Indian Journal of Geo-Marine Sciences
Vol. 39(4), December 2010, pp. 509-515
Wave refraction and energy patterns in the vicinity of Gangavaram,
east coast of India
K. V. S. R. Prasad, S. V. V. Arun Kumar*, Ch. Venkata Ramu & K. V. K. R. K. Patnaik
Department of Meteorology and Oceanography, Andhra University, Visakhapatnam– 530003, India
*[E-mail: [email protected]]
Received 20 August 2010; revised 21 December 2010
Wave energy distribution along Gangavaram, east coast of India has been carried out for the predominant waves
representing southwest monsoon (June-September), northeast monsoon (October-February) and storm period (March-May
and October) using a wave refraction model. Model computes refraction coefficient, shoaling coefficient, breaker heights
and breaker energies along the coast. During all seasons, higher wave energy pattern is observed in the region to the south of
the port but towards north, complex wave conditions exist due to rocky headlands and promontories and as a result wave
breaking transpires at deeper depths. Low wave energy conditions are observed very near to north breakwater during all the
seasons and even during storms. During storm conditions wave energies amplify along the coast. South breakwater of the
port is under the region of convergence during southwest monsoon and for the storms approaching in south-south-east
direction. Numerical wave refraction studies facilitate the coastal engineers and scientists to understand the coastal
processes.
[Keywords: Wave refraction, Refraction model, Nearshore, Gangavaram coast]
Introduction
Wave refraction phenomenon is an important
process responsible for effecting changes in coastal
configuration. Along the east coast of India, wave
refraction studies were conducted using numerical1,2,3
and traditional methods4,5. In the north coastal sector
of Andhra Pradesh, these studies are very meager.
Visakhapatnam, the city of destiny consisting of two
ports: one near Dolphin’s nose and the other is newly
constructed Gangavaram port at a distance of just
15 km southwards. These ports require frequent
monitoring of wave conditions, littoral transport and
bathymetry changes in order to maintain the ports and
facilitate navigation. The present study deals with
wave refraction and distribution of nearshore wave
energy patterns in the neighborhood of Gangavaram.
Gangavaram is located in the industrial nerve
center of north coastal Andhra Pradesh (latitude 17°
37' N and longitude 83° 14' E). The coast here forms a
bay between Yarada hill at north and Mukkoma hill at
south, and comprises of promontories, pocket beaches
as well as open sandy beaches are peculiar for coastal
studies. A creek in between these two hills forms
Balacheruvu lagoon, where the natural port of
Gangavaram has been developed mainly to cater to
the needs of the adjoining Visakhapatnam steel plant
in the south (Fig. 1). It is one of the deepest natural
ports in India with a depth of about 20 m. The
stations are identified on the location map showing
S1 to S13 on the southern side of the port and N1 to
N9 towards north of the port.
Climate in this region is mainly controlled by the
Indian monsoons. The swell waves are having periods
5–10s6,7 approaching from SSE and E directions
during southwest monsoon and northeast monsoon
respectively. During storm conditions (considering
before and after storms also), the wave periods of 810s are predominant and sometimes 12-18s 8 are also
observed in both the seasons. Though the lower
periods are dominant the higher periods are the ones
which are important as far the energy distribution is
concerned3. The sea is rough during June to
September with wave heights ranging from 1 to 3 m,
and wave heights, of the order of 0.5 to 1 m prevail
during October to December, except during the
cyclone periods.
Materials and Methods
Based on wave atlas prepared for the Indian coast
and past studies5,7, the predominant deep water wave
directions E and SSE, with periods 8 and 10 s
representing southwest (June-September) and north-
510
INDIAN J. MAR. SCI., VOL. 39 NO. 4, DECEMBER 2010
Fig. 1—Location map and study area.
east monsoon (October-January) respectively are
considered. In the Bay of Bengal, the frequency of
storms is more during March-May and October4 with
wave periods 14s. Naval Hydrographic Charts 3002
and 3035 were considered for extracting digital
bathymetry using Arc Map software. Numerical
refraction procedure is adapted from Skovgaard et al.9
and Mahadevan3. Similar numerical refraction studies
were previously carried by many researchers1,2,3,10,11
for the Indian coast. In this study, we computed the
nearshore wave energy and breaker conditions in
addition to refraction and shoaling coefficients.
Assumptions of model
The computation of wave ray pattern is based on
linear small amplitude wave theory applied for
shallow waters. Accordingly the wave speed depends
on the depth of the water in which it propagates. This
model computes wave speed at every grid point under
the following assumptions:
1. Wave energy transmitted between adjacent
orthogonal remains constant. This supposes
that the lateral dispersion of energy along the
wave front, reflection of energy from sloping
bottom, and the loss of energy by friction and
other processes are negligible.
2. The direction of wave advance is
perpendicular to the wave crest.
3. Waves have small amplitude, constant period
and long crest
4. The speed of a wave with a given period at any
location depends only on the depth of water at
that location
5. Changes in bottom topography are gradual
6. Effects of current and winds are considerably
negligible.
Model details and computation of wave refraction
The wave refraction pattern for the given area can
be computed based on wave orthogonals and the
1
refraction coefficients, Kt =
, which can be
β
obtained by solving the set of differential equations
d 2β
dβ
+ p(s)
+ q( s ) β = 0
2
ds
ds
where, p( s) = −
q(s) =
1
(cosθC x + sin θC y ) and
C
1
(sin 2 θC xx − sin 2θC xy + cos 2 θC yy )
C
In order to solve these equations Runga-Kutta-Gill
integration procedures are used3. The procedure for
wave refraction input details are explained in
PRASAD et al.: WAVE REFRACTION AND ENERGY PATTERN
Appendix-I. The numerical computation requires the
water depths Hij at each grid point for computation of
wave speed Cij. Other input data needed are the deep
water wave characteristics such as the wave period (T
in seconds), wave direction (αο in degrees) and height
(ho in meters). Therefore Cij can be calculated using
the formula
Cij =
 2πH ij
gT
tanh
 Cij T
2π





The model computes the wave speed and wave
height ( hij = ho K r K s ) by an iterative procedure at all
grid points, starting from the deep water of the model
domain; the deep water wave speed provides the
initial approximation for the iterative procedure. For
computing the wave speed at subsequent grid point
and considered, the calculated at the previous points
serves as the initial approximation. For depths less
than L/2, where L is the deep water wave length, the
wave speed was computed from Cij = gH ij ,
whenever Hij is less than 0.1 m; Cij was assumed to be
zero. It is necessary to calculate the partial derivatives
of the wave speed with respect to x and y grid points.
With grid spacing as the unit of measurement for
length in the horizontal plane, the finite difference
forms of the differential coefficients are
 ∂C   Ci +1, j − Ci −1, j   ∂C   Ci , j +1 − Ci , j −1 

=
,  ∂y  = 

2
2
 ∂x  
 


 ∂ 2C 
 ∂ 2C 
 2  = Ci +1, j − 2Ci , j + Ci −1, j ,  2 
 ∂x 
 ∂y 




= Ci , j +1 − 2Ci , j + Ci , j −1
 ∂ 2 C   Ci −1, j +1 − Ci −1, j +1 − Ci +1, j −1 + Ci −1, j −1 

 

 ∂x∂y  = 
4


 
Initial conditions
The integration of the differential equations is
started offshore in the deep water where the wave
rays are parallel, and proceeds towards the shore. The
integration step size is expressed as a fraction of the
spatial grid spacing, the unit of measurement for
lengths in horizontal plane. As the x-axis of the
coordinated system is in deep water, the origin for the
wave rays may be selected at equal intervals along the
x-axis. Where the refraction coefficient is 1, i.e., β =
1, and its derivative is zero θ will be equal to the
511
deep water wave direction. These conditions along
the x-axis form the initial conditions for the
differential equations.
Termination of model and extraction of breaker parameters
The model computes the wave speed, wave angle
(with respect to x-axis), refraction coefficient (Kr),
shoaling coefficient (Ks) and height (hij) at every grid
point and terminates under one/all of the following
conditions:
(a) Wave steepness hi j / L greater than 1/7
(b) Breaking depth db equals 1.28 hb
(c) Wave ray reaches zero depth or any negative
depth value (denotes land). Whenever breaking
depth is reached, the model automatically
extracts the near shore wave height (breaker
height hb), breaker angle (αb), shoaling
coefficient (Ks), refraction coefficient (Kr) and
computes the near shore breaker energy using
1
the formula, Eb = ρghb 2 .
8
Results and Discussion
Wave refraction and Energy distribution
Southwest monsoon period – The refracted wave
orthogonals for the SSE direction and for the periods
8 and 10s are shown in Fig. 2. For 8s wave period,
convergence is observed near the south breakwater
and further southwards in the Appikonda beach at
station S4 (Fig. 2a) with breaker heights 1.0-1.5 m
(Fig. 3a) and nearshore breaker energy is about
3.675 × 103 J/m2 (Table. 1). Waves of 10s period
show more convergence of energy along the coast
(Fig. 2b) than that of 8s period with highest wave
energy of 4.485 × 103 J/m2 at station S4. But, here the
convergence is just shifted southward (from S3 to
S4). In the north of the port, divergence is observed
with breaker heights 0.5-1.0m (Fig. 3a&b) having
energies ranging between 1.482 × 103 J/m2 (at station
N5) to 4.456 × 103 J/m2 (at station N2). The shoaling
and refraction coefficients vary along this coast
ranging between 1.0-1.25 and 0.5-1.5, respectively.
The waves in the northern region are breaking at
deeper depths than usual due to the presence of rocky
headlands and promontories of Yarada hills. During
this season, most of the wave energy is concentrated
in the southern portion of the port and the
south breakwater is likely to be in the region of
INDIAN J. MAR. SCI., VOL. 39 NO. 4, DECEMBER 2010
512
Fig. 2—Wave refraction for SSE waves: (a) 8s period (b) 10s period.
Fig. 3—Variation of breaker parameters for SSE waves: (a) 8s and (b) 10s.
Table 1—Near shore wave energy Eb (× 103 J/m2) for different wave conditions
Station
ID
T = 8 sec
E waves
SSE waves
T = 10 sec
E waves
SSE waves
T = 14 sec
E waves
SSE waves
S7
S6
S5
S4
S3
S2
S1
1.207
3.294
2.411
2.407
1.892
1.534
3.117
1.622
3.675
1.343
2.460
1.537
1.292
3.417
3.141
2.474
2.186
1.616
3.666
1.328
4.485
1.236
4.022
1.548
6.870
17.328
13.762
8.064
7.988
5.713
12.361
4.551
11.908
4.223
15.804
6.380
N1
N2
N3
N4
N5
N6
1.819
1.229
1.537
1.214
-
1.694
3.109
3.253
2.701
1.504
1.548
1.647
2.094
1.732
1.697
-
1.537
4.456
2.484
3.008
1.482
1.545
5.570
8.045
9.000
5.643
-
3.632
17.854
8.405
18.025
4.922
5.406
PRASAD et al.: WAVE REFRACTION AND ENERGY PATTERN
convergence. Because of the steep foreshore in the
south the breaking waves may be plunging to surging
type. Due to the presence of the shoals in the north
and premature breaking, the waves seem to be less
intense but due to rocky headlands all around, not
safe for swimming. Recent news papers reported
many deaths at the headland of Yarada hill (stations
N3-N5) not due to rip currents but only due to sharp
rocky bed.
Northeast monsoon period – Wave orthogonals
approaching the coast from E direction and for the
periods 8 and 10s are shown in Fig. 4. Waves of 8s
period show convergence near the south side of the
513
port (station S1) where rocky promontories exist and
further southwards in the Appikonda beach (station
S6) with breaker heights 0.8-1.2m and wave energies
3.117 × 103 J/m2 and 3.294 × 103 J/m2 respectively
except at rocky outcrops it is around 1.5 m (Fig 5a).
The wave convergence has shifted from the port
break water during south-west monsoon to the station
S1 during the season. At Appikonda beach secondary
wave convergence is observed. Waves of 10s period
show more convergence of energy along the coast
than that of 8s period as in case of southwest
monsoon. In the north of the port, divergence is
observed with breaker heights 0.5-0.8 m (Fig. 5) and
Fig. 4—Wave refraction for E waves: (a) 8s period (b) 10s period.
Fig. 5—Variation of breaker parameters for E waves: (a) 8s and (b) 10s.
514
INDIAN J. MAR. SCI., VOL. 39 NO. 4, DECEMBER 2010
with energies 1.647 × 103 J/m2 (at station N1) and
2.094 × 103 J/m2 (at station N2). During this season, a
consistent convergence is also observed at the tip of
the south breakwater. The intensity of wave
convergence and wave energy is reduced when
compared with that of south-west monsoon season.
Storm
period
–
The
occurrence
of
storms/cyclones is higher during pre monsoon
(March-May) and post monsoon (October).
Vulnerability of the coast depends on the extent of
wave effect during storm periods4. So, wave
refraction is also considered for 14s i.e., longer wave
periods with 2 m deep water wave height for E (pre
monsoon) and SSE (post monsoon) waves and shown
in Fig. 6 (a & b). Wave orthogonals are converging
near the south of port (station S1) and very far
southwards in Appikonda beach (station S6) for E
waves and for SSE waves the convergence pattern is
shifted in the areas where there is divergence for E
waves. For SSE waves, intense convergence patterns
are observed at stations S4, S2 and south breakwater.
During these conditions, breaker heights are observed
to be reaching 3-4m (Fig. 7). In the northern portion,
waves are converging slightly at station N5 and the
remaining is unaffected. In the south for storms
approaching from east, wave energies are very higher
of the order 17.328 × 103 J/m2 at station S6 and it
reduces to 5.713 × 103 J/m2 at station S1 nearer to
Fig. 6—Wave refraction for storm conditions of 14s period: (a) E waves (b) SSE waves.
Fig. 7—Variation of breaker parameters for (a) E waves and (b) SSE waves during storms.
PRASAD et al.: WAVE REFRACTION AND ENERGY PATTERN
south breakwater. In the north, for the same wave
approach wave convergence is slight with higher
value at station N3 and N4. Whereas for waves
approaching from south-south-east direction, the
energies are surprisingly lower of 4.551 × 103 J/m2
at station S5 and also observed that the convergence
pattern is shifted towards north during this condition.
For stations N1 to N4 wave divergence is clearly
observed as in case of seasonal waves (8s and 10s).
Conclusions
During all seasons, higher wave energy pattern is
observed in the region to the south of the port but
towards north complex wave conditions exist due to
rocky headlands and promontories and as a result
wave breaking transpires at deeper depths. Low wave
conditions are observed very near to north breakwater
during all the seasons, even during storms. This may
be attributed to the presence of shoals in the vicinity.
During storm conditions wave energies amplifies
along the coast but for E waves it is much higher than
that of SSE waves on either side of the port South
breakwater is under the region of convergence during
southwest monsoon and storms approaching in southsouth-east direction. Numerical wave refraction
studies facilitate the coastal engineers and scientists
to understand the coastal processes and this model
can be successfully adapted to any type of coast.
Acknowledgements
Authors are grateful to Prof. B. S. R. Reddy, for
his constructive suggestions during the progress of
this work and acknowledges Dr. B. R. Subramaniam,
Director, and Dr. V. Ranga Rao, Scientist, ICMAM
PD for constant encouragement and collaboration.
Author (S. V. V. Arun Kumar) sincerely
acknowledges C.S.I.R, New Delhi for providing
research fellowship.
References
1
2
3
4
Angusamy, N., Udayaganesan, P. & Victor Rajamanickam,
G., Wave refraction pattern and its role in the redistribution
of sediment along southern coast of Tamilnadu, India, Indian
J. Mar. Sci., 27(1998) 173-178.
Chandramohan, P., Longshore sediment transport model
with particular reference to Indian coast, Ph.D. thesis, IIT
Madras, India, 1988, pp. 210.
Mahadevan R, Numerical calculation of wave refraction,
(National Institute of Oceanography, Goa, India, Technical
Report No. 2/83) 1983, pp. 28.
Prasad, K.V.S.R., Arun Kumar, S.V.V., Venkata, Ramu, Ch.
& Sreenivas, P., Significance of nearshore wave parameters
5
6
7
8
9
10
11
515
in identifying vulnerable zones during storm and normal
conditions along Visakhapatnam coast, India, Natural
Hazards, 49(2)(2009) 347 – 360, doi :10.1007/s11069-0089297-4.
Reddy, B.S.R., Venkata reddy, G. & Durga Prasad, N., Wave
conditions & wave-induced longshore currents in the
nearshore zone off Krishnapatnam, Indian J. Mar. Sci.,
8(1979) 61-67.
Chandramohan, P., Narasimha Rao, T.V., Panakala Rao, D.
& Prabhakara Rao, B., Studies on nearshore processes at
yarada beach (South of Visakhapatnam harbour), east coast
of India, Indian J. Mar. Sci., 13(1984) 164–167.
Chandramohan P., Sanil Kumar, V. & Nayak, B.U., Wave
statistics around the Indian Coast based on ship observed
data, Indian J. Mar. Sci., 20(1991) 87–92.
Sanil Kumar, V., Ashok kumar, K. & Raju, N.S.N., Wave
characteristics off Visakhapatnam coast during a cyclone,
Curr. Sci., 86(11)(2004) 1524-1529.
Skovgaard, O., Jonsson, I.G. & Bertelsen, J.A., Computation
of wave heights due to refraction and friction, J. Waterways,
Harbour Coastal Eng. Div. ASCE, 1(1975) 15-32.
Chandramohan, P., Sanil Kumar, V. & Nayak, B.U., Coastal
processes along the shorefront of Chilika lake, east coast of
India, Indian J. Mar. Sci., 22(1993) 268–272.
Sajeev, R., Chandramohan, P. & Sanil Kumar, V., Wave
refraction and prediction of breaker parameters along the
Kerala coast, India, Indian J. Mar. Sci., 26(1997) 128-134.
Appendix-I
The data input (Input.dat) required to trace the
orthogonals and to calculate the refraction, shoaling
coefficients are:
1. Total number of grid point along the x-axis:
NX
2. Total number of grid point along the y-axis:
NY
3. Whether depths at grid points printed (1) or
not: PRNT=1 or 0 (default 1)
4. Number of orthogonal to be traced: NSET
5. Origin of the domain: X1, Y1
6. Orthogonal spacing: SPAC
7. Maximum number of possible steps involved
in integration: MAX
8. Integration step size: ISTEP
9. Deep water wave period: T (in seconds)
10. Deep water wave direction with respect to xaxis of the model domain: THE (in degrees)
11. Deep water wave height: H (in meters)
Input.dat
36
140
6000
8
36 1
0 0 0.25
0.05
80 1.0