Basics of Ocean Waves and significance

in Marine operations

Training Course on
‘Remote Sensing of Potential Fishing Zones and
Ocean State Forecast’

March 27, 2014

K G Sandhya
Scientist
INCOIS 1
 Ocean surface waves
Ocean surface waves are the result of forces acting
on the ocean surface.
Predominant natural forces are:
 Stress from atmosphere (through wind)
 Earthquakes
 Gravity of earth and celestial bodies (moon & sun)
 Coriolis force (due to earth’s rotation)
 Surface tension
Characteristics of the waves depend on the controlling
Forces.
2
Classification of ocean waves by wave period
(Munk, 1951)
Forcing: earthquake
wind moon/sun

Restoring:
gravity
surface tension Coriolis force

2
1

10.0 1.0 0.03 3x10-3 2x10-5 1x10-5

Frequency (Hz)
3
 Properties 7-1
of ocean waves

Waves are the undulatory motion of a water surface.

• Two general wave categories:

– Progressive waves (waves that move forward
across the ocean surface)
• Surface waves
• Internal waves
• Tsunamis
– Standing waves
• Seiches (water surface “seesawing” back and forth)

4
 A simple sinusoidal wave

The shape of a wave is defined by the vertical

displacement of the water surface from the
undisturbed sea level, as a function of both time and
space. 5
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
6
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
7
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T) Mean Sea Level (MSL)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
8
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth. 9
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T) L
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth. 10
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
T
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
11
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and 12
depth.
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
13
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth. 14
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C)
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
15
 OCEAN SURFACE WAVES
• Wave profile (η)
• Wave crest
• Wave trough
• Wave amplitude (a)
• Wave height (H)
• Wave length (L)
• Wave period (T)
• Wave frequency (f)
• Wave number (k)
• Angular wave frequency (σ)
• Wave celerity (C): (C=L/T).
• α and β: horizontal and vertical water particle
displacements respectively, and functions of time and
depth.
16
 Some Definitions
• Crest – highest point of a wave
• Trough – lowest point of a wave
• Wave Height (H) – vertical distance between the crest
and the trough
• Amplitude (a) – half the wave height
• Wavelength (L) – horizontal distance between two crests
or two troughs
• Wave period (T) – The time in seconds between
successive wave crests as they pass a stationary point on
the ocean surface
• Wave frequency – reciprocal of wave period (number of
crests passing a fixed point in one second)
F = 1/T 17
 Some Definitions
• Wave number – number of waves per unit distance
k = 2π/L
• Angular frequency – number of radians per second.
ω = 2π/T
• Wave celerity – is the speed of a wave and is given by the
distance travelled by a crest or trough per unit time,
commonly referred to as wave speed or phase speed
C = L/T
• Wave steepness – is the ratio of wave height to wave
length
S = H/L

18
 CLASSIFICATION OF WAVES ACCORDING
TO PERIOD (SHORT, INTERMEDIATE AND LONG WAVES)

Wave Classification (Ippen 1966)

Range of h/L Range of kh= 2πh/L Types of waves
(water depth
/wave length)

0 to 1/20 0 to π/10 Long waves (shallow-water

wave)

1/20 to ½ π/10 to π Intermediate waves

½ to ∞ π to ∞ Short waves (deepwater waves)

19
 Dispersion relation
The variation of wave speed with wave length is called dispersion and
the functional relation is called dispersion relation. It could be expressed
as
ω2 = gk tanh(kh),
where h is the water depth.
For deep water, h is large, hence
⇒ ω2 = gk
Substituting for ω and k, we have

Further, the wave speed c = L/T = ω/k =

The wave speed in deep water is not a function of water depth, but a
function of wave number. Thus deep water waves are dispersive. 20
 Dispersion relation

For shallow water, h is small and hence

=> ω2 = gk2h

Further, the wave speed c = L/T = ω/k =

The wave speed in shallow water is a function of water

depth, but not a function of wave number. Thus shallow

water waves are non dispersive.

21
 Wave profile
Wave profile of a simple sinusoidal wave could be
expressed as
η(x,t) = a sin (kx- ωt),
where a is the amplitude of the wave. The greater the
wave’s amplitude is, the more energy the wave carries.
If we consider a snapshot at time t=0 then the wave profile
is frozen as
η(x) = a sin (kx).
If a waverider buoy is recording the waves at x=0, then the
wave profile will be varying in time as
η(t) = a sin (- ωt).

In reality, simple sinusoidal waves are never observed in

ocean, only swells passing through an area with no wind
will come close. 22
 Orbital motion of water particles
• As waves pass, wave form and wave energy move
rapidly forward, not the water.
• Water molecules move in an orbital motion as the wave
passes
• In deep water, the horizontal and vertical displacements
of water particles are approximately the same (It is a
circular orbit)
• In shallow water, the motion of water particles follows
an elliptical path
• The wave speed (speed of the wave profile or phase
speed, L/T) is much higher than the speed of individual
water particles (equal to πH/T)
23
Water Particle Trajectories

Water Particle Trajectories for Long and Short

Waves as A Function Depth
24
Water Particle Trajectories

25
 How Water Moves in a Wave
Water particle motion in a progressing wave

26
Actual orbit of a water particle and the Stoke’s drift
In reality water particles do not return exactly to the starting point
of its path. It ends up in a slightly advanced position in the
direction in which the waves are travelling, and thus a small net
forward shift remains. This shift is more for steeper waves.

27
 Energy in waves
 Waves disturb the water and hence there is kinetic
energy associated with the waves, which moves along
with the wave.
 Waves also displace water particles in the vertical and
hence affect the potential energy of the water column,
which also moves along with the wave.
 Wave energy moves at the group velocity, which is the
speed of the group of waves.
 The total energy of a simple linear wave is
where ρ is the density of water.
28
WAVE BEHAVIOR
7-3IN SHALLOW WATER
 WAVE BEHAVIOR IN SHALLOW WATER
• When the water depth h > L/2, water particle
motion is negligible.
• When waves propagate into shallow water, they
begin to “feel” the bottom.
• Wave period remains constant.
• Wave speed decreases and hence wave length
decreases (since L=cT).
• Wave height increases. (By law of conservation of
energy - when group velocity and wave length
decrease, energy in each wave length must
increase => H increases, as )
WAVE BEHAVIOR IN SHALLOW WATER

Near shore Wave Processes 31

WAVE REFRACTION

Wave Refraction in a bay 32

 WAVE REFRACTION
• Refraction may occur when the waves begin to “feel” the bottom.
When waves enter water of transitional depth, if they are not
travelling perpendicular to the depth contours, the part of the
wave in deep water travel faster than the part in shallower water.
This causes the crest to turn parallel to the bottom contours.
• Generally any change in wave speed (eg. due to gradients of
surface currents) may lead to refraction, irrespective of the water
depth.

Refraction along an
irregular shoreline

33
 WAVE REFRACTION
Refraction along
a straight beach
with parallel
bottom contours

Refraction by a Refraction by a
submarine ridge submarine canyon

34
WAVE DIFFRACTION

35
 WAVE DIFFRACTION
• Diffraction usually happens when waves encounter an obstacle, such
as a breakwater or an island.
• The waves bend around an obstacle to reach the lee side of the
obstacle. The waves can affect the lee side of a structure, although
their heights are much reduced.
• Diffraction of water waves is the simplest kind as it happens in two
dimensions. Diffraction along with interference can create patterns
like that shown below.

36
Wave diffraction at Channel Islands breakwater
(California) (from CERC, 1977)

37
WAVE BREAKING

38
 WAVE BREAKING
• Speed of water particles at the crest is slightly
greater than that at the trough. The steeper the
waves this effect is more.
• Once (wave height/wave length) becomes equal to
1/7, the forward speed of the water particles at the
crest becomes equal to the wave propagation
speed.
• Once this happens, the water particles plunge
forward out of the wave or the waves break.

39
 Wave Steepness and breaking
• The limiting angle at the crest of a deep-water wave is 120
degrees.

• At this point the steepness (S) of the wave is 1:7.

• According to Stoke’s theory waves can not attain a height more

than L/7 without breaking. At this point, the forward and
backward slopes of the wave meet in the crest at an angle of
120°. To break in deep water, steepness must exceed 0.142
which is a ratio of 1:7. 40
WAVE BREAKING

41
 Types of Wave Breaking

Plunging and Collapsing Breaker

Spilling Breaker

Surging Breaker

42
 LONG WAVES and EXTREME WAVES
IN NATURE

• TIDAL WAVES
• SWELL WAVES
• SEICHES (RESONANCE OF BASINS)
• FREAK WAVES

43
 Wind seas and Swells

Sea state is a combination of wind seas and swells.

storm center wave crests

wind direction

wind sea swell

fetch
Fetch is the distance over which wind blows over the water surface 44
Chaotic Sea exhibiting complex surface wave forms

SWELL WAVES
 wind sea
• Waves growing under the influence of wind
• Poorly organized, short-crested waves of irregular
size and spacing
swell
• Waves outside the generating area
• Well organized, regularly-spaced wave train
• If height of the wind waves is Hsw and height of swell
waves is Hss then total wave height = (Hsw2 + Hss2)1/2

46
 Difference Between Sea (Wind) and Swell Waves
• "Sea (Wind) Waves" are produced by local winds and
measurements show they are composed of a chaotic mix of height
and period. In general, the stronger the wind the greater the amount
of energy transfer and thus larger the waves are produced.
• As sea waves move away from where they are generated they
change in character and become swell waves.
• "Swell Waves" are generated by winds and storms in another area.
As the waves travel from their point of origin they organize
themselves into groups (Wave trains) of similar heights and periods.
These groups of waves are able to travel thousands of miles
unchanged in height and period.
• Swell waves are uniform in appearance, have been sorted by period,
and have a longer wave length and longer period than sea waves.
Because these waves are generated by winds in a different location,
it is possible to experience high swell waves even when the local
winds are calm. 47
 Wave fields on the
ocean – a composition
of simple waves

The sea surface obtained

from the sum of many
sinusoidal waves

Any observed wave pattern in the

ocean could be shown to comprise a
number of simple waves, which differ
from each other in height, wavelength
and direction.
48
How a complex wave pattern could be analyzed???

49
 Wave groups and group velocity
• Individual waves in a group travel at the velocities
corresponding to their wave lengths, but the wave group as a
coherent unit travels at its own velocity - the group velocity

• Group velocity is the velocity with which the wave energy is

propagated

• Group velocity can be written as

Cg = dω/dk

• Deep water group velocity is

Cg = C/2
=> The wave group travels slower than the fastest wave in
the group.

• Shallow water group velocity is

Cg = C 50
This is since the shallow water waves are non dispersive.
Wave record - the motion of water
surface at a fixed point

51
 Wave spectrum - Concept
Wave spectrum is the
distribution of wave energy over
various frequencies (or
directions or frequencies and
directions). It is obtained by the
FFT of the wave record.

The concept of wave spectrum

is used in modelling the sea
state.

Example of a wave
spectrum with
corresponding wave record.
52
Wave parameters derived from the spectrum
• The form of a wave spectrum is usually expressed in
terms of the moments of the spectrum. The nth order
moment of the spectrum is given by

where E(f)df represents the variance ai2/2 contained in the

ith interval between f and f+df and ai is the amplitude of the
ith component in a wave record.

• Zeroth moment m0 is the area under the spectral curve. It

could be derived that the significant wave height Hm0
(average (1/3)rd of the highest waves) is
53
Wave parameters derived from the spectrum

Peak wave frequency fp is the wave frequency

corresponding to the peak of the spectrum.

Peak wave period Tp is the period corresponding to fp ,

Tp = 1/fp

Mean wave period

54
 Wave spectrum
Model forms for wave spectra:
Usually expressed as: E(f); E(f,θ) (or) E(k)
As wave number and frequency are connected by the dispersion
relation.

(i)Philips spectrum:
Usually used to represent the high frequency part of the spectrum,
above the spectral peak. In general form:

g2 
E ( f ) = 0.005 5  g
Earlier used for
f  if f ≥   representing the
 u tail
= 0 (else) 
 Wave spectrum
(ii) Pierson-Moskowitz spectrum (1964):
Used for a fully developed sea (equilibrium state when fetch and
duration are unlimited):
Originally developed based on sub-set of 420 (1955 – 1960) wave
measurements with ship-borne wave recorder (Tucker, 1956).
4
 g 
α g2 −0.74 
 2π fU 
E( f ) = e
(2π ) f
4 5

where, α = non-dimensional quantity = 0.0081

U = wind speed at 19.5 m above seasurface
Peak frequency of P-M spectrum:
g
=
f p 0.877 ; H1/3 0.0246 U 2 (for fuly developed seas)
2π U
Hence, H1/3 = 0.04 f p-2
 Wave spectrum
(ii) JONSWAP spectrum (1973):
JONSWAP (1973) gave a description of wave spectra growing in
fetch limited condition. Basic formulation of the spectrum is
expressed in terms of peak frequency rather than wind speed.
4
 f 
α g2 −1.25 
 fp 
E( f ) = e .γ ( f )
 
(2π ) f
4 5

where, γ is peak enhancement factor which modifies the

interval around spectral peak making it more sharper than
PM spectrum (otherwise shape is similar). Using JONSWAP
results, Hasselmann (1976) proposed a relation between variance
and peak frequency for wide range of growth stages.
0.0414 f p−2 ( f pU )1/ 3
H m0 ≈ H1/3 =
(or) f p = 0.148 H m−0.6
0 U 0.2

where, U = U10 (wind speed at 10 m height)

General form of a JONSWAP
spectrum as a function of f/fp

58
A Rogue wave occurs 7-3when there is a momentary
appearance of an unusually large wave formed by
constructive interference of many smaller waves.
Rogue7-3
(freak) waves
 Standing
7-4 Waves

• Standing waves or seiches consist of a water

surface “seesawing” back and forth.
• Node : No vertical movement
– Located in centers of enclosed basins and toward
the seaward side of open basins.
• Antinodes: Points where there are the
maximum vertical displacement of the surface
as it oscillates.
– Antinodes usually located at the edge of the
basin.
61
 7-5
Tsunamis were previously called tidal
waves, but are unrelated to tides.
• Tsunamis consist of a series of long-period waves characterized
by very long wavelength (up to 100 km) and high speed (up to 760
km/hr) in the deep ocean.
• Because of their large wavelength, tsunamis are shallow-water to
intermediate-water waves as they travel across the ocean basin.
• They only become a danger when reaching coastal areas where
wave height can reach 10 m.
• Tsunamis originate from earthquakes, volcanic explosions, or
submarine landslides.

62
Generation of a Tsunami

63
Tsunami damage

64
Wave measurement
- Waverider buoys
• Small & light, deployable from small
vessels

• The non-directional wave buoy

measures its own vertical acceleration
on a gravity stabilised platform.

• The directional wave rider buoys

measures tilt (pitch and roll) in addition
to vertical acceleration.
 Wave height measurements using
satellites

Altimetry is meant to retrieve

“Sea Surface Height”

This is the difference between

the satellite-to-ocean range
(calculated by measuring the
signal’s round-trip time) and
the satellite’s position on
orbit with respect to an
arbitrary reference surface (a
raw approximation of the
Earth’s surface, called the
reference ellipsoid)

Animation

66
 SIGNIFICANCE OF THE STUDY OF WAVES
•Waves are one of the most important parameter in coastal
oceanography, ocean engineering and coastal management.

•Accurate measurement and long term observations of waves are

required to understand the wave climate of a region.

•The waves in harbours and near shore regions affect the marine
operations as well as the fishing activities.

•Coastal engineers require the wave climate to design harbours,

jetties, ports, oil rigs and other coastal protection structures.

•Studies of sediment transport and coastal erosion require accurate

knowledge of the wave climate of location.

•Waves also play a significant role in flux exchange and air sea
67
interaction.
68