some studies on wave refraction in relation to beach erosion along the kerala coast
TRANSCRIPT
S O M E S T U D I E S ON W A V E R R F R A C T I O N I N R E L A T I O N T O B E A C H E R O S I O N
A L O N G T H E K E R A L A C O A S T
BY P. K. DAS, V. HARIHARAN AND V. V. R. VARADACHAm [Indian Ocean Physical Oceanographic Centre of the lndian Ocean Expedition
( C.S.LR.), Ernakulam]
Received Jtme 11, 1965
(Communicated by Dr. N. J K. Panikkar, r.A.SC.)
ABSTRACT
Using the British Admiralty bathymetric charts ofl" the West Coast of India and employing the graphical method of constructing wave refrac- tion diagrams, an attempt is made to study the behaviour of the short- period waves (4, 5 and 6 seconds) which are found to affect the coast generally in the neighbourhood of Cochin Port entrance. Nineteen stations, at intervals of roughly one mile, are ehosen ~round the three- fathom line in this area. Considering a probable field of approach of deep-water waves, limited to a cone of 90 o, five directions of approach are chosen at intervals of 225z ~ in the range of 202{ ~ to 292�89 o. Refraction diagrams are prepared for these dilections and periods, and from these, the refraction functions and directional parameters ate evaluated for each station. The possible directions of flow of long-shore current and the areas vulnerable to erosion and sedimentation are investigated.
INTRODUCTION
THE area under investigation is a twenty-mile stretch along the Kerala Coast, the neighbourhood of the entrance to the Cochin Port--which has to be maintained at a constant depth of 6 fathoms by frequent dredging as ir often gets chocked with accretion materials. There is evidence of beach erosion at some points that are covered by this study.
The seaward stretch of the area of investigation extends from the coast to the fifty-fathom line. The bathymetric charts used for the purpose are British Admiralty Charts Nos. 749 and 750 (with recent corrections) which have been enlarged for conveniente. A preliminary statistical analysis of the wave periods, reported in the lndian Daily Weather Reports for the years
192
Wave Refraction in Relation to Beach Erosion along the Kerala Coast 193
1961 and 1963, shows that waves with periods around "5 seconds or less form a large percentage of the waves affecting the coast. Hence, this study is limited to waves of periods 5 and 6 seconds. However, 4-second period waves are also considered as the lower limit. Waves of periods less than 4 seconds are not considered because it is felt that their effect is small. Five directions for deep-water waves approaching the shore are chosen at intervals of 22�89 ~ in the range of 202�89 ~ to 292�89 o. Angles beyond these limits are left aside due to the involved drafting errors and the possibilities of diffraction.
Nineteen stations marked, alphabetically, A to S are chosen roughly at intervals of one mile along and around the three-fathom line. Stations A to F are to the south of Cochin Port entrance while G to S ate on the northern side of it.
METHOD OF STUDY
With the introduction of high speed electronic computers in recent years, the construction of wave refraction diagrams has become much simplified and less time-consuming (Dorrestein, 1960; Griswold, 1963; H. M. Iyer et aL, 1963). But the basic procedures remain almost the same as that followed in the present study, where the method suggested by Arthur, Munk and Isaacs (1952) and adopted by Pierson, Neumann and James (1955) is used to prepare refraction diagrams and to evaluate the refraction and di_rec- tion functions R and �91 respectively. These functions have been used to gain ah understanding of beach erosion and sedimentation along the coast.
In this study, the ocean waves are represented as simple sine waves. A single wave train is considered at a time in which the crests are parallel in deep water. This is followed up to the coast in aH successive refractions at different depths at intervals of one fathom applying Snell's/Law---Ca/c --- sine aa/sine ~ ; where ca and c are the wave celerities in deep and shallow waters and aa a n d a are the angles made by the wave ray with the normal to the contour on the deep and shallow sides respectively. A wave ray is a line perpendicular at every point to the successive wave crests and shows the direction of propagation of wave energy which again is supposed not to cross any wave ray. For a particular wave crest, as many wave rays of orthogonals are drawn as practicable, which are equally spaced in deep water where all the orthogonals are parallel to each other. These are followed up to the points of interest (stations A to S here), when, after successive refractions, the spacing between the adjacent wave rays and their directions are changed, depending upon the nature of the bathymetry. The changes in direction and spacing between adjacent wave rays are used to evaluate the refraction
TABL
E l
Refr
actio
n fu
nctio
ns
Dir
ecti
ons
~\
Per
iod
~
\ \
Sta
tion
s ~
~
A
.~
C
..
D
.,
H
.,
J ..
K
..
L
..
M
..
N
..
P .~
Q
o.
R
..
S ..
28~*
4 Se
c.
5 Se
c.
6 Se
c.
.~
..
..
0.7
27
3
..
..
0-68
61
0'6
56
0
..
0.7
32
7
0"65
44
..
0"77
05
0.7
15
8
..
0.8
30
7
0.8
17
2
0.94
51
..
..
..
.,
..
..
0.81
87
0.8
45
4
0.9
56
6
0,8
11
5
0.8
22
6
0.8
65
2
0.8
14
6
0.75
4~
1.11
53
0.5
61
7
0.9
37
0
0.7
64
7
0.8
56
7
0.5
1�9
1
0.59
58
0.65
19
0.56
48
0.61
38
0-78
71
0.85
71
0"77
99
0.63
71
0.3
72
8
0.7
72
0
1.12
25
0.86
31
0.7
42
0
0.9
45
9
1.0
07
0
1.89
40
0.75
45
0.8
30
6
..
225
~
4 S
ec.
5 S
ec.
6 S
ec.
0.8
20
8
..
..
0.78
71
..
..
0.8
30
2
0.7
42
9
0.8
48
5
0.8
11
9
0.76
68
0.8
65
5
0.8
25
6
0"82
25
1.01
87
0.8
94
6
0.88
70
1.09
87
..
.,
.~
0.78
31
0.8
42
4
1.11
75
0.85
41
0.91
01
1.01
10
0.8
53
7
0.8
41
2
1.02
56
0.9
31
0
1.14
20
1.61
92
0.4
87
5
0.6
30
8
0.6
�91
0.71
63
0.6
19
3
0.59
49
0.80
68
0.7
73
9
0.9
09
3
0.8
35
5
0"86
33
1.30
86
0.7
78
2
0.8
38
4
0.8
15
6
0.9
38
0
0.71
91
0.81
39
0.88
48
0.94
25
1"28
30
0"86
53
1.0
80
4
1.23
10
247�
89 ~
4 Se
c.
5 Se
r 6
Ser
0.8
54
3
0.98
71
..
0.8
34
2
0"74
38
0.7
31
7
0.8
47
0
0.86
13
0.9
76
7
0.8
47
7
0.8
29
2
1.1
60
2
0.84
90
0.8
86
0
0.9
06
0
0,89
73
0.7
89
6
1.08
45
0,77
75
..
..
..
0.77
38
0.9
89
8
0.8
29
5
0.84
41
1.06
54
0.8
97
8
0"95
94
1.44
41
0.91
79
0-90
17
1-07
32
0.71
21
0.67
85
0.5
75
8
0.83
44
0.82
91
1.23
65
0.8
31
3
0.7
56
7
0.9
35
7
0.8
76
5
0-91
36
1,14
08
0.75
97
0.6
94
2
0.9
59
8
0.9
48
3
0.9
98
9
1.51
91
..
0.9
88
2
1.08
29
..
0.7
76
2
0"89
50
270
~
4 Se
c.
5 Se
c.
6 Se
c.
0.8
51
8
0.97
27
1.23
79
0-8
41
7
0.7
;80
0
.91
20
0.8
37
4
0.8
07
7
0.9
50
9
0-8
29
2
0.7
84
6
0.9
45
1
0.8
26
9
0.98
05
1.0
53
8
..
0.94
12
1.2
85
0
0,7
41
0
0.9
15
2
1.06
67
0.71
17
0.74
59
0.8
30
2
0.76
46
0.8
65
4
1.1
32
5
0.9
31
4
0.9
68
7
1.2
16
8
0.8
22
6
0.7
23
8
1,02
33
0"76
44
0.8
18
2
1.16
83
0.7
95
1
0.7
29
4
0.8
19
8
0.7
84
0
0.8
25
2
0.9
83
6
0.85
11
0.8L
48
1.0
79
2
0.8
01
2
0.9
44
2
1.2
51
2
0.9
05
3
0.9
16
5
1.12
20
0.7
39
8
0.7
03
3
0.7
28
9
0.8
70
3
0.8
69
5
1.44
60
292~
*
4S
ec.
5 Se
c.
6 Se
c.
0.82
41
0.8
08
4
0.8
02
0
0.8
20
2
0.73
61
0.8
84
6
0-8
01
7
0.7
33
6
0.8
57
6
0.7
96
2
0.6
70
9
0.8
29
3
..
0.7
44
3
1-03
05
0.
7752
.
..
.
0.6
60
9
0.5
78
4
0.8
35
7
0'4
36
0
0.4
66
3
0.76
42
0.7
74
7
0.6
50
0
0.86
08
~).8
865
1.00
55
1"0~
15
0.8
47
0
0.7
23
0
0.80
10
0.7
17
0
0.5
85
6
0.8
10
5
0.6
95
4
0.6
68
0
0.93
31
0"69
33
0.79
69
0.95
01
0-8
07
4
1"14
75
1.04
80
0.7
29
8
0.7
0�9
1
0.82
40
0.9
27
3
0.96
11
0.8
41
0
0"79
72
0"68
64
..
..
..
..
�9
TA
BL
E I
I
Dir
ectio
n fu
nctio
ns
Dir
ecti
on
s
A
..
'B
..
]~
..
F ..
G
O
..
~
Q
..
~2
�89
o
4 S
ec.
5 So
=.
6 S
ec.
~
..
.~
+
8.6
0
..
..
+
7-1
0
+1
2.9
0
..
+
6.4
0
+1
2.7
4
..
+
5.2
6
+1
0.1
3
..
225
~
4 S
o:.
5
Sec
. 8
Sec
.
+
4.5
0
..
..
+
3.4
0
..
..
+
2.7
0
+
8.5
6
+1
0.5
8
+
2.4
0
+
6.2
2
+
7.6
5
+
1.6
0
+
4.3
3
+
3,$
8
247�
89 ~
4 S
ec.
5 S
ec.
6 S
e=.
+8
.90
+
4.5
7
..
+1
.60
+
3.3
0
+
6.6
3
+1
-16
+
2.5
0
+
2.4
6
+0
.30
+
0.9
8
+
0.1
0
-0.7
0
-0.6
7
- 1
.90
4 S
r
-0.8
0
.o
--2
"40
--3
.10
- 5
.02
27
0 ~
5 S
ec.
6 S
ec.
+
3.5
6
+
7.4
0
+1
2.0
5
.,
,.
..
+
5"0
0
+
8.9
8
+1
2.4
0
+
7.7
0
+1
0"8
0
+1
6"0
4
+1
3.3
7
+1
5"4
7
+2
8.5
0
+1
3'4
0
+2
4.5
7
+3
T.7
0
+1
1"4
0
+2
0.4
8
+2
2-6
0
+
9.1
0
+1
4.7
5
+2
0.2
0
+
9.4
8
+1
6.5
8
+1
8.5
0
+1
1.I
0
+1
5.7
5
+1
7.1
0
+
7.4
0
+1
0.7
5
+1
4.4
0
+
8.2
0
+1
9.0
0
+2
1.6
0
+10
"88
+
16
.9|
..
+
1.2
0
+
3.6
0
+
3"3
0
,,
,o
,,
+
1,5
0
+
3-9
0
+
7.1
4
+
0.1
5
+
3.6
8
+
1.9
2
+
8.0
0
+
3.0
8
+
6.0
3
+1
0.5
5
+
9.7
0
+1
8.2
5
+1
3.6
0
+1
5.0
0
+2
3.0
0
+
5,1
0
+
9.5
0
+1
1.2
8
+ 4.
44
+ 5.
10
+ 6.
84
+
4.0
6
+
3"6
0
+
7.6
0
+
5"2
0
+
5.0
6
+1
0.7
8
+
5"9
0
+
4.2
5
+
4.6
0
+
8.0
0
+
4.8
8
+ll
.B0
+
5.9
0
+
7.7
4
+1
1.7
0
-0.2
0
-0.3
3
- 2
.83
+4
.10
.
..
.
-3.8
0
-3.4
0
--
3.2
6
-3.6
0
-4"3
7
- 6
.64
-0"3
0
-2.2
8
- 0
.20
+6
.60
+
5.7
3
+1
3'8
2
+6
.30
+
3.8
0
+
9.1
0
-0.4
0
+0
.63
-
1.0
3
-0"5
0
-1.0
0
- 2
.12
+1
.00
-1
.22
-
1.0
4
+0
.08
--
1.4
5
+
0.8
7
+0
'06
-1
.55
-
2"5
2
+2
-50
+
1.6
4
+ 9.
83
+0
.50
+
1.8
4
+ 8"
00
,~
--1
.50
--7
.30
-9-4
2
-2.9
0
+3
"10
+0
.63
-4.7
0
- 6
"04
-|.3
O
--4
"70
-2"7
6
- 1.
40
-3.8
0
+1
.47
+
0
.48
--1
.40
-
2.7
2
-2.9
0
- 5
"70
--4
.72
--
8
.60
--6
"60
-
6"8
0
-5.6
3
--
8.4
0
-4.0
0
- 8
.15
- 5
.25
-7.5
7
-2.3
7
-3-0
0
-1.6
7
- 3
.67
-4.4
2
-10
.92
-6.1
5
-10
.02
-4.3
8
--1
0.3
0
-4.1
0
-10
.76
-0'3
0
- 6
.30
-1.8
0
- 7
'02
-2.1
2
- 7
.50
--
2.1
0
--
3-9
4
--
6.7
9
- 2
"80
-
7.4
5
-11
"24
--
3.4
0
- 9
,83
--
13
"80
--
4.6
0
--1
2.2
8
-16
.82
- 4
-80
--
12
.24
-1
6"0
0
,,
,.
,.
f - 7
.80
-1
5.7
3
-18
.37
-13
.70
-1
6"8
0
-23
.27
-2
6.1
7
-17
.95
-1
4.8
0
-21
.47
-2
4.2
0
7.7
6
- 7
.00
-1
1.1
0
-12
.50
I
+
1.7
2
- 1
.90
-
5.6
0
- 8
.30
--
5"1
0
-10
"27
-1
5'0
2
-11
.52
--
16
.75
--
21
'67
-I1
.10
-1
8"0
5
-22
"67
--
8.8
0
-16
.07
-1
7.9
0
.,
..
..
..
~
292�
89
4S
ec
. 5
Se
c.
6 S
ec.
#
--1
4.2
5
--1
7.2
5
-10
,40
-1
4.3
5
~-,
-11
.62
..
196 P . K . DAS AND OTHER$
funetion R and the direetion function 0 at the point of interest. The refrac- tion function R is given by
R = [K (F, 0)] ~ = ba/b
c 4rrd 1 ca + L sinh4~rdc A
L c
where L i s the wavelength, d the depth at the point of interest, Ca and C the wave veloeities in deep water and at depth d respectively, ba and b the distances between two adjaeent orhogonals in deep water and at depth d respeetively and F is the frequency of the wave. The refraetion parameters are evaluated for each wave period and direction from the refraetion diagrams prepared for each individual direction and pe¡
DI$CUSSlON
In Tables I and II are presented the refraction and direetion funetions, worked out at each of the stations for varying directions and periods. Table III presents the angles that the refraeted rays make with the normal drawn to the contour at the point of interest. Figures 1 to 5 show the varia- tion of R values from stations A to S with different direetions and periods, while the arrow marks at eaeh station show the probable direction of flow of long-shore currents de¡ from the direction function 0.
The distribution of R values along the stations is directly related to the amount of energy associated with the changed wave height whieh can be shown to be equal to a/R times the deep-water wave height. Hence the convergence and divergente of wave rays, deciding R and 0 values a t a particular station, speak for the accumulation and dissipation of wave energy as no energy crosses wave rays. The change in wave direction at the station, seen as the change from the deep-water direction, and the angle made by the wave ray with the normal drawn at the station to the mean contour on the landward side, gives us an idea about the direction of flow of the longshore eurrent. The accumulated water, after the breaking of the waves, goes back as ¡ current and the like, only after an alongshore flow f o r a certain distance, depending upon the angle at which the wave breaks with the shore line (Per Bruun, 1963). Suggestions are therefore made regarding the longshore drift of beach materials along with the longshore eurrents at eaeh station. The general partero of erosion and accumulation of beaeh material follows from this.
TA
BLI~
III
Dir
ectio
n fu
nctio
ns
(Wit
h r
espe
ct t
o t
he
No
rmal
at
th
e S
tati
on
co
nto
ur)
I D
irec
tion
s I
202�
89 ~
225
~ 24
7�89
~ 27
0 ~
292�
89 ~
~..
~--
+
4 Se
c.
5 Se
c.
6 S
ec.
4 Se
c.
5 Se
c.
6 Se
c.
4 Se
c.
5 S
ec.
6 Se
c.
4 Se
c.
5 Se
c.
6 Se
c.
4 Se
c.
5 Se
c.
`6 S
ec.
Sta
tion
s
A
..
B
C
D
E
F G
H
I J K
L M
N
O
P Q
R
S
..
.o
..
+5
5.9
0
. .
..
.
+4
7-4
0
+4
1.6
0
..
+4
4.1
0
+3
7.7
6
..
+4
5.2
4
+4
0.3
7
..
+2
5.9
4
+2
2.1
0
+1
7.4
5
.,
.o
4 6.
50
+ 2
.52
--
0
.90
+4
0.8
0
+3
7.7
0
+3
2"4
6
+7
3.1
3
+7
1"0
3
+5
8.0
0
"'i
45
8.1
0
+4
6"9
3
+3
3.8
0
11
+2
3"1
0+
14
"02
+1
0"9
0
+4
5"4
0+
39
-75
+3
4.1
3
+4
2.0
2
+3
5.9
2
+3
3.0
0
I-[-
57"4
0 +
52
.75
+
51
.40
!!
+4
3.1
0
+3
6"1
0
+3
9.7
7
+5
6-|
0
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198 P . K . DAS AND OTHERS
z o
I i l
~11 14
o - I 1
T DIRECTION-202~ I
4 M C O N a S
. . . . . . . 5 "
I
�9 ~ .,'̂ ',r , / ~ * ' -
�9 , Ÿ
. v
A B C D " E F G H 1 J K I. H N O P O R 5 STA TtO~IS
F i n . 1.
i . r
z i o ~. s.a u
o�91
o,4
�9
DIRECTION-225"
. . . . . . . . ,5 - ~ "
t ~
_ _ - - ! ^~ i ~ / ,,,..-*-: ~'.. ~ ..., ',,~, .., '~ . , ' ,..-"
W A B C D E F G H I J K L M N O P Q R S
STATIONS Fin . 2.
:1 o
I ' l b
I ' 6
o
t�91
o t
i
DII~ ECTION_: 247~ ~ 4 S r 1 6 2
. - - - , - - - + $ �9
~ . , , �9 - , r �9 - , -e & -
,. ., ' \~ , , . "
A ; ' , / \ ~ i ,.,
...J \ /, ,.,,..I \ /"'-. ~, I ' V 'V ,'---.,,:,,.
V
A B C D E F G H ] J K L H N O P O R S STATIONS
1~O. 3,
Wa~
I � 9 1
I � 9 1
o
i~ 1.2
z
0. �91
0.4
Y.,II
t . i
, z o ff ,.a u
= mO
O6
O4
R~fracttbn in Relation to Beach Erosion along the Kerala Coast
DIRECTION- 270"
t
199
A B C D E F G H I J K L M N O P O R $
STATIONS FZG. 4.
DIREC TION- 292 Yz" � 9 _ �91 s | r
S �9
- 5 " " ' - ~ / " " . ~ " ~ ' - ~ / - " ~ ' / - ~ " ~
A B C D E F G H I J K L H N O P O R S
STATIONS FIG. 5.
As has been reasoned before, the lower period waves, like the ones considered in this study, are the most expected to have any continuous action on the shore line, barring the occasional storms that may send swells or long- period waves to make sudden impacts on it. In the latter tases, however, the energy associated with the waves at breaking would be very high, com- pared with those of the lower periods studied here. The refraction functions therefore would be much higher than those found in the cases of 4, 5 and 6- second waves. The waves however will break at greater depths and the total effect may vary much from those expected from the wave actions of lower periods.
In general, the refraction functions show increasing values from 4-second to 6-second waves as expected. The interesting point, however, is that while this is gradual between 4 to 5 seconds it is abrupt between 5 and 6-seeond periods. Thus, it is felt that any significant wave action in this region may be due to waves around 5 seconds and above as the energy associated with
200 P . K . DAS ANO OTrtEaS
still lower periods will be too low to have much bearing on this. This is readity verifiable from Figs. 2, 3 and 4 where the R values for 4 and 5 seconds seem to lŸ ctose to each other and follow closely for the directions 225 o, 247�89 ~ and 270 o. In the limiting cases, i.e., for angles 202�89 ~ and 292�89 ~ (as can be seen from Figs. 1 and 5) this trend in R values is not well defined, probably because of the drafting errors associated with the acute angles of approach of the waves.
The R functions as can be seen from Table I are mostly less than 1.0 for 4 and 5-second waves for aU the deep-water directions. This shows a lessening of wave height in general and therefore dissipation of the energy associated, which may be aggravated by local wind actions, bottom percola- tions and the like. The 5-second waves, however, show R function to be more than the value of 1.0 at certain stations with varying directions such as at K and S for 225 ~ and at J and O for 292�89 o. This, however, will be dis- cussed later along with the station characteristics. For 6-second period, R mostly shows a value of more than 1.0 for all the five directions of approach and at particular stations (Table I) this is seen to be persistent a s a t stations E and F on the southern side of the fairway channel and at J, K and O on the northern side of ir. The maximum and mŸ R values for all the periods and the directions are found to be 1-89 (at station R for period 6 seconds and direction 202~ :~) and 0- 37 (at station P for period 5 seconds and direction 202�89176
In Table II are presented the net changes in directions, brought in by refraction, indicated with positive and negative signs so that when added algebraically to the corresponding deep-water directions, the direction of arrival of the wave at the point of interest is obtained. It is seen from this table that up to a deep-water direction of 225 ~ the changes in direction are positive whereas beyond 270 ~ the changes are negative for all stations, i.e., up to 225 ~ the angles increase and beyond 270 ~ they dccrease. The deep- water direction 247�89 ~ shows the characteristics of transition and the changes in angles here are mixed indifferently for different periods. There are posi- tive and negative values for each of the periods which vary stationwise. Table III shows the angles that the refracted wave rays make with the landward drawn normals to the depth contours of the station. A rough idea of the alongshore currents can be had from the values given in the table.
In the perspective of the above fmdings the following suggestions, regard- ing the longshore flow, may be offered. For 202�89 ~ deep-water direction, the northerly component of longshore current is predominant for all the periods excepting a singular case in station I for 6-second waves. For the direction 29210 the direction of flow of longshore currents is towards south
Wave Refraction in Relation to Beach Erosion along the Kerala Cbast 201
in general excepting at station K where for 5 and 6-second wave periods the flow is found to be northerly. While these ate as expected, being the two marginal directions, the refracted wave rays for the other deep water direc- tions show very interesting behaviour as seen from Figs. 2, 3 and 4. Here, the longshore current is not northerly of southerly at all stations as in the marginal cases. The changes in longshore currents associated with the changes in peak and trough values of refraction functions give very plausible hints for the erosion and accumulation patterns for different waves.
FI~. 6.
As seen from Figs. i to 5 there exist alternate peaks and troughs for R in the northern side of the fairway channel, i.e., from stations G to S. This however is less true for the southern side, i.e., for the stations A to F. Thus the southern side is likely to be less disturbed by the wave refraction and this is more so as there are few cases of energy concentration due to conver- gence of wave rays. The situation at station F is somewhat different and the high value of R here is mostly associated with an increased wave height which is due to the shoal immediately south of the channel (Fig. 6). Con- sidering the directions 225 o, 247�89 ~ and 270 ~ for stations F to S ir is found that the longshore current directions diverge out from certain stations and converge to certain others (Figs. 2 to 4). And this whole process
202 P . K . DAS AND OTHERS
keeps changing with changing deep-water directions and periods. These factors can most presumably be utilised to explain the facts that (i) the fairway channel gets filled up with materials carried in from both sides and (ii) erosion takes place ncar the stations J and K (Narakkal area). In Fig. 6 ir can be seen that at station K the wave rays converge strongly.
Station O is another point where a peak for R exists due to convergence of wave rays for all the directions and periods with the only exception for a deep-water direction of 270. ~ Hence, this point is also susceptible to denudation due to wave action.
SUMMARY The investigations on wave refraction along the coast near Cochin
reveal the possible existence of areas of concentration of wave energy at certain points and removal of the material of the beach, which is disturbed near areas of concentratŸ of wave energy by diverging longshore currents and the accretion of materials at certain regions due to converging long- shore currents. The accretion of material in the fairway channel near Cochin and the reported beach erosion near Narakkal (situated near station K in Fig. 6) could be clearly explained with the help of this investigation.
ACKNOWLEDGEMENT
The authors wish to acknowledge gratefully the facilities extended by Prof. M. S. Krishnan, formerly Director of National Geophysical Research lnstitute and Dr. N. K. Panikkar, Director, Indian Ocean Expedition. Two of the authors (P. K. Das and V. Hariharan) wish to express their sincere thanks to Shri V. S. Rama Raju for the encouragement given and facilities provided during the work and to Shri K. V. Sundara Ramam for his very helpful suggestions during the initial stages of this study.
[. Arthur, R. S. et al.
2. Bruun Per.
3. Dorrestein, R.
4. Griswold, G. M.
5. Iyer, H. M. and Punton, N. V. W.
6. Pierson, W. J., Jr., et al.
REFERENCES .. "'The direct construction of wave rays,'" Transaetions of
the American Geophysical Union, 1952, 33, 855-65. .. "Longshore currents and longshore troughs,'" J. Geophys.
Res., February 15, 1963, 68 (4), .. "Simplified method of determining refraction coefficients
for sea-waves," lbid., February 1960, 65 (2). .. "Numerical calculation of wave refraction," lbid., March
1963, �91 " A computer program for plotting wavefronts and rays
from a point source in dispersive mediums," Ibid., June I, 1963, 68 (l 1).
.. "Practical methods for observing and forecasting Ocean Waves by means of wave spectra and statistics,'" U.S.H.O. Publication No. 603, 1955.
398-66. Printed at The Bangalore Press, Mysore Road, Bangalore City, and Published by B S. Venkatachar, Editor, "Proccedings of the Indian
Academy of Sciences," Bat~galore