.wave deformation

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

2C1C

2sin1sin=

θθ (1)

TLC = (2)

where C1, and C2, are the celerities at the points where θ has the values θ1 and θ2.

This is the Snell’s law. The simplest case of plane waves approaching a straight coast

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


2

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

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

refracted in such a way that the angle θ 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

θθ

= (3)

Where bo is 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. 2 shows 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 refraction

diagrams.


3

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 = 21 ⎥

⎦

⎤⎢⎣

⎡+

kd2sinhkd21

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

0C0E0b21

0P = ( 5)

(4)= (5) leads to

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

C0C

b0b

n1

21

0EE (6)

we know 02H

2H

8/02Hg

8/2Hg

0EE

=ρ

ρ= (7)

Hence b0b

.C0C

n1

21

0HH

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= (8)

Where,

==⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

sKC0C

n1

21 Shoaling coefficient and

b0b

= RK = Refraction coefficient

KS is 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 height

when 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.


4

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 and

normal to the breakwater as shown. The regions may be idealized as follows.

(1) The region 0 < θ < θo is 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 θo < θ < (θo + π) 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 (θo + π) < θ < 2π is 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, Kd which the ratio of diffraction wave height to incident wave

height are shown in Table.1 in 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.


5

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

2r2

16r2

Lx β

+β

= …. (10)

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

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

the complete solution.

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

plots showing the wave height distribution inside Madras Harbour, Tangassery

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

Diffraction of waves around breakwaters


6

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, 142.0LH> (Deep waters)

142.0LH> 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. 2T2g

oLπ

=

We know at point of breaking

8.0bdbH=

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.

T2g2gH

161

oC2oHg

161

oEπ

ρ=ρ=

and bgdb2gH81

bCb2Hg81

bE ρρ ==

As Eo = Eb and db = Hb/0.8

5/2g25.1

gT2

oH2bHoH

⎭⎬⎫

⎩⎨⎧

=π

or 5/1

oHoL

50.0bHoH

⎟⎟⎠

⎞⎜⎜⎝

⎛=


7

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

found out as follows.

( ) ( ) 2/1ocos2/1gd2/oCoHbH θ= (11)

where Hb is breaker wave height

Based on the condition Hb = 0.78d,

2/1

ocosbgd2

oCoHbKd

⎥⎥⎦

⎤

⎢⎢⎣

⎡= θ (12)

Solving for db yeilds

5/2

2ocosoCo2H

5/4K5/1g

1bd ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛=

θ (13)

where

( ) ( )2gTbH

mambk −=

( ) ⎟⎠⎞⎜

⎝⎛ −−= m19e0.18.43ma

( )1m5.19e156.1mb

−⎟⎠⎞⎜

⎝⎛ −+=

and m = Beach slope.

As m approaches zero k=0.78.

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

shore to the shoreline is 5/2

ooo2

5/45/1b

b 2cosCH

kmg1

md

x ⎟⎟⎠

⎞⎜⎜⎝

⎛ θ== (14)

Finally, 5/2

ooo25/1

bb 2cosCH

gkxkmH ⎟⎟

⎠

⎞⎜⎜⎝

⎛ θ⎟⎟⎠

⎞⎜⎜⎝

⎛== (15)


8

Medium Slope

Medium H/L

Steep H/L

Steep Slope

Low H/L

White water Foam

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

oLoH

tan⟨=

θ

N2 = 4.0

oLbH

tan⟨=

θ

N1 > 0.5 to 3.3 N2 > 0.4 to 2.0

N1 >3.3 N2 >2.0


9

Galvin (1972) suggested that breaker type depend on the parameter θ2tanoL

oH

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

with plane concrete beaches, he obtained.

Parameter Spilling Plunging Surging

(Ho/Lo)/(tan2β)

(Ho/Lo)/(tanβ)

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 = ω/k

Where

C : wave celerity

ω : wave frequency

0.09 0.00048

4.8 0.011


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k : wave number, 2π/L

so ω = 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 ω*, given by:

ω* = C*k = (C+U) k = ω + k *U

Where ω* : 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,

( ) ( )kgC,

*kg*C 22 == (17)

Hence, from eq. (16)

( )C

*CU*k

k2C

2*C +==

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


11

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++=

2/1

CU411

21

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. Then C* and L* decreases.

If the opposing current has a magnitude exceeding a quarter of the wave velocity in

still water, i.e. U < 4c− then no solution of eq. (18) is possible. At this critical

point, U = -C/4 and C* = C/2, while, the local group velocity U2*CCG −== . Thus,

the local CG is equal and opposite to the current velocity and no wave energy can be

propagated against the current. The waves would, in fact, steepen and break before

the limiting velocity is reached.

The change in wave height can be obtained from the principle of conservation of

wave action, defined by E/σ. In the present case the principle leads to the equation

o2CE

21const*CC

21U*E ==⎟

⎠⎞

⎜⎝⎛ + ∗ (19)

where E* and E are the energy per unit area of the waves in the presence and absence

of the current respectively. Since,

o2ga21E,

2*ga21*E ρ=ρ=

it can be shown that

( )[ ] 2/1U2*C*C

oCa*a

+= (20)


12

Where a* and a are the corresponding wave amplitudes. With an opposing current,

U<0, it is seen that the amplitude increases and approaches infinity as the

velocity U approaches –C/2 the limit found above. If the depth is shallow (d/L <

1/20), equation (16), (19) are still valid, that C* and C must be obtained from the

equation involving tanh kd.

Calculation of Wavelength when both waves and currents coexist. In Shallow Waters ( )( ) aTgdL U +−= αδcos In deep waters

( ) 2

0

2

cos411

41

2

2/

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −++=

=

=

CLL

gTC

gTL

o

ao

ao

U αδ

π

π

Lo : Deep water length, L : Wave length

G : gravitational Constant, Ta : absolute period

U : Current velocity

(δ-α) : Angle between wave orthogonal and current direction = 0 if waves & current are in

the same direction

d : Water depth

Co : Deep water celerity

Refraction of waves entering a current

In a general case, the waves may approach the current at an angle and the current

velocity may vary across it. Fig.13 illustrates the case of waves travelling at an angle

θ to the y axis meeting a current U(y), the velocity of which varies in the y direction.

In this case the waves may be refracted as well as undergo changes in the velocity

and wave length. Let k, C and θ be the values of wave number, wave velocity and


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angle to the y axis where U = 0 and k*, C* and θ* be the corresponding values where

the current has a finite value, U. Applying the same principles to this case, we have

k*sinθ* = k sinθ (Snell’s law) (21)

k*(C* + Usinθ*) = k* C* = σ (22)

Assuming that the waves are in deepwater, eq.(17) holds. Then from Eq. (17), (21)

and (22) it may be shown that

θsin

CU1

1C*C

⎟⎠⎞

⎜⎝⎛−

= (23)

2

sinCU1

k*k

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−= θ (24)

sinθ*= ( ) ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

− 2sinC/U(1

sin

θ

θ (25)

Since sinθ* cannot exceed unity, there is an upper limit to U for which a solution exists, i.e.,

( )θθ−

≤sinsin1

CU 21

(26)

At this limit sinθ* = 1, so that θ* = π/2 and the waves are totally reflected by the current.

If the current is in the opposite direction i.e., U< 0 there is no limit on its magnitude but as U

approaches zero. In other words, the wavelength becomes very small and the direction of

propagation of the waves is nearly normal to the current.

The change in wave height may be obtained as,

2121

2sin2sin

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∗

∗∗

θθ

EE

aa (27)

As in the previous case considered, in shallow water, the appropriate equation for the wave

velocity c must be used in place of eq. (17) and equations (23) onward will be modified.


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When a spectrum of waves encounters a current, the various components must be considered

separately, since the changes depend on the velocity in still water, which is a function of

frequency. The waves of shorter wavelength will be affected to a greater extent. If the wave

spectrum enters an adverse current, either head-on or obliquely, the shorter waves may reach

breaking point producing a very choppy sea. The longer waves although modified may be

able to pass through the current without breaking so that the wave field emerging on the far

side will appear smoother.

REFERENCES

1. Bowden, K.F., ‘Physical oceanography of coastal waters’, Ellis Horwood

Limited, 1983.

2. Shore protection Manual, U.S. Army Coastal Engineering Research Centre,

Vol.II, 1983.

3. Dean, R.G. and Dalrymple R.A., ‘Water wave Mechanics for Engineers and

Scientists’. 1983.

4. Wiegel, R.L. ‘Oceanographical Engineering’.Prentice Hall Inc

5. Penney, W.G. and Price, A.T. (1952) “The diffraction theory of sea waves and

the shelter afforded by breakwaters”, Philos. Trans. Roy. Soc. A, Vol. 244 (882)

pp 236-253.


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WORKED OUT EXAMPLES

Problem 1

A 10 second period wave having a height of 5m propagates from deep to shallow

waters. Assuming that the bottom contours are parallel to each other, compute the

wave height at a water depth of 10m.

Solution

Lo = 1.56 T2 = 1.56 * 100 = 156 m

0642.0Ld

o

=

Corresponding ,1082.0Ld= (from wave table)

.sec/m6.15TLC o

o ==

.sec/m24.910

4.92C,m4.92L ===

31.110*065.0*2kd2,065.0L2K ===π

=

88.0kd2sinh

kd2121n =⎥

⎦

⎤⎢⎣

⎡+=

9828.088.0*2

124.96.15

n21

CC

HH o

o

==

Therefore wave height,H at d = 10m is 0.9828 * Ho = 4.91 <Ho


16

Problem 2

A deep water wave of height 3.5m and period 10 secs is refracted so that the distance

between the orthogonals is reduced by fifty percent at the depth of 10m and reduced

by 20% at 5m water depth. What will be the height of the wave here assuming no

energy losses.

Solution

rKsKoH

H=

2bob,cesin2

bob

rK,CoC

n1

21

sK ====

m1562T56.1oL ==

1082.0Ld,064.0

oLd

==

9828.0CoC

n21

sK ==

389.12*9828.0oH

H==

Since Ho = 3.5

H=3.5 * 1.389 = 4.86 m >3.5(Ho) for d=10m

Where as breaking wave height Hb > 0.78d For the above problem, if refraction is not considered, H / Ho =Ks =0.9828, Hence H=0.9828* Ho= 3.43m < Ho of 3.5m (b) H at d=5m


17

Kr = 8.0

1

8.0

18.66.15

9349.0*21

5.3H

=

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/Lo values 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 .1278 25 .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


18

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 tanβ = 6/500 = 0.012. Lo = 1.56T2 = 1.56(72) = 76.44m. (a) Komar & Gaughen(1973):

( ) 5/1oL/oH

563.0

oHbH=


19

( )m22.3

5/144.76/3

563.0x3bH ==

Breaker depth = bd78.0bH = Xb = 344m (b) LeMehaute & koh(1967):

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

oHbH −= β

( ) 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

oHbH −= β

( ) ( ) 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

oLbH

βπ

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

+−−=

0.7859 =e-0.0592 b; for Hb = 2.782m


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db= 4.0693m (f) We have

[ ] ( ) 2/1ocos2/1gd2/oCoHbH θ=

[ ] ( ) 2/180cos2/16x81.92/7x56.1x3bH = m054.1bH =

5/2

2/ocosoC2oH

5/4k5/1g

1bd ⎥⎦

⎤⎢⎣⎡= θ

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 = tanβ 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 = ½ = 0.5m

by substituting the values of C*, Co , U we get the value of a* a* = 0.8 m.

*CT

*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 = ½ [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* = 2π /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 at breaking (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

bbb L

hLH

π2tanh142.0 (2)

Where Lb is 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 of

the 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.

71m41

oLoH

oH76.0bH−

⎟⎟⎠

⎞⎜⎜⎝

⎛= (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/Hb and m.

07.085.640.1

1≤

−= mfor

mhH 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 slopes

obtained 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

Lh

LHo

bob

π (6)

♦ Weggel (1972), proposed an empirical formula for computing breaking wave height

from 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.

( )[ ]

( )[ ]mhgTmTh

Hb

bb 19exp175.43

5.19exp1/56.12

2

−−+−+

= (7)

♦ Komar and Gaughan (1972), used linear wave theory to derive a breaker height formula

from 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

oob L

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 formula was proposed

25.0

2.0−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

o

oob L

HmHH (9)

♦ Madsen (1976), combined the formulas of Galvin (1969) and Collins (1970) to be ( ) 10.04.6172.0 <+= mformhH bb (10) ♦ Black and Rosenberg (1992) found that the formula of Madsen (1976) gives good

predictions for individual breaking wave height in laboratory and field experiments. ♦ Ostendorf and Madsen (1979), modified the formula of Miche (1944) by including the

beach slope in to the formula. After calibrating to the laboratory data, the Miche (1944)’s formula was modified to be

( ) 1.02

58.0tanh14.0 ≤⎥⎦

⎤⎢⎣

⎡+= mfor

Lh

mLHb

bbb

π (11.1)

( )( ) 1.02

1.058.0tanh14.0 ≤⎥⎦

⎤⎢⎣

⎡+= mfor

Lh

LHb

bbb

π (11.2)

♦ Sunamura (1980), conducted an empirical formula based on an analysis of various

laboratory data (Iversen, 1952; Bowen et al ., 1968; Goda, 1970 and Sunmaura, 1980) and obtained the following formula

6/1

00 /1.1

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

LHmhH bb (12)


25

♦ Singamsetti and Wind (1980), conducted a laboratory experiment and proposed two empirical formulas based on their own data. The experiments cover a range of 1/40<m>1/5 and 0.02 <H0 /L0<0.065.

254.0

0

0031.00575.0

−

⎥⎦

⎤⎢⎣

⎡=

LH

mHHb (13)

and

13.0

0

0155.00937.0

−

⎥⎦

⎤⎢⎣

⎡=

LH

mhHb (14)

♦ Ogawa and Shuto (1984), obtained the following formula from the same data sets as

Goda (1970). The formula is limited to use for the range of 1/100 < m<1/10 and 0.003<H0L0<0.065.

25.0

0

009.0068..0

−

⎥⎦

⎤⎢⎣

⎡=

LH

mHHb (15)

♦ Larson and Kraus (1989), developed a breaking criterion based on the large wave tank

data of Kajima et al. (1983). The breaking height index Hb/hb was related to the deepwater wave steepness and the local beach slope seaward of the breaking point.

21.0

00 /14.1

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

LHmhH bb (16)

♦ Hansen (1990), used the laboratory data from Van Dorn (1978) and unpublished data of ISVA to plot the relationship between Hb/hb and mLb/hb and proposed the following empirical formula

2.0

05.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

b

bbb h

LmhH (17)

♦ Smith and Kraus (1990), proposed 2 empirical formulas based on the analysis of 11

sources of laboratory data performed on plane beach conditions. The experiments cover a range of 1/80 <m<1/10 and 0.001 <Ho/Lo<0.092.

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧

−−−−+

=o

obb L

Hm

mhH 43exp10.5

60exp112.1 (18)

and

( )m

o

oob L

HmHH

88.030.0

47.234.0+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+= (19)

♦ Gourlay (1992), proposed an empirical formula based on seven sources of laboratory data

(Bowen et al., 1968; Smith, 1974; Visser, 1977; Gourlay, 1978; Van Dorn, 1976; Stive,


26

1984; and Hansen and Svendsen, 1979). The experiments cover a range of 1/45 <m<1/10 and 0.001 <Ho/Lo<0.066. The data was used to plot the relationship between Hb/Ho and Ho/Lo, the curve fitting yields

28.0

478.0−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

o

oob L

HHH (20)

CONCLUSIONS A total of 574 cases of wave height data from 24 sources of published experimental results were used to examine and compare the accuracy of the 20 existing breaker height formulas by Rattanapitikon and Vivattanasirisak(2000). It was reported by them that the formula of Eq(8) gives the overall predictions with a maximum error of 11.2 %. The best formulas for predicting the breaking wave heights on the bottom slopes of m = 0, 0 < m ≤ 0.07, 0.07 < m ≤ 0.1, and m > 0.10 are the formulas of Eqtions (8), (6),(11)and (8), respectively. REFERENCES 1. Collins, J.I. and Weir, W. (1969). Probabilities of wave characteristics in the surf zone,

Tetra Tech. Report, TC-149, Pasadena, California, USA. 2. Galvin, C.J. (1969). Breaker travel and choice of design wave height, J. Waterways and

Harbors Division, ASCE, Vol.95, No. WW2: 175-200. 3. Goda, Y. (1970). A synthesis of breaker indices, Trans. JSCE, Vol.2: 227-230. 4. Gourlay, M.R. (1992). Wave set-up, wave run-up and beach water table: Interaction

between surf zone hydraulics and groundwater hydraulics, Coastal Engineering, Vol. 17: 93-144.

5. Komar, P.D. and Gaughan, M.K. (1972). Airy wave theory and breaker height prediction, Proc. 13rd Coastal Eng. Conf., ASCE: 405-418.

6. Larson, M. and Kraus, N.C. (1989). SBEACH: Numerical model for simulating storm-induced beach change, Report 1, Tech. Report CERC-89-9, Waterways experiment Station, US Army Crops of Engineers.

7. Le Mehaute, B. and Koh, R.C.Y. (1967). On the breaking of waves arriving at an angle to the shore, J. Hydraulic Research, Vol.5, No. 1: 67-88.

8. Madsen, O.S. (1976). Wave climate of the continental margin: Elements of its mathematical description, D.J. Stanley and D.J.P. Swift (Editors), Marine Sediment Transport in Environmental Management. Wiley, New York: 65-87.

9. McCowan, J. (1894). On the highest waves of a permanent type, Philosophical Magazine, Edinburgh, Ser. 5, Vol.38: 351-358.

10. Miche, R.(1944). Mouvements ondulatoires des mers en profondeur constante on decroissante, Ann.des Ponts et Chaussees, Ch.114: 131-164, 270-292 and 369-406.

11. Okawa, Y. and Shuto, N. (1984). Run-up of periodic waves on beaches of non-uniform slope, Proc. 19th Coastal Eng. Conf., ASCE, pp. 328-344.

12. Ostendorf, D.W. and Madsen, O.S. (1979). An analysis of longshore current and associated sediment transport in the surf zone, Report No. 241, Dept. of Civil Eng., MIT.

13. Rattanapitikon,W. and Vivattanasirisak,T.(2000). Examination of Breaker Height Formulas. Proc.12th Congress of the Asia & Pacific Division of IAHR, Bangkok, 13-16 Nov,pp 425-434.


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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 model for eroding beaches, Proc. 17th Coastal 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 Kd with distance parameter, βr


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Fig.8 Isolines of approximate diffraction coefficients for normal wave incidence and a breakwater of length ten tomes the wave lengths. (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 θ=300 with respect to North

Fig.11 Wave height contours inside Tangassery harbour for incident wave of T=8s, H=2m and θ=450 with 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 θ=900 with respect to North

Fig.12b Wave height contours inside Visakhapatnam harbour ( Proposed configuration) for incident wave of T=7s, H=1.2m and θ=900 with respect to North


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Fig.13 Refraction of waves entering a current


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.wave deformation

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