. KUNDU
744
where
(K
2 f
G (o,y;o,)
o
sin
k-k
+
k) (K
cos
k-k cos k)
dk
exp{-K(y+)},
2i
(I
G2(o,y;o,n
_K 2
(3.2)
K),
Ky)(l
G l(o,y;o,n)
(K sin ky-k cos ky) (K sin kn-k cos k
k (k 2 + K 2)
f
K
i exp {-K (y
z
(-I)
+ n)}
(2n-l)
1.3.5
n
dk
n(/+l))
+ (l-Ky)(l-Kn) { + /-I
+
sin
k(k 2 + K 2)
o
n=2
2
n
2n-2
n!
etc.
Thus to solve the IE (2.11) we assume a similar expansion for f(y) given by
fo (y)
f(Y)
+ E2 gn
fl(y)
+
2f2(y)
+ 0 ( En
E,
g)
(3.3)
where f (y) satisfies the IE
n
b
f
fn (y) Go(’Y;’)
a
Here
-
u (n)
o
Un (n)
and
Un(n),
dy
a <
(3.4)
< b, n--0,1,2
exp (-K)
(3.5)
b
f
a
fo
Gn(’Y;’)
(y)
dy, n
1,2
and f (y) has integrable singularities at the two ends. Closed form solution of
n
(3.4) is not possible, and instead, as in Packham and Williams [2], we solve it
a+b.
approximately by exploiting the assumption that the depth of submergence h( --)
of the midline of the gap is large compared to the breadth b-a
2c say.
Now the kernel in the IE (3.4) can be written as
G (o,y;o,n)
o
n [y-n[ + g(o,y;o,n)
g(o,y;o,n) is regular.
where
Then (3.4) is
b
b
f f n (y) n
a
[y-HI
u (n)
n
dy
+ f
a
f (y)g (o,y;o,)dy, a <
n
n
< b.
Since 2c is small compared to h, we can approximate the right side of
its value at
n
h, so that
(3.6)
(3.6) by
(3.6) reduces to
b
a
where
f f n (y)
An -Un(h)
n [y-[
dy
+ g(o,h;o,h)
c n,
A
n’
a <
n
0,1,2
< b
(3.7)
TRANSMISSION OF OBLIQUELY INCIDENT SURFACE WAVES THROUGH A GAP
745
b
and
c
/
n
f (y) dy.
n
a
Now the solution of (3.7) when f (y) has integrable singularities is given by
n
(cf. Cooke [9])
A
f (y)
n
(3.8)
n
{ (y-a) (b-y)
- ---
# i.
provided
u (h)
Thus
For
c
n
n
n
0,
c
Un(h)
0.
n
g(o,h;o,h)
u (h)
o
o
n
b
Also
b-a
f
b-a
g(o,h;o,h)
fo (y) Gn(’Y;’h)
a
dy
G (o,h;o,h)c
n
o
G (o,h;o,h)c
so that
4.
c
n
n
b-a
n
o
n > 0
g(o,h;o,h)
TRANSMISSION AND REFLECTION COEFFICIENTS.
+
Making
(2.9) as
II
=,
T
in (2.7),
in (2.8) and noting the form of
and using (2.5) and (2.6) we obtain respectively
-2i (I
f
exp (-Ky) f(y) dy
a
b
e2)- f
+ 2i(i
R
b
e2)
exp (-Ky) f(y) dy
a
so that
so that
Now
T+R= I.
Noting (3.3) we can expand
T
To +
R
Ro
To + Ro
2 ne
T
e2 T
+
+ e2 ne R + 2
i, T
+ R
0, T
R2
2
+ 0(
-2i exp
TI
-2i exp (-Kh)c I,
T2
-2i exp
and
n
e, e )
+ 0(g En e, B )
+ R2
(-Kh)Co,
20
etc.
2
T
(-Kh)( + c2),
0, etc.
R
as
G(x,y;,n) in
_
74b
B. N. MANDAL AND P. K. KUNDU
Following Evans [I0]
X
where
i(x)
Thus
T
2
(et/t)
i
ri} exp(-2Kh)
(2Kh)
dt.
:i+e 2Kh nh+ i (2Kh)
o
TI
T
2{E
n(2h)
g(o,h;o,h)
-
(l-Kh)
2i
T
=
2
2
exp(-2Kh) T o
T
[I
etc.
We may note that
ITo[
G2(o,h;o,h)]
exp (2Kh)
coincide with the result given by Tuck
[I] and Packham and
Williams [2] for a narrow gap.
5.
DISCUSSION.
The approximate expressions for the transmission and reflection co-efficients
are obtained here under the assumption that the gap is narrow.
of the ratio
.c
For a fixed value
0.15 (as considered by Packham and Williams [2]) numerical
,, , ,
0.I, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4 and 1.6
calculations are performed for Kh
0
and the angle of incidence a
D
I0
a up to 30
graphically represented against
mentioned above (cf. fig.l,2 and 3).
15
mo
25
and BO
ITI
and
IRl
are
for the different values of Kh
It is observed that for a fixed
the trans-
mission co-efficient first increases while the reflection co-efficient decreases and
then the transmission co-efficient decreases while the reflection co-efficient
increases for different values of Kh.
result of Packham and Williams
infinite depth of water.
This has already been demonstrated in the
[2] and Tuck [I] for the case of normal incidence and
However for a fixed Kh, the transmission co-efficient
increases and the reflection co-efficient decreases as
increases.
This type of
phenomenon was also observed in connection with the scattering of an obliquely
incident surface wave train by a fixed vertical partially immersed barrier
(cf. Mandal and Goswami [4], Evans and Morris [II]), a submerged plate (cf. Mandal
and Goswami [5]), a submerged barrier extended infinitely downwards (cf. Mandal and
Goswami
[6]) and a half immersed circular cylinder (cf. Mandal and Goswami [7]).
All
the curves for different fixed values of Kh exhibit the behavior that more energy is
transmitted through the gap, when the wave.train is obliquely incident than when it
is normally incident on a barrier with a narrow gap.
For other small values of
2c
-h-’
numerical calculations are also performed and
similar qualitative nature of the transmission and reflection co-efficients are
observed.
It may be noted that we have restricted our analytical as well as numerical
calculations correct up to four decimal places and up to a
30
However as
sin a < i, in principle, there is no difficulty to make calculations for more
0
values of a.
n
and
gn
However in that case we need to consider higher order terms involving
En (n
f(y), T and R.
4) in the expansion of the kernel of the IE and similar terms in
747
TRANSMISSION OF OBLIQUELY INCIDENT SURFACE WAVES THROUGH A GAP
0.7225
0.7100
685O
0.7550
0.6600
0.7350
0.6350
0.6100
0.7150
\
0.5860
0.6950
o
5
15
10
20
25
oK,ANGLE OF INCIDENCE(deg)
I:’ 9.
0.6650
,0.9900
0.6050
0.9500
0.5/.50
0.9100
0.4850
0.8700 6
0.L250
0.8300
----e-
0.3650
.... )..K_.h
0.7900
=0.6
0.7500
0.3050
0
5
10
15
20"
’:K,,ANGLE OF INCIDENCE(deg)
F{9.2
25
":
0
B. N. MANDAL AND P. K. KUNDU
748
0.2300
0.9975
O. 2000
0.9925
0.1700
’0.9875
0.925
-e-
0.1100
h-=.>-
....
c)-_
-0.9775
Kh=l.6
0.0800
0
10
15 ’
20
25"
. 30
D.9725
,ANGLE OF INCIDENCE (deg)
ACKNOWLEDGEMENT.
This work was carried out under the project No.F.8-8/85(SR III)
dr.3 May, 1985, sponsored by U.G.C., New Delhi.
REFERENCES
i.
TUCK, E.O. Transmission of water waves through small aperatures, J. Fluid Mech.
49 (1971), 65-74.
2.
PACKHAM, B.A. and WILLIAMS, W.E. A note on the transmission of water waves
through small aperatures, J. Inst. Maths. Applics. I0 (1972), 176-184.
3.
PORTER, D.
4.
MANDAL, B.N. and GOSWAMI, S.K. A note on the diffraction of an obliquely incident surface wave by a partially immersed fixed vertical barrier,
Appl. Sci. Res. 40 (1983), 345-353.
5.
6.
MANDAL, B.N. and GOSWAMI, S.K. The scattering of an obliquely incident surface
wave by a fixed vertical plate, J. Math. Phys. 25 (1984), 1780-1783.
MANDAL, B.N. and GOSWAMI, S.K. A note on the scattering of surface waves
obliquely incident on a submerged fixed vertical barrier, J. Phys. Soc.
Japan 53 (1984), 2980-2987.
7.
MANDAL, B.N. and GOSWAMI, S.K.
8.
96 (1984), 359-369.
LEVINE, H. Scattering of surface waves by a submerged circular cylinder,
J. Math. Phys. 6 (1965), 1231-1243.
9.
The transmission of surface waves through a gap in a vertical
barrier, Proc. Camb. Phil. Soc. 71 (1972), 411-421.
Scattering of surface waves obliquely incident
on a fixed half-immersed circular cylinder, Math. Proc. Camb. Phil. Soc.
COOKE, J.C. The solution of some integral equations and their connection with
dual integral equations and series, Glasgow Math. J. ii (1970), 9-20.
10.
EVANS, D.V.
ii.
EVANS, D.V. and MORRIS, C.A.N. The effect of a fixed vertical barrier on
obliquely incident surface waves in deep water, J. Inst. Maths. Applies.
9 (1972), 198-204.
Water-wave transmission through barriers with small gaps, J.
Math. Ii (1977), i-I0.
E.ngg.