3.Wave Deformation.pdf

8/10/2019 3.Wave Deformation.pdf

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Wave Deformation

Prof.V.Sundar, Dept of Ocean Engg, IIT Madras, India

1

WAVE DEFORMATION

Prof. V. Sundar, Department of Ocean Engineering, Indian Institute of Technology

Madras, INDIA

1 GENERAL:

Wave deformation may occur due to

(i) Lateral diffraction of wave energy

(ii)

By the process of attenuation

(iii) Air resistance encountered by the waves or by directly opposing winds

(iv)

By the tendency of the waves to overrun the currents.

The important phenomenon in regard to wave deformations taking place in

the near shore zone is refraction, diffraction and reflection. This chapter deals with

these aspects together with breaking of waves and the changes taking places in

propagation of waves when it encounters currents.

2 WAVE REFRACTION

Variation in wave celerity occurs along the crest of a wave moving at an angle

to under water contours because that part of the wave crest in deeper waters will be

moving faster than the part in shallower waters. This variation causes the wave crest

to bend towards the alignment with the contours. The bending effect of the wave

called refraction, depend on the relation of water depth to wave length. If the angle of

incidence is defined as the angle between normal to the wave front and the normal to

the bottom contours, then the change in the angle with changing wave celerity is

given by

2C

1C

2sin

1sin =

(1)

T

LC= (2)

where C1, and C2, are the celerities at the points where has the values 1and 2.

This is the Snells law. The simplest case of plane waves approaching a straight coast

over a uniformly sloping bottom is shown in Fig.1.


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Waves approaching the shore at right angles for which = 0, are slowed

down but not refracted. Waves approaching at an oblique angle, however, are

refracted in such a way that the angle is decreased, corresponding to the wave

fronts becoming more nearly parallel to the shore. If two rays, defined as orthogonal

to the wave fronts, a distance b, apart are considered. It can be shown that

ocos

cosobb

= (3)

Where bois the distance between the same orthogonal in deep waters.

A plot showing the coordinate positions of the orthogonal is for a given location with

defined bathymetry (depth contours) is called as a refraction diagram. Fig. 2shows a

typical refraction diagram. By drawing a refraction diagram the following

interpretation can be made.

(i)

Convergence of orthogonal (eg: towards a headland)

(ii) Divergence of orthogonal (eg: approaching a bay between two

headlands).

Convergence leads to increase in wave heights or Energy, resulting in shore erosion.

Divergence leads to decay in wave heights or energy, resulting in accession or

deposition.

The basic assumptions for construction of Refraction Diagram are,

Wave energy between the wave rays or orthogonal is conserved.

The direction of the advance of waves is given by the direction of orthogonal.

The speed of a given wave is a function of only depth.

The bottom slope is gradual.

Waves are constant period, small amplitude.

Effects of currents, winds and Reflection from beaches are negligible.

Graphical methods or computer program may be used for drawing the refractiondiagrams.


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In refraction analyses, it is assumed that for a wave advancing toward shore, no

energy flows laterally along a wave crest; that is, the transmitted energy remains

constant between orthogonal. The average power transmitted by a wave is given by

CEnbP= (4)

Where n =2

1

+

kd2sinh

kd21

For deep water conditions, (2kd/sinh 2kd) 0 and hence

0C0E0b2

10P = ( 5)

(4)= (5) leads to

= C

0C

b

0b

n

1

2

1

0E

E

(6)

we know

02H

2H

8/02Hg

8/2Hg

0E

E=

= (7)

Henceb

0b.C

0C

n

1

2

1

0H

H

= (8)

Where,

==

sK

C

0C

n

1

2

1Shoaling coefficient and

b

0b = RK = Refraction coefficient

KSis given in the wave tables as a function of d/L Eq. (8) enables the determination

of wave heights in transitional or shallow waters, knowing the deepwater wave heightwhen the relative spacing between orthogonal can be determined. Refraction diagram

for Paradeep Port due to waves of period 8 Sec approaching from South is shown in

Fig.3.


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3 WAVE DIFFRACTION

Diffraction of water waves is a phenomenon in which energy is transferred laterally

along a wave crest. It is most noticeable where a barrier such as a breakwater

interrupts an otherwise regular train of waves. In such a case, the waves curve around

the barrier and penetrate into the sheltered area.

A simple illustration is presented in Fig.4, in which case a wave propagates normal to

a breakwater of finite length and diffraction occurs on the sheltered side of the

breakwater such that a wave disturbance is transmitted into the geometric shadow

zone.

In order to understand more clearly refer to Fig.5, the origin of the coordinate system

is taken to be at the tip of the breakwater with the x and y-axes running parallel andnormal to the breakwater as shown. The regions may be idealized as follows.

(1)

The region 0 < < ois the shadow zone of the breakwater in which the

solution consists of only the scattered waves. In this area the wave crests

form circular arcs centered at the origin.

(2)

The region o< < (o+ ) is the one in which the scattered waves and

the incident waves are combined. It is assumed that the wave crests are

undisturbed by the presence of the breakwater.

(3)

The region (o+) < < 2is the region in which the incident waves and

the reflected waves are superimposed to form an oblique incidence and a

partial standing wave for normal incidence. Values of diffraction

coefficient, Kdwhich the ratio of diffraction wave height to incident wave

height are shown in Table.1in terms of polar coordinates normalized with

respect to wave length.

Calculation of diffraction effects is important for several reasons. Wave height

distribution in a harbor or sheltered bay is determined to some degree by the

diffraction characteristics of both the natural and manmade structures affording

protection from incident waves. Proper design and location of harbor entrances to

reduce such problems as silting and harbor resonance also require a knowledge of the

effects of wave diffraction.


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The wave diffraction is governed by the Helmholtz equation given by

( ) 0y,xF2k2y

F2

2x

F2=+

+

(9)

F(x,y) is complex, and contains both amplitude and phase information. Fig.6

represents wave fronts and isolines of relative wave height for y > 0.

The solution for F(x,y) for engineering applications according to Penny & Price

(1952) can be obtained as follows.

First for large y, the relative wave height approaches one half on a line separating the

geometric shadow and illuminated regions (See Fig. 7). Second, for y/L > 2, Isolines

of wave height behind a breakwater may be determined in accordance with the

parabolic equation.

Ly

2r

2

16r

2

Lx += . (10)

in which R is obtained from Fig. 7for any value of relative wave height. R = H/HI.

The dashed lines in Fig.6compare several isolines obtained by eq.10 with those from

the complete solution.

Sample wave diffraction plots for typical cases are shown in Figs.8and .9. Typical

plots showing the wave height distribution inside Madras Harbour, Tangassery

Fishing Harbour, kerala and Visakhapatnam Harbour are shown in Fig.10, 11and 12.

Diffraction of waves around breakwaters


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4. BREAKING OF WAVES

A ocean wave would break leading to the dissipation of energy under the conditions

listed below.

When horizontal particle velocity at the crest exceeds the celerity of the wave When vertical particle acceleration is greater than acceleration due to gravity When crest angle is less than 120o

When the wave steepness, 142.0L

H> (Deep waters)

142.0L

H> tanhkd (shallow waters)

When wave height is greater than 0.78d.

The Wave height at the breaking point:

If it is assumed that the waves at the point of breaking can be described by

linear shallow water wave theory, a rough estimate of the wave height at breaking can

be obtained. 2T2

goL

=

We know at point of breaking

8.0bd

bH =

The energy dissipation from deep water to the point of breaking is assumed

to be small. The shore-normal energy. Flux at deep water Eo is therefore equal to

energy flux at breaking point, Eb.

T2

g2gH

16

1oC

2oHg

16

1oE

==

and bgdb2

gH8

1bCb

2Hg

8

1bE ==

As Eo= Eband db= Hb/0.8

5/2g25.1

gT

2oH2

bH

oH

=

or

5/1

oH

oL50.0bH

oH

=


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The breaker depth and the distance from the shore to the region of wave breaking is

found out as follows.

( ) ( ) 2/1ocos2/1

gd2/oCoHbH = (11)

where Hbis breaker wave height

Based on the condition Hb= 0.78d,

2/1

ocosbgd2

oCoHbKd

= (12)

Solving for dbyeilds

5/2

2ocosoCo

2H

5/4K

5/1g

1bd

= (13)

where

( ) ( )2

gT

bHmambk =

( )

= m19e0.18.43ma

( )1

m5.19e156.1mb

+=

and m = Beach slope.

As m approaches zero k=0.78.

For a plane beach where d = mx and m = tan , the beach slope, the distance from the

shore to the shoreline is

5/2

ooo2

5/45/1

bb

2

cosCH

kmg

1

m

dx

== (14)

Finally,

5/2

ooo25/1

bb2

cosCH

g

kxkmH

== (15)


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Medium Slope

Medium H/L

Steep H/L

Steep Slope

Low H/L

White waterFoam

Mild Slope

Taking the beach slope, wave direction also into consideration has derived the above

formulation. Formulae for predicting the breaker depth proposed by several

Investigators are given in Appendix "A".

5. Types of Breakers

Breaking waves may be classified as follows

1.

Spilling breakers: Low steepness waves on mild slopes which break by

continuous spilling of foam down the front face sometimes called as white water.

Breaking is gradual.

2.

Plunging breakers: Medium steepness waves on medium steepness beaches curl

over. Waves break instantly.

3. Surging breakers: These occur with the steepest waves on the steepness

Beaches. The base of the wave surges up the beach generating considerable

Foam. It builds up as if to form the plunging type.

N1= 5.0

oL

oH

tan=

N2 = 4.0

oL

bH

tan=

N1> 0.5 to 3.3

N2> 0.4 to 2.0

N1>3.3N2>2.0


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Galvin (1972) suggested that breaker type depend on the parameter2tanoL

oH

Where is the beach slope and Ho/Lois the deep water wave steepness. In lab tests

with plane concrete beaches, he obtained.

Parameter Spilling Plunging Surging

(Ho/Lo)/(tan2)

(Ho/Lo)/(tan)

These limits are only approximate. They were obtained for slopes of 0.05, 0.1 and 0.2

and may not apply outside that range. More fundamentally, the transition from one

type of breaker to another is very gradual and estimation of the limit is highly

subjective.

Galvin (1972) also describes a fourth category of breaking waves called as

Collapsing breakers. The collapsing breaker comes between the plunging and

surging breakers.

6. WAVES ON CURRENTS

Very often there will also be a current flowing in the direction of or at an angle to the wave

propagation. This current will influence some of the wave characteristics, and, in their turn,the waves will influence certain current characteristics.

Clearly, if the current is in the same direction as the wave propagation direction, the

wave height will decrease and the wavelength will increase. If the current is in the

opposite direction, the wave height will increase and the wavelength will decrease.

The wave celerity, in the absence of current, is given by:

C = /k

Where

C : wave celerity

: wave frequency

0.09

0.00048

4.8

0.011


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k : wave number, 2/L

so = C k

For currents and waves C will change to , given by

C * = C + U

Where C*: wave celerity in case of waves on a current

U : is the current velocity component in the direction of the wave propagation.

The frequency will change to *, given by:

* = C*k = (C+U) k = + k *U

Where * : wave frequency in the case of waves and a current

This component, kU, is often called the Doppler effect.

If waves travelling initially in water with no mean motion enter a current, certain

changes take place in the wavelength and height of the waves. Let us consider the

case of waves propagating in the x direction encountering a current, U in the same

direction. On the assumption that the number of wave crests is conserved, it may be

shown that

K*(U + C*) = k C (16)

Where k, C refer to the wave number and velocity of propagation in the absence of

the current and k*, C* to their values when waves are superposed on the current.

For simplicity, if the waves are assumed to be in deep waters, then,

( ) ( )k

gC,

*k

g*C

22 == (17)

Hence, from eq. (16)

( )C

*CU

*k

k

2C

2*C +==

This may be written as quadratic equation in C*/C with the solution


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++=2/1

C

U411

2

1

C

*C (18)

when the current is flowing in the same direction as that of waves, travelling, i.e.

U >0, the velocity C* and the wave length L* increases compared to their

respective values in still water.If the waves encounter an adverse current, i.e.,U 0.78d

For the above problem, if refraction is not considered,

H /Ho=Ks=0.9828,

Hence H=0.9828* Ho= 3.43m < Hoof 3.5m

(b) H at d=5m


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Kr=8.0

1

8.0

1

8.6

6.15

9349.0*2

1

5.3

H=

H=3.5*1.24 = 4.34m > 0.78d (=3.9m) for d=5m

Problem 3

For a wave with T = 10 secs, Ho= 4m, evaluate the variation in the wave

height when the wave approaches shallow waters. Also determine the

depth of water in which the wave would break theoretically.

Solution

Ho = 4 m, T = 10 sec., Lo = 156 m. The variation of wave height as the wave

approaches shallow waters, is given in the table below. The method involves the

computation of d/Lovalues for progressive waves. With this information it is possible

to obtain the d/L values and the shoaling coefficient. The depth in which the wave

would break is computed from the Miche steepness equation i.e.,

.kdtanh142.0LH

The wave would break at d 5 m.

d(m) d/Lo d/L L H/Ho H H/L 0.142

tanh kd

70 0.45 0.4531 154.491 0.9847 3.938 0.02549 .1390

60 0.385 0.3907 153.57 .9728 3.891 .02534 .1379

50 .32 .3302 151.423 .9553 3.821 .02524 .1357

45 .288 .3014 149.303 .9449 3.779 .02531 .1338

35 .224 .2455 142.566 .9242 3.697 .02593 .127825 .160 .1917 130.412 .9130 3.652 .0280 .1169

15 .0964 .1380 108.696 .9358 3.743 .03444 .09786

6 .0386 .08175 73.395 1.072 4.228 .05842 .06608

4 .0257 .06613 60.487 1.159 4.636 .07664 .05504


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Problem 4:

For a wave with period T= 10sec , deep water wave height H0= 3.5m and beach slope

m=1/15 calculate the breaking wave height using various formulae .

Solution

Sl.No Formulae for Breaking wave height Breaking

wave

height(Hb)

m

1 Le Mehaute and Koh(1967)

Hb=0.76*H0*(H0/L0)^(-0.25)*m^(1/7)

4.668

2 Komar and Gauhgen(1972)

Hb=0.56*H0*(H0/L0)^(-0.2)

4.189

3 Sumura and Horikowa(1974)

Hb=H0*m^0.2 *(H0/L0)^(-0.25)

5.262

4 Singamsetti and Wurd(1980)

Hb =0.575*H0*m^0.031*(H0/L0)^(-0.254)

4.854

5 Ogawa and Shuto(1984)

Hb=0.68*H0*m^0.09*(H0/L0)^(-0.25)

4.819

Problem 5:

Given T = 7 sec, Ho= 3m, Deep-water wave direction = 80o. Find Breaker height and

breaker depth using all formulae.

Solution:

Given tan= 6/500 = 0.012.Lo= 1.56T

2= 1.56(7

2) = 76.44m.

(a) Komar & Gaughen(1973):

( ) 5/1oL/oH

563.0

oH

bH =


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( )m22.3

5/144.76/3

563.0x3bH ==

Breaker depth = bd78.0bH = Xb = 344m

(b)

LeMehaute & koh(1967):

( ) ( ) 4/1oL/oH.7/1tan76.0

oH

bH =

( ) 4/144.76/34/1)012.0(x3x76.0bH =

m74.2bH =

m6.290bx,m50.378.0/74.2bd ===

(c)

Sunamura(1983):

( ) ( ) 4/1oL/oH.5/1tan

oH

bH =

( ) ( ) 25.044.76/32.0012.0x3bH =

m297bx,m56.3bd;783.2bH ===

(d)

Munk(1949):

( ) 3/1oL/oH3.3/oHbH =

( ) 3/144.76/33.3

3oH =

m675.2bH =

m43.3bd =

bx = 286m

(e)

Godal(1975):

+= 3/4tan151oL/bd5.1exp1A

oL

bH

( ){ }[ ]0027.0*15144.76/bd712.4exp117.044.76

bH +=

0.7859=e-0.0592

b; for Hb= 2.782m


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db= 4.0693m

(f) We have

[ ] ( ) 2/1ocos2/1gd2/oCoHbH =

[ ] ( ) 2/180cos2/16x81.92/7x56.1x3bH =

m054.1bH =

5/22/ocosoC

2oH5/4k5/1g

1bd

=

where m = beach slope

K = b(m) a (m) Hb/ gT2

a(m) = 43.80[1.0 e-19m

]

b(m) = 1.56[1.0+ e-19.5m]-1

m = tan and m = 0.012

a(m) = 43.80 [1.0 e-19x0.012

]

a(m) = 8.93, b(m) = 0.87 and k= 0.850

db= 1.70m

db= m Xb, where Xb= distance from the shore to breaker zone

Xb= 141.60m.

Problem 6:

Given wave height (H) = 1m, d=4m, wave period T = 8 sec,

Current velocity U = 1 m/ sec.

Calculate the amplitude and length of the waves when currents are in the same

direction & also in opposite direction. Also calculate the critical velocity at which

wave would tend to break.


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Solution:

Wave amplitude and length when currents in the same direction

we have ( )

++= 2/1C/U4112/1

C

*C (1)

where C*= wave velocity in the presence of currents

U = current velocity

Lo= 1.56T2= 1.5x82= 99.84 m/sec, U = 1 m/sec

d/Lo= 0.0400, from wave tables d/L = 0.08329

L = 48.025 m

we have C = L/T = 48.025/8 = 6.00m/sec.

Substituting these values in equation (1)

We get C*= 6/2 [1+(1+4x1/6)1/2]

C*= 6.87 m/sec

further we have

( )[ ] 2/1*** 2// UCCCaa o +=

Co = Lo/ T = 12.48m/sec

a = H/2 = = 0.5m

by substituting the values of C*, Co, U we get the value of a

*

a*= 0.8 m.

*C

T

*L=

L*= 6.87 x 8

= 54.96m

when the waves travel in the same direction of current L*increases


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(b) wave amplitude and length when currents in opposite direction:

we have C*/C = [1+ (1-4U/C)

1/2]

where U = -1 m/sec and C=6.00m/sec

we get C*= 4.732m/sec

then a*/ao= Co/ [C*(C*-2U)]1/2

a = 1.73 m.

L*= 2/K*but K*g/(C*)2

K*= 0.438 and L*= 2 x 3.14/0.81 = 14.38m

when the waves travel in opposite direction L*decreases.

At U = C/4 the waves would steepen and tends to break, then this will be the critical

velocity for breaking.

the critical velocity U = C/4 = 6/4 = 1.5m/s


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APPENDIX "A"

Formulae for prediction of Breaker Depth/ height

The Majority of the existing formulas represent a relationship between the breaking wave

height (Hb) and the variables at the breaking or deepwater conditions, i.e., water depth atbreaking (hb), wavelength at breaking (Lb), bottom slope (m), deepwater wavelength (Lo), and

deepwater wave height (Ho). A brief review of the 20 breaker wave height formulas, that may

be used in general cases, are described as follows.

McCowan (1894),Hb= 0.78hb (1)

Miche (1944),

=

b

b

bbL

hLH 2

tanh142.0 (2)

Where Lbis the wavelength at the breaking point. Danel (1952)suggested changing the

coefficient from 0.142 to 0.12 when applying to the horizontal bottom.

Le Mehaute and Koh (1967)proposed an empirical formula based on three sources ofthe experimental data (Suquet, 1950; Iversen, 1952; and Hamada, 1963). The

experiments cover a range of 1/50 < m < 1/5 and 0.002 < Ho/ Lo< 0.093.

71m

41

oL

oHoH76.0bH

= (3)

Galvin (1969),performed laboratory experiments with regular wave on plane beach

combined his data with the data of Iversen (1952) and McCowan (1894). The breaking

criterion was developed by fitting empirical relationship between hb/Hband m.

07.085.640.1

1 = mform

hH bb (4.1)

07.092.0

>= mforh

H bb (4.2)

Collins and Weir (1969),derived a breaking height formula from linear wave theory and

empirically included the slope effect into the formula. The experimental data from three

sources (Suquet, 1950; Iversen, 1952; and Hamada, 1963)were used to fit the formula.

Hb= hb(0.72 + 5.6m) (5)

Goda (1970), analyzed several sets of laboratory data on breaking waves on slopesobtained by several researchers (Iversen, 1952; Mitsuyasu, 1962; and Goda, 1964)and

proposed a diagram presenting criterion for predicting breaking wave height. Then Goda(1974)gave an approximate expression for the diagram as

( )

+= 3/41515.1exp117.0 m

L

hLH

o

b

ob

(6)

Weggel (1972), proposed an empirical formula for computing breaking wave heightfrom five sources of laboratory data (Iversen, 1952; Galvin, 1969; Jen and Lin, 1970;


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Weggel and Maxwell, 1970; and Reid and Bretschneider, 1953). The experiments

cover a range of 1/50 < m < 1/5.

( )[ ]( )[ ]mhgT

mThH

b

b

b19exp175.43

5.19exp1/56.12

2

+

+= (7)

Komar and Gaughan (1972),used linear wave theory to derive a breaker height formulafrom energy flux conservation and assumed a constant Hb/hb. After calibrating the

formula to the laboratory data of Iversen (1952), Galvin (1969), and unpublished data

of Komar and Simons (1968), and the field data of Munk (1949),the formula was

proposed to be

5/1

56.0

=

o

o

obL

HHH (8)

Sunamura and Horikawa (1974),used the same data set as Goda (1970) to plot the

relationship between Hb/Ho, Ho/Lo, and m. After fitting the curve the following formulawas proposed

25.0

2.0

=

o

o

obL

HmHH (9)

Madsen (1976),combined the formulas of Galvin (1969) and Collins (1970)to be( ) 10.04.6172.0


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Singamsetti and Wind (1980),conducted a laboratory experiment and proposed twoempirical formulas based on their own data. The experiments cover a range of

1/401/5 and 0.02


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1984; and Hansen and Svendsen, 1979). The experiments cover a range of 1/45


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14.Singamsetti, S.R. and Sind, H.G. (1980). Characteristics of breaking and shoaling

periodic waves normally incident on to plane beaches of constant slope, Report M1371,

Delft Hydraulic Lab., Delft, The Netherlands.

15.Smith, J.M. and Kraus, N.C. (1990). Laboratory study on macro-features of wave

breaking over bars and artificial reefs, Technical Report CERC-90-12, WES, U.S. Army

Corps of Engineers.

16.

Sunamura, T. (1980). A laboratory study of offshore transport of sediment and a modelfor eroding beaches, Proc. 17

thCoastal Eng. Con., ASCE: 1051-1070.

17.

Sunamura, T. and Horikawa, K. (1974).Two-dimensional beach transformation due to

waves, Proc. 14th

Coastal Eng. Conf., ASCE, pp. 920-938.

18.Weggel J.R. (1972).Maximum breaker height, J. Waterways, Harbors and Coastal Eng.

Div., Vol. 98, No. WW4: 529-548.


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Fig.1 Refraction of waves approaching a straight coast

Fig.2 Refraction diagram, wave currents increase where wave orthogonals

converge, and decrease where orthogonals diverge


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Fig.3 Wave refraction diagram for Paradeep Port for waves from the

south with period of 8 secs

Fig.4 Diffraction of wave energy

Fig.5 Diffraction of waves by an impermeable breakwater


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Fig.6 Wave fronts and contour lines of maximum wave heights in the lee of a rigid breakwater, and waves

being incident normally. ( ____ ) exact solution. ( ------- ) approximate solution of Penny and Price (1952)

Fig.7 Variation of Kdwith distance parameter, r


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Fig.8 Isolines of approximate diffraction

coefficients for normal wave incidence and a

breakwater of length ten tomes the wavelengths. (Penney and Price (1952))

Fig.9 Isolines of approximate diffraction

coefficients for normal wave incidence and a

breakwater gap width of 2.5 wave lengths.(Penney and Price (1952))


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Fig.10 Wave height contours inside Madras harbour for incident wave of

T=10s, H=1m and =300with respect to North

Fig.11 Wave height contours inside Tangassery harbour for incident wave of T=8s, H=2m and

=450with respect to North


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Fig.12a Wave height contours inside Visakhapatnam harbour(existing configuration as on 1994) for

incident wave of T=7s, H=1.2m and =900with respect to North

Fig.12b Wave height contours inside Visakhapatnam harbour ( Proposed configuration) for incident

wave of T=7s, H=1.2m and =900with respect to North


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Wave Deformation


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Fig.13 Refraction of waves entering a current


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Wave Deformation


3.Wave Deformation.pdf

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