GENERAL ¨ ARTICLE

Finite Amplitude Ocean Waves

Waves with Peaked Crests and Broad Troughs

K K Varma

When amplitudes are small, ocean waves are rep-

resented by sinusoidal functions. There are, how-
ever, situations when this treatment of waves is
inadequate. In such situations, finite amplitude
wave theories that deal with waves having peaked
crest have to be used. In this article, selected fea-
tures of a variety of such wave profiles are pre-
sented. K K Varma was Professor
and Dean i/c, College of
1. Introduction Fisheries, Kerala
Agricultural University.
Waves are the undulations of the sea surface. The most After retirement in 2007,
commonly observed waves on ocean surface are those he worked for 3 years as a
generated by wind forcing. In the beginning, small rip- contract faculty and for 1
ples appear on the sea surface and these grow further year as Visiting Professor,
under UGC scheme, in the
by extracting energy from prevailing winds. There are Department of Physical
different ways in which waves are classified and one of Oceanography, Cochin
them is based on its relative amplitude, as small am- University of Science and
plitude waves and finite amplitude waves. This article Technology. He has also
provides a brief introduction to finite amplitude wave worked in the National
Institute of Oceanography
theories. Some of the general characteristics of waves as and in Engineers India
well as the importance of finite amplitude wave theories Ltd. He has carried out
are touched upon. research on several aspects
of physical oceanography.
2. Small Amplitude Waves
The topmost and the lowest levels of the waves are re-
spectively called the crest and trough of the wave. The
horizontal distance between successive identical points,
say two crests, is the wavelength (L). Wave height (H) is
Keywords
the vertical distance between crest and trough. The ver- Nonlinear waves, particle mo-
tical distance from mean position to crest or to trough tion, wave steepness, Stokes
is the amplitude (a), which in this case is half of wave drift, translatory wave, offshore
structures.

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 GENERAL ¨ ARTICLE

As the waves height. The surface profile is given by

progress, the water η = a sin (kx − σt) or,
particle motion is π
either circular or if there is phase shift by , then,
2
elliptical. η = a cos (kx − σt) .
Here η is the departure from undisturbed level, a is the
amplitude, k is wave number (1/L or 2π/L, L is wave-
length), and σ is frequency (1/T or 2π/T , T is the pe-
riod). The wave speed is given by L/T . This is an
example of a sinusoidal wave. In the context of waves
on the surface of fluid, these waves (also known as airy
waves) are based on Bernoulli’s equation for irrotational
fluid motion
∂ϕ P 1
− + + (u2 + w2) + gz = 0 , (1)
∂t ρ 2
where ϕ is the potential function of flow, p is pressure,
ρ is the density of fluid, u and w are the horizontal and
vertical velocities, respectively, g is the acceleration due
to gravity, and z is the depth. The higher order terms
of velocities can be neglected for small amplitude waves
and then the equation becomes
∂ϕ P
− + + gz = 0 (2)
∂t ρ
Hence, small amplitude waves are also called linear waves.
Most of the aspects of the ocean waves can be explained
by the small amplitude wave theory.
Let us now see the water particle motion due to waves.
While wave energy is carried by the wave as it progresses
forward, the water particles oscillate up and down. How-
ever, it is not merely an up and down movement. It is
either circular or elliptical movement. If the depth of
the water column is more than half of wavelength, then
waves are known as deep water waves. In the case of
such waves, particle motion is circular. On the other
hand, if the depth of the water column is less than half
of wavelength and more than 1/20 of wave length, they

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(a) (b)

Shallow water waves or intermediate

water waves Deep water waves

are known as intermediate or transitional water waves Figure 1. Motion of particles

and if the depth of the water column is less than 1/20 of in small amplitude waves.
wavelength, they are called shallow water waves. In the (adapted from [1]). Radius
case of both these waves, the particle motion is elliptical. of the circular motion of the
Particle motions are shown in Figure 1. deep water waves (a) de-
creases exponentially with
The velocity of waves is generally referred to as wave depth. In (b) the ellipse be-
celerity. For small amplitude waves, celerity, c is given comes flatter as depth in-
by creases, and at the sea bot-

tom, the movement is to and
g fro.
c= tanh kh (3)
k
k is the wave number and h is the depth of the water
column.
g
This equation gets simplified to c = k
for deep wa-
ter waves because tanh kh can be approximated as 1
√ h. For shallow water waves, celerity becomes
for large
c = gh as the hyperbolic term tends to kh and for
intermediate water waves, (3) is to be used. Deep and
intermediate water waves are dispersive as the velocity
of these depends on wavelength. This is not the case
with shallow water waves and they are nondispersive.

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In the case of finite 3. Finite Amplitude Waves

amplitude waves, the
The basic feature of the finite amplitude waves is that
amplitude cannot be
the amplitude cannot be considered as small in compar-
considered as small
ison with either the wavelength or the depth of water
in comparison with
column. Therefore, the higher order terms in (1) cannot
wave length or water be neglected and thus, they are also known as nonlinear
depth. waves. A consequence of considering the full form of (1)
is that the wave profiles are not sinusoidal.
One of the main characteristics of the finite amplitude
waves is that they do not have sinusoidal shape and in-
stead have peaked crests and broad troughs. Also, the
upward amplitude, i.e., towards the crest is more than
the downward amplitude, i.e., towards the trough. The
retention of nonlinear terms makes the solution of the
equation difficult. As a result of this, there is no unique
theory that is applicable to all depth regions from deep
to very shallow water. It must be noted that in the case
of small amplitude waves, a single theory is applicable
irrespective of water depth. On the other hand, different
finite amplitude wave types are applicable to waves at
different depths. For deep water locations, Stokes wave
or trochoidal wave is applicable. If water depth is less,
it is Cnoidal wave; and in very shallow waters, solitary
wave is found to be suitable (see Table 1). Another as-
pect is that different wave parameters gain importance
in different depth regions. If the depth is more than half
wavelength, then the important parameters are wave-
length and wave steepness, given by H/L. In regions
Table 1. Applicability of dif-
ferent finite ampliude wave
theories.
Water depth in relation Theories
L, H and h are wavelength,
wave height and water depth
to wave length
respectively.
*
It must be noted that the h > L/10 Stokes and trochoidal
depth limits are not exact L/10 > h > L/50 Cnoidal
and theories will overlap at h < L/50 Solitary
the fringe regions.

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Figure 2. Trochoidal wave

(adapted from [2]).The small
circle is part of the large circle
that rolls below the plane. The
path traced by the point A has
broad troughs and peaked
crests.

with depth between half and 1/20 of wavelength, H/L

and water depth, h are important, and in still shallower
regions, the depth of water and H/h become important
parameters. Some basic features of these three types of
finite amplitude waves are now discussed.
3.1 Trochoidal Wave
This type of wave derives its the name from its shape.
Trochoid is the shape traced by a point on a circle as
the circle rolls over or under a surface (Figure 2). In the
figure, the rolling circle and the point on it are shown.
If the distance of the point from the centre of the circle
decreases, the shape tends to be sinusoidal, while if it
increases to the radius of the circle, the crest becomes
too pointed. It may be noted that these waves are also
called Gerstner waves (after F Gerstner who propounded
the theory behind these waves).
2
The height of the crest is H2 + πH 4L0
and the depth of
H πH 2
trough is 2 − 4L0 , where L0 is deep water wavelength
and H is wave height. From the above, it is clear that
the vertical distance from the mean level to the crest
is more than half wave height, but the vertical distance
from mean level to trough is less than half wave height.
Though this theory gives an exact solution, it does not
satisfy the irrotationality condition.
3.2 Stokes Wave
This wave has a broad trough and somewhat peaked
crest (Figure 3a). Stokes presented a method for the
solution of the above nonlinear equation of velocity

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Figure 3. Different finite am-

plitude wave profiles (adapted
from [1]).

potential. Hence, these are known as Stokes waves.

The equation is solved as a series, using perturbation
method. It is assumed that the solution can be repre-
sented in terms of a power series expansion of a small
quantity and this quantity is related to wave steepness,
which is the ratio of wave height to wavelength. The
sum up to the nth order term is the nth order solution.
Solution is, therefore, not unique and as the order in-
creases, the resulting wave becomes more peaked. The
solution satisfies the irrotationality condition. It may be
noted that the method requires a number of numerical
calculations, which increases as the order of the solution
increases. Though Stokes presented the second order
solution, subsequent researchers have worked out higher
order solutions.
The first order solution gives results for wave profile and
wave celerity that are similar to those for small ampli-
tude waves. In the case of second order solution, though
the expression for wave celerity does not change, the
wave profile is complicated and is given by
H
η= cos (kx − σt)
2
π H 2 cosh kh(2 + cosh 2kh)
+ cos2 (kx − σt) . (4)
8 L (sinh kh)3

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3.3.1 Stokes Drift: In the case of Stokes waves, there There will be a net
is a net transport of water as the wave passes. At this transport of water
stage, it is necessary to consider the particle motion as due to the Stokes
presented earlier for small amplitude waves, i.e., the wa- wave.
ter particles return to their original position at the end
of the passage of a sinusoidal wave. In other words,
particle motion is in closed circles in deep water waves.
However, on examining the horizontal and vertical com-
ponents of particle motion for Stokes wave, it can be
seen that the horizontal component has a non-periodic
term. Hence, the circle is not a closed one and there
is net transport of water in the direction of progress of
wave. This is known as Stokes drift. One of the expres-
sions for Stokes drift presented by Weigal [3] is
   πH  4πZ
u = πHT L
e L , (5)
where, H, T and L are height, period and length of the
wave, respectively; and z is the depth at which drift is
estimated. The depth axis is positive upwards. Thus,
the drift decreases rapidly with the depth. It can be said
that the Stokes drift at surface is about 1% of the wind
speed that generates the waves.
3.3 Cnoidal Wave
If the depth decreases, the theories mentioned earlier are
not the appropriate ones to be used. It is the cnoidal
wave theory that is more applicable. In this theory, Ja-
cobian elliptical function (cn) is used. Hence, this is
called as cnoidal, a word analogous to sinusoidal. Ja-
cobian elliptical function is periodic and its modulus
lies between 0 and 1, both inclusive. The shape of the
cnoidal wave is characterized by very flat troughs and
peaked crests (Figure 3b). The wave profile is given by
 
2 2π 2π
η = zt + Hcn [2K(k) x− t , k] . (6)
L T
Here, cn is the Jacobian elliptical function, K(k) is the
elliptical integral of first order with modulus k, zt is the

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Solitary wave is a height of trough from sea bottom and H is the wave
single wave crest height. When the modulus k is zero, the wave profile
alone, that becomes sinusoidal. At the other extreme, i.e., when k
translates forward. is 1, the period will be ∞. However, a small reduction in
k gives finite period (for example, if k = 0.9999, period
becomes 7π).
3.4 Solitary Wave
For even shallower conditions, the limiting case of in-
finite wavelength is approached. This is the limiting
case of cnoidal wave mentioned earlier and then cnoidal
wave becomes a solitary wave. All the waves that have
been considered earlier are oscillatory in nature. Soli-
tary wave, on the other hand, is translatory. The profile
of this consists of only a crest and the entire wave will be
above mean level (Figure 3c). Thus, the wave translates
forward. The wave profile is


3 H
η = h + sech2 (x − ct) . (7)
4 h3

Here, h is the depth of the water column and c is the

velocity of propagation.
This kind of wave was first observed by John Scott Rus-
sell in 1834. Russell was an engineer and naval archi-
tect. On observing a boat being pulled rapidly by horses
through a narrow channel, Scott saw that the water that
accumulated just in front of the boat rose as a single
wave crest and moved forward without change of shape
or reduction of speed. He called it a ‘wave of transla-
tion’. This chance observation occurred in a channel in
Edinburgh and was subsequently verified by studies in
a wave tank.
4. Further Development of Theories
Stokes and cnoidal wave theories were propounded around
the middle and second half of 19th century. Though
solutions of orders higher than the second give more

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Figure 4. The regions of pos-

sible wave theories.
Adapted from [5].
Courtesy: J D Fenton.

accurate results, these involve cumbersome calculations.

However, in the second half of the 20th century, several
studies on theoretical and laboratory aspects of higher
order waves have been carried out. One of the impor-
tant developments is the Fourier method of solution. In
this method, the full nonlinear equation is solved. The
Fourier coefficients are obtained numerically with the
help of a computer program. The main advantage is
that the entire depth region from deep water to very
shallow water is covered in the solution. The regions of
applicability of different theories can be delineated with
better accuracy (see Figure 4). In the figure, the closed
circles are from Williams [4] and the curve is fitted by
Fenton [5]. The fitted curve in this figure indicates the
maximum H/h (wave height/water depth) ratio, beyond
which the wave breaks. Wave breaking occurs when the
wave becomes unstable and the crest tumbles forward.
In the deep water region, h is large and therefore L/h ra-
tio is small. Then the wave breaks as the ratio of wave
height to wavelength rises to 0.141. At the left-hand
end of the figure, the fitted curve corresponds to this
situation. On the other hand, as the depth decreases,
wave-breaking limit is reached according to the ratio
H/h, which increases with decrease in depth. For soli-
tary wave (at the right-hand end of figure), the limiting
value of this ratio is 0.83.

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5. Why are Finite Amplitude Waves Important?

The study of finite amplitude waves is important as
these theories have different applications. One aspect is
the waves in the generating area. As mentioned earlier,
the wind waves are generated by the transfer of energy
of winds blowing over the sea. As the waves grow in size,
the gravity tries to restore the sea surface back to normal
condition. Gravity being the restoring force, these are
known as gravity waves. The small amplitude and finite
amplitude waves considered here are gravity waves. At
the regions of wave generation, waves of different charac-
teristics will be present simultaneously giving a chaotic
appearance to the sea surface. Finite amplitude waves,
such as the higher order Stokes waves, which break when
they become unstable, will also be present here. This oc-
curs when the breaking condition for deep water waves,
mentioned earlier, is reached and then white caps appear
on the sea surface. Sharp-crested waves that are always
present in the wave-generating area during strong winds
can be seen in Figure 5. In order to understand the
transfer mechanism of the energy of wind to waves and
between different waves and also to delineate the wave
generation processes, the finite amplitude wave theories
are very important.

Figure 5. Waves in the gen-

erating area (NOAA).
Source:Commons.wikimedia.org/
wiki/file/00810.jpg.
Copyright: NOAA.

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Another aspect is the designing of marine structures. Acknowledgments

These structures are necessary for extraction of oil and
natural gas, minerals, deep sea nodules, etc. Similarly, The author thanks John D
harnessing wave energy or for harvesting wind energy, Fenton, Guest Professor,
by establishing wind mills in coastal waters also requires Faculty of Civil Engineer-
the construction of suitable structures. Use of the ap- ing, Vienna University of
propriate nonlinear wave theory depending on the area Technology for going
of application, gives better estimates of wave forcing, through the manuscript.
wave breaking, etc. Wave-induced forcing needs to be He is thankful to the
estimated according the appropriate wave theory for the anonymous referee for
reliable design of marine structures. Using wave data helpful comments.
and information on water depth, diagrams such as Fig-
ure 4 can be used to decide which wave theory is appro-
priate.
6. Conclusions Address for Correspondence
K K Varma
Finite amplitude waves, which are also known as non- Nediyal
linear waves, have peaked crests and flat troughs. These Thannipparamu Madhom
Edappally
types of waves are also present on the sea surface, along
Kochi 682 024, India
with linear small amplitude waves. Depending on the Email:
relative depth of the water column, different finite am- varmacochin@yahoo.com
plitude wave theories become suitable.

Suggested Reading

[1] U S Army Corps of Engineers, Coastal Engineering Manual, Part II, Chapter 1. pp.II-1.1-1.121, 2008.
[2] W Bascom, Waves and Beaches, The dynamics of the ocean surface, Doubleday, Garden City, N Y, 268 p.196.
[3] R L Wiegel, Oceanographical Engineering, Prentice Hall Inc. pp.11–132, 1964.
[4] J M Williams, Limiting gravity waves in water of finite depth, Phil. Trans. Roy. Soc. London, Vol.A31, pp.139–
188, 1981.
[5] J D Fenton, Nonlinear wave Theories, The Sea, Vol.9, Ocean Engineering Science, Eds. B Le Méhauté and D
M Hanes, New York, Wiley, pp.1–17, 1990.
[6] R J Dean and P A Eeagleson, Finite amplitude waves, Estuary and coastline hydrodynamics, Ed. A T Ippen Mc
Graw-Hill book company, pp.93–132, 1966.
[7] B Kinsman, Wind Waves, their generation and propagation on ocean surface, Prentice Hall Inc, NJ, p. 675, 1965.
[8] J D Fenton, Coastal and ocean engineering http://johndfenton.com/Lectures/Coastal-and-Ocean-Engineer-
ing/Coastal-and-Ocean.pdf
[9] J D Fenton and M M Rienecker, A Fourier method for solving nonlinear water--wave problems: application
to solitary-wave interactions, J. Fluid Mech., Vol.118, pp.411–443, 1982.

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