.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
2C
1C
2sin
1sin
=
θ
θ
(1)
T
L
C = (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.

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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
cos
obb
θ
θ
= (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.

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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 =
2
1
⎥
⎦
⎤
⎢
⎣
⎡
+
kd2sinh
kd2
1
For deep water conditions, (2kd/sinh 2kd) → 0 and hence
0C0E0b
2
1
0P = ( 5)
(4)= (5) leads to
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
C
0C
b
0b
n
1
2
1
0E
E
(6)
we know
0
2H
2H
8/0
2Hg
8/2Hg
0E
E
=
ρ
ρ
= (7)
Hence
b
0b
.
C
0C
n
1
2
1
0H
H
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
= (8)
Where,
==⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
sK
C
0C
n
1
2
1
Shoaling coefficient and
b
0b
= 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.

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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.

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5
The wave diffraction is governed by the Helmholtz equation given by
( ) 0y,xF2k
2y
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.
L
y
2
r
2
16
r
2
L
x 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

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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, 142.0
L
H
> (Deep waters)
142.0
L
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. 2T
2
g
oL
π
=
We know at point of breaking
8.0
bd
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.
T
2
g2gH
16
1
oC2
oHg
16
1
oE
π
ρ=ρ=
and bgdb
2gH
8
1
bCb
2Hg
8
1
bE ρρ ==
As Eo = Eb and db = Hb/0.8
5/2
g25.1
gT
2
oH2
bH
oH
⎭
⎬
⎫
⎩
⎨
⎧
=
π
or
5/1
oH
oL
50.0
bH
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.
( ) ( ) 2/1
ocos
2/1
gd2/oCoHbH θ= (11)
where Hb is breaker wave height
Based on the condition Hb = 0.78d,
2/1
ocos
bgd2
oC
oHbKd
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
= θ (12)
Solving for db yeilds
5/2
2
ocosoCo
2H
5/4K5/1g
1
bd
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
θ
(13)
where
( ) ( )
2gT
bH
mambk −=
( ) ⎟
⎠
⎞⎜
⎝
⎛ −−= 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 β, the beach slope, the distance from the
shore to the shoreline is
5/2
ooo
2
5/45/1
b
b
2
cosCH
kmg
1
m
d
x ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ θ
== (14)
Finally,
5/2
ooo
25/1
bb
2
cosCH
g
k
xkmH ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ θ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
== (15)

Wave Deformation
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
oL
oH
tan
〈=
θ
N2 = 4.0
oL
bH
tan
〈=
θ
N1 > 0.5 to 3.3
N2 > 0.4 to 2.0
N1 >3.3
N2 >2.0

Wave Deformation
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)/(tan2
β)
(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

Wave Deformation
10
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,
( ) ( )
k
g
C,
*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

Wave Deformation
11
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
++=
2/1
C
U4
11
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. 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 <
4
c−
then no solution of eq. (18) is possible. At this critical
point, U = -C/4 and C*
= C/2, while, the local group velocity U
2
*C
CG −== . 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
o
2
CE
2
1
const*CC
2
1
U*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,
o
2ga
2
1
E,
2*ga
2
1
*E ρ=ρ=
it can be shown that
( )[ ] 2/1U2*C*C
oC
a
*a
+
= (20)

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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
cos
411
4
1
2
2/
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
++=
=
=
C
LL
gT
C
gTL
o
a
o
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

Wave Deformation
13
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
C
U
1
1
C
*C
⎟
⎠
⎞
⎜
⎝
⎛
−
= (23)
2
sin
C
U
1
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.,
( )
θ
θ−
≤
sin
sin1
C
U
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
2sin
2sin
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= ∗
∗∗
θ
θ
E
E
a
a
(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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14
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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15
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
0642.0
L
d
o
=
Corresponding ,1082.0
L
d
= (from wave table)
.sec/m6.15
T
L
C o
o ==
.sec/m24.9
10
4.92
C,m4.92L ===
31.110*065.0*2kd2,065.0
L
2
K ===
π
=
88.0
kd2sinh
kd2
1
2
1
n =⎥
⎦
⎤
⎢
⎣
⎡
+=
9828.0
88.0*2
1
24.9
6.15
n2
1
C
C
H
H o
o
==
Therefore wave height,H at d = 10m is 0.9828 * Ho = 4.91 <Ho

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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
rKsK
oH
H
=
2
b
ob
,cesin2
b
ob
rK,
C
oC
n
1
2
1
sK ====
m1562T56.1oL ==
1082.0
L
d
,064.0
oL
d
==
9828.0
C
oC
n2
1
sK ==
389.12*9828.0
oH
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

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17
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/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.,
.kdtanh142.0
L
H
≥
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

Wave Deformation
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):
( ) 5/1
oL/oH
563.0
oH
bH
=

Wave Deformation
19
( )
m22.3
5/144.76/3
563.0x3
bH ==
Breaker depth = bd78.0bH =
Xb = 344m
(b) LeMehaute & koh(1967):
( ) ( ) 4/1
oL/oH.7/1tan76.0
oH
bH −= β
( ) 4/144.76/34/1)012.0(x3x76.0bH −=
m74.2bH =
m6.290bx,m50.378.0/74.2bd ===
(c) Sunamura(1983):
( ) ( ) 4/1
oL/oH.5/1tan
oH
bH −= β
( ) ( ) 25.044.76/32.0012.0x3bH −=
m297bx,m56.3bd;783.2bH ===
(d) Munk(1949):
( ) 3/1
oL/oH3.3/oHbH =
( ) 3/144.76/33.3
3
oH =
m675.2bH =
m43.3bd =
bx = 286m
(e) Godal(1975):
⎥⎦
⎤
⎢⎣
⎡
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞⎜
⎝
⎛ +−−= 3/4tan151oL/bd5.1exp1A
oL
bH
βπ
( ){ }[ ]0027.0*15144.76/bd712.4exp117.0
44.76
bH
+−−=
0.7859 =e-0.0592
b; for Hb = 2.782m

Wave Deformation
20
db= 4.0693m
(f) We have
[ ] ( ) 2/1
ocos
2/1
gd2/oCoHbH θ=
[ ] ( ) 2/180cos
2/1
6x81.92/7x56.1x3bH =
m054.1bH =
5/2
2/ocosoC2
oH
5/4k5/1g
1
bd
⎥⎦
⎤
⎢⎣
⎡= 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.

Wave Deformation
21
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.
*C
T
*L
=
L*
= 6.87 x 8
= 54.96m
when the waves travel in the same direction of current L*
increases

Wave Deformation
22
(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

Wave Deformation
23
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
b
bb
L
h
LH
π2
tanh142.0 (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.
71m
41
oL
oH
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.
07.0
85.640.1
1
≤
−
= mfor
m
hH bb (4.1)
07.0
92.0
>= mfor
h
H b
b (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
( )
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
+−−= 3/4
1515.1exp117.0 m
L
h
LH
o
b
ob
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;

Wave Deformation
24
Weggel and Maxwell, 1970; and Reid and Bretschneider, 1953). The experiments
cover a range of 1/50 < m < 1/5.
( )[ ]
( )[ ]mhgT
mTh
H
b
b
b
19exp175.43
5.19exp1/56.1
2
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
o
ob
L
H
HH (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
o
ob
L
H
mHH (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
( ) 1.0
2
58.0tanh14.0 ≤⎥
⎦
⎤
⎢
⎣
⎡
+= mfor
L
h
mLH
b
b
bb
π
(11.1)
( )( ) 1.0
2
1.058.0tanh14.0 ≤⎥
⎦
⎤
⎢
⎣
⎡
+= mfor
L
h
LH
b
b
bb
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
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
LH
m
hH bb (12)

Wave Deformation
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
0031.0
0575.0
−
⎥
⎦
⎤
⎢
⎣
⎡
=
L
H
mHHb (13)
and
13.0
0
0155.0
0937.0
−
⎥
⎦
⎤
⎢
⎣
⎡
=
L
H
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
009.0
068..0
−
⎥
⎦
⎤
⎢
⎣
⎡
=
L
H
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
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
LH
m
hH 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
b
bb
h
L
mhH (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
o
bb
L
H
m
m
hH 43exp10.5
60exp1
12.1
(18)
and
( )
m
o
o
ob
L
H
mHH
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,

Wave Deformation
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
o
ob
L
H
HH (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.

Wave Deformation
27
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.

Wave Deformation
28
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

Wave Deformation
29
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

Wave Deformation
30
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

Wave Deformation
31
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))

Wave Deformation
32
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

Wave Deformation
33
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
Fig.12b Wave height contours inside Visakhapatnam harbour ( Proposed configuration) for incident
wave of T=7s, H=1.2m and θ=900

Wave Deformation
34
Fig.13 Refraction of waves entering a current

Wave Deformation
35

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