Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
CHAPTER 1: INTRODUCTION- HAZARDS-OCEANS-WAVES
1.1. NATURAL HAZARDS:
Hazards are unpreventable natural events that, by their nature, may expose our Nation's
population to the risk of death or injury and may damage or destroy private property, societal
infrastructure, and agricultural or other developed land. Hazards include earthquakes,
volcanoes, floods, droughts, earthquakes, hurricanes, volcanic eruptions.
1.1.1. Floods
Floods are the most common and widespread of all natural disasters--except fire. Floods
have been an integral part of the human experience ever since the start of the agricultural
revolution when people built the first permanent settlements on the great riverbanks of Asia
and Africa. Seasonal floods deliver valuable topsoil and nutrients to farmland and bring life
to otherwise infertile regions of the world such as the Nile River Valley. Flash floods and
large 100-year floods, on the other hand, are responsible for more deaths than tornadoes
or hurricanes. In the United States alone over the past 60 years, an average of 127 people
have lost their lives due to floods. Floods can be slow, or fast rising but generally develop
over a period of days.
1.1.2.Volcanic Eruptions
Volcanic eruptions are one of Earth's most dramatic and violent agents
of change. Not only can powerful explosive eruptions drastically alter
land and water for tens of kilometers around a volcano, but tiny liquid
droplets of sulfuric acid erupted into the stratosphere can change our
planet's climate temporarily. Eruptions often force people living near
volcanoes to abandon their land and homes, sometimes forever. Those
living farther away are likely to avoid complete destruction, but their
cities and towns, crops, industrial plants, transportation systems, and
electrical grids can still be damaged by tephra, lahars, and flooding.
The bed effects of volcanic eruption:
(1) pyroclastic eruptions can smother large areas of landscape with hot ash, dust, and
smoke within a span of minutes to hours;
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
1
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
(2) red hot rocks spewed from the mouth of a volcano can ignite fires in nearby forests and
towns, while rivers of molten lava can consume almost anything in their path as they
reshape the landscape;
(3) heavy rains or a rapidly melting summit snowpack can trigger lahars-sluices of mud that
can flow for miles, overrunning roads and villages; and
(4) large plumes of ash and gas ejected high into the atmosphere can influence climate,
sometimes on a global scale.
According to a new USGS report on NVEWS, since 1980, 45 eruptions and 15 cases of
notable volcanic unrest have occurred at 33 U.S. volcanoes. Volcanic activity since 1700
A.D. has killed more than 260,000 people, destroyed entire cities and forests, and
severely disrupted local economies
for months to years. Even with our
improved ability to identify hazardous
areas and warn of impending
eruptions, increasing numbers of
people
face
certain
danger.
Scientists have estimated that by the
year 2000, the population at risk from
volcanoes is likely to increase to at
least
500
million,
which
is
comparable to the entire world's
population at the beginning of the
seventeenth
century!
Clearly,
scientists
face
a
formidable
challenge in providing reliable and
timely warnings of eruptions to so
many people at risk.
1.1.3
Hurricane
Few things in nature can compare to the destructive force of a hurricane. In fact, during
its life cycle a hurricane can expend as much energy as 10,000 nuclear bombs!
Hurricane winds blow in a large spiral around a relative calm centre known as the "eye."
The "eye" is generally 20 to 30 miles wide, and the storm may extend outward 400 miles.
As a hurricane approaches, the skies will begin to darken and winds will grow in strength.
As a hurricane nears land, it can bring torrential rains, high winds, and storm surges.
August and September is peak months during the hurricane season that lasts from June
1 through November 30.
The term hurricane is derived from Huracan, a god of evil recognized by the Tainos, an
ancient aborigines Central American tribe. In other parts of the world, hurricanes are
known by different names. In the western Pacific and China Sea area, hurricanes are
known as typhoons, from the Cantonese tai-fung, meaning great wind. In Bangladesh,
Pakistan, India, and Australia, they are known as cyclones, and finally, in the Philippines,
they are known as baguios.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
2
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
At the surface, hurricanes can diminish rather quickly given the right conditions. These
conditions include: (1) the storm moving over cooler water that can't supply warm, moist
tropical air; (2) the storm moving over land, again cutting of the source of warm, moist air;
and finally (3) moving into an area where the large-scale flow aloft is not favorable for
continued development or sustainment.
.
Figure 1.1.3.1: A Photograph taken during a Hurricane
1.1.4
Drought
Agricultural disasters may not be as dramatic as a volcanic eruption or a hurricane, but
they are by far the most damaging. Worldwide, since 1967, drought alone has been
responsible for millions of deaths and has cost hundreds of billions of dollars in damage.
Many different climatic events can trigger crop failures including excess rainfall leading to
flood damage or crop disease, heat waves, drought, fire, unexpected cold snaps, severe
storms, climate-related disease outbreaks, and insect invasions. Large-scale weather
patterns such as El Niño, La Niña, and the Pacific Decadal Oscillation affect agriculture
world-wide by changing rainfall patterns. The average El Niño costs about $2 billion in
agricultural losses in the United States.
Nationwide losses from the U.S. drought of 1988 exceeded $40 billion, exceeding the
losses caused by Hurricane Andrew in 1992, the Mississippi River floods of 1993, and
the San Francisco earthquake in 1989. In some areas of the world, the effects of drought
can be far more severe. In the Horn of Africa the 1984–1985 drought led to a famine
which killed 750,000 people.
Drought can be defined according to meteorological, hydrological, or agricultural criteria.
Meteorological drought is usually based on long-term precipitation departures from
normal, but there is no consensus regarding the threshold of the deficit or the minimum
duration of the lack of precipitation that make a dry spell an official drought.
Hydrological drought refers to deficiencies in surface and subsurface water supplies. It’s
measured as stream flow, and as lake, reservoir, and ground water levels.
Agricultural drought occurs when there is insufficient soil moisture to meet the needs of a
particular crop at a particular time. A deficit of rainfall over cropped areas during critical
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
3
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
periods of the growth cycle can result in destroyed or underdeveloped crops with greatly
depleted yields. Agricultural drought is typically evident after meteorological drought but
before a hydrological drought.
1.1.5
Earthquakes
One of the most frightening and destructive phenomena of nature is a severe earthquake
and its terrible after-effects. An earthquake is a sudden movement of the Earth, caused
by the abrupt release of strain that has accumulated over a long time. For hundreds of
millions of years, the forces of plate tectonics have shaped the Earth as the huge plates
that form the Earth's surface slowly move over, under, and past each other. Sometimes
the movement is gradual. At other times, the plates are locked together, unable to
release the accumulating energy. When the accumulated energy grows strong enough,
the plates break free. If the earthquake occurs in a populated area, it may cause many
deaths and injuries and extensive property damage.
Although we still can't predict when an earthquake will happen, we have learned much
about earthquakes as well as the Earth itself from studying them. We have learned how
to pinpoint the locations of earthquakes, how to accurately measure their sizes, and how
to build flexible structures that can withstand the strong shaking produced by
earthquakes and protect our loved ones
An earthquake is a sudden shaking of the ground. They generate seismic waves which
can be recorded on a sensitive instrument called a seismograph. The record of ground
shaking recorded by the seismograph is called a seismogram. The Earth's outermost
surface is broken into 12 rigid plates which are 60-200 km thick and float on top of a
more fluid zone, much in the way that icebergs float on top of the ocean
Faults are narrow zones in the Earth, usually extending no more than about 10 miles
deep, which separate rigid crustal blocks.
A well known fault is the San Andreas Fault which separates the Pacific plate from the
North American plate. The Pacific plate has San Fransicso and Los Angeles on it, while
the North American plate contains the rest of California and the U.S. The Pacific plate is
moving to the northwest at a rate of about 4 inches per year.
An earthquake is a sudden shaking of the ground. They generate seismic waves which
can be recorded on a sensitive instrument called a seismograph. The record of ground
shaking recorded by the seismograph is called a seismogram.
1.2
OCEANS:
Oceans cover about 70% of the Earth's surface. The oceans contain roughly 97% of the Earth's
water supply. The ocean appears blue because it is reflecting the blue colour of the sky. On a
grey, cloudy day, the ocean appears grey.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
4
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Figure 1.2.1: Earth’s Oceans
The oceans of Earth serve many functions, especially affecting the weather and temperature.
They moderate the Earth's temperature by absorbing incoming solar radiation (stored as heat
energy). Moving ocean currents distribute this heat energy around the globe. This heats the land
and air during winter and cools it during summer.
The Earth's oceans are all connected to one another. Until the year 2000, there were four
recognized oceans: the Pacific, Atlantic, Indian, and Arctic. In the spring of 2000, the
International Hydrographic Organization delimited a new ocean, the Southern Ocean (it
surrounds Antarctica and extends to 60 degrees latitude).
Ocean
Pacific
Ocean
Atlantic
Ocean
Indian
Ocean
Southern
Ocean
Arctic
Ocean
Table 1.2.1: Characteristics of Oceans
Area (square miles)
Average Depth (ft)
Deepest depth (ft)
64,186,000
15,215
Mariana Trench, 36,200 ft deep
33,420,000
12,881
Puerto Rico Trench, 28,231 ft
deep
28,350,000
13,002
Java Trench, 25,344 ft deep
the southern end of the South
7,848,300 sq. miles 13,100 - 16,400 ft deep
Sandwich Trench, 23,736 ft (7,235
(20.327 million sq km ) (4,000 to 5,000 meters)
m) deep
5,106,000
3,953
Eurasia Basin, 17,881 ft deep
The Pacific is by far the most active tsunami zone, according to the U.S. National Oceanic and
Atmospheric Administration (NOAA). But tsunamis have been generated in other bodies of
water, including the Caribbean and Mediterranean Seas, and the Indian and Atlantic Oceans.
North Atlantic tsunamis included the tsunami associated with the 1775 Lisbon earthquake that
killed as many as 60,000 people in Portugal, Spain, and North Africa. This quake caused a
tsunami as high as 23 feet (7 meters) in the Caribbean.
The Caribbean has been hit by 37 verified tsunamis since 1498. Some were generated locally
and others were the result of events far away, such as the earthquake near Portugal. The
combined death toll from these Caribbean tsunamis is about 9,500.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
5
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Large tsunami waves were generated in the Marmara Sea in Turkey after the Izmit earthquake
of 1999.
1.3 WAVES, CLASSIFICATION, TRANSFORMATIONS:
1.3.1
Basic Wave Parameters
Wave parameters, which are necessary to describe the wave profile, are called the
“basic wave parameters”. These are: wave amplitude or wave height, wave period or
wavelength, and the water depth. A relation sheep exist between the wavelength and the
wave period, and the water depth. For the reason, either the wavelength or the wave
period can be considered as a basic parameter. If the sea is sufficiently deep, then the
water dept does not affect the wave motion. For this special case, the number of basic
wave parameters is only two (i.e. wave height and wave period). All characteristics of the
wave motion can be computed in terms of the basic parameters.
Wave Profile
For the simplicity of treatment and in most engineering problems a waveform is used.
This idealized wave is called sinusoidal wave because its shape agrees with the
trigonometric sine function (or equivalently with the cosine function). The profile of a
sinusoidal wave is shown in Fig. 1.3.1.1.
Figure 1.3.1.1: The profile of a 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. It is called the wave profile
or the waveform. The profile of the sinusoidal wave is given as
  a sin 2 ( x / L  t / T )
Here the dependent variable η shows the wave profile, the independent variables x and t
are the space and time coordinates respectively; a, L, and T are the wave parameters
which will be defined below.
Definitions:
Definitions for the wave parameters shown in Fig. 1.3.1.1 are as follows
Wave profile (η): vertical displacement of the sea surface from the still water level
(SWL) as a function of time and space.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
6
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Wave crest: the highest point of wave profile.
Wave trough: the lowest point of wave profile.
Wave amplitude (a): the vertical distance from the still water level to the wave crest.
Wave height (H): the vertical distance from wave trough to wave crest. It is equal to
twice the wave amplitude. (H=2a)
Wave length (L): the horizontal distance between two successive crest and troughs.
Wave period (T): the time interval between the passages of two successive crests past
a fixed point.
Wave frequency (f): the number of waves to pass a given point per unit time. It is equal
to reciprocal of wave period (f=1/T).
Wave number (k): 2Π times the number of waves per unit horizontal distance. It is equal
to reciprocal of wavelength time’s 2Π (k=2Π/L).
Angular wave frequency (σ): it is equal to wave frequency times 2Π (σ =2Πf=2Π/T).
Wave celerity (C): the speed at which a waveform moves. Since the wave moves one
wavelength during a wave period, the wave celerity is equal to the ratio of wavelength to
wave period (C=L/T).
α and β: are the horizontal and vertical water particle displacements respectively, that
are functions of time and depth.
1.3.2
Basic Equations of the Wave Motion
THE VELOCITY POTENTIAL:
The simplest and general most useful theory is the small amplitude wave theory first
present by Airy (1845).
Solving the Laplace Equation develops the small amplitude wave theory for twodimensional periodic waves:
 2  2

0
x 2 y 2
(1a)
With the bottom and surface conditions, the following velocity potential is obtained.

a cosh( y  d )
cos( kx  t )
k
sinh d
(1.b)
For a progressive wave travelling in positive x direction the velocity potential as given in
Eq. (1b) correspondence a wave profile is
  a sin( kx  t )
Similarly to the velocity potential as given in Eq. (1c) corresponding to a wave profile is
  a cos( kx  t )
or by shifting the origin by L/4

ga cosh k ( y  d )
sin( kx  t )

cosh kd
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
(1c)
where k=2  /L and  =2  /T
7
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
1.3.3
Chapter 1
Progressive Waves
Waveform, which moves relative to the fluid, is called a progressive wave. The speed of
the profile movement is equal to the wave celerity, C. It is clear that in progressive wave,
a wave crest or a wave trough is located every point within a wavelength during a wave
period. Eq (1) in the preceding section represents a progressive wave.
1.3.4. Wave Length and Wave Celerity
The relation between wavelength, wave period and water depth is written as
L
gT 2
2d
tan(
)
2
L
(2)
where L is the wavelength, T is the wave period, d is the still water depth, g is the
gravitational acceleration equal to 9.81 m/sec2 , and tanh is the hyperbolic tangent
function.
Eq (2) is an implicit equation that is unknown variable L appears both in the left and right
hand sides of the equation. For given T and d values, to obtain L it may require to carry
out several trial calculations. However, for convince, solutions are all ready given in
graphical form, or in tables.
Wave celerity is equal to the ratio of wavelength to wave period as
C=L/T (3)
Thus using Eqns. (2) and (3) we get
C
gT
tanh( 2 d / L)
2
gL
C
tanh( 2d / L)
2
(4.a)
1/ 2
(4.b)
1.3.5 Constancy of Wave Period
For a simple harmonic wave train, the wave period is independent of depth. The number of
waves which have entered to the region 1 is n1 while, the number of waves which has left
the region 2 is n2 ifT1 is the period of waves entering to the region 1, then n1=Δt/T1, T 2 is
the period of waves leaving the region 2, then n 2 =Δt/T 2 .
Then (Number of waves accumulated within the region) =n1-n2=Δt(1/T1-1/T2) where the
time increment Δt→∞
The number of waves accumulated within the region will be ± ∞ depending onT1   T2.
This is physically unrealistic. Then only realistic possibility is T1=T2=T this result holds for
any depth d.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
8
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
1.3.6
Chapter 1
Classification of Waves According to Period (Short, Intermediate and Long Waves)
Waves are classified as short waves, also referred to as ‘deepwater waves,’ when the
relative depth is greater than approximately 1/2. Coastal waves described in Part II-2 are
generallyof this class. The geometry of short waves implies wave steepness great
enough to cause them to break. The class of waves between short (deep) and long
(shallow) are referred to as ‘intermediate waves.’ Table 1.3.6.1 (Ippen 1966) summarizes
wave classification criteria according to relative depth and the wave parameter kh
defined below.
Table 1.3.6.1: Wave Classification (Ippen 1966)
Range of d/L
Range of kh=2  d/L
0 to 120
0 to  /10
 /10 to 
 to 
1/20 to ½
1/2 to 
Types of waves
Long
waves
(shallow-water
wave)
Intermediate waves
Short waves (deepwater waves)
Applying the relative depth and wave number parameter to the characteristics of long
waves can be seen in simplification to progressive small-amplitude wavw theory
solutions. For example the wave celerity ,wave length,horizontal(x-direction) and vertical
velocities can be written as
u=
agk cosh k ( h  z )
sin(kx- t )

cosh kh
w=-
agk sinh( k (h  z)
cos(kx  t )

cosh kh
(5a)
(5b)
where k is the wave number (2  /L),  is the angular frequency (2  /T) where T is the
period of the wave ), a is the amplitude of the wave , g the acceleration of gravity, h is the
total depth, and z is the depth measured downward from the quiescent fluid surface.
Additionally, one important difference between long waves and short waves can be seen
in the computed orbital velocities. Figure 1.3.6.1 shows water particle trajectories for
long,short,and intermediate waves as a function depth. As can be seen ,and computed
from equations 5a and 5b, the horizontal velocity of a long wave is maintened throughout
the water column, from the surface to the bottom. In the case of short waves, the
strength of the horizontal and vertical component decreases with depth to the point that
waves do not inducude bottom currents. The fact that long waves affect the bottom is
important in that bottom sediments can beeroded and transported by tidal or other longwave currents. For example, tidal flood and ebb currents contribute to the transport of
sediments to form enn and flood shoals.
For “shallow water” conditions tanhk= kd and the wave celerity becomes
C  gd 
1/ 2
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
9
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
In this case the wave celerity is independent of wavelength. Similarly for “shallow water”
conditions wavelength reduces to
L   gd  T
1/ 2
(5.d)
Figure 1.3.6.1: Water Particle Trajectories for Long, Short and Intermediate Waves as A
Function Depth
Waves don't actually move water from one place to another. Water in a wave moves up
and down. The water moves up as a crest passes, and moves down as a trough passes.
Each molecule of water draws a circle as one complete wave (crest to crest) passes.
In the cartoon at right, the waves are moving from left to right. As the wave crest
approaches the seagull, the bird moves UP. As the trough approaches, the bird moves
DOWN.
There is a small amount of side-to-side motion (more with longer wavelengths), but
mainly the water motion is up and down. This is called orbital motion.
The circle (=orbit) of water motion at the surface sets in motion another orbit of water
beneath it. However, some of the energy from the first orbit is lost, so the next orbit is
somewhat smaller, as shown below.
Figure 1.3.6.2: How Water Moves in a Wave
Eventually the orbit of motion gets so small, it disappears altogether, and there is no
motion. This water depth is called wave base and is usually ~1/2 the wavelength.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
10
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
This means that waves in deep water do not touch bottom!!! They do not touch bottom
unless the water depth is less than 1/2 the wave's wavelength.
Waves start out in the open ocean, and eventually move toward land, across the
continental shelf and up onto the beach.
As waves approach the beach, water depth decreases (it gets shallow!) and waves start
to touch bottom where water depth is around 1/2 the wavelength.
When waves begin to touch the seafloor, friction starts to distort the perfect circles of the
orbits. The orbits change from circles to ellipses (oval-shaped) as shown in the cartoon at
right.
Only where water is shallower than the wave base can waves move sediment around. In
shallow water, waves can erode sediment or deposit it.
It depends on the size of the wave and the size of the sediment (sediment texture).
Waves tend to be bigger in the winter because big winter storms are more frequent than
hurricanes. Beaches tend to be eroded during the winter, but during the summer smaller,
more gentle waves will bring the sand back to be deposited. This is a natural cycle, but
try telling that to someone who just paid $100,000 for a little lot on the beach, and sees
what happens after the first winter storm!!!
Beaches tend to be eroded in winter because waves are bigger during the winter,
because there are more big storms.
Beaches tend to be deposited in the summer because big storms are less frequent, and
the waves are usually smaller.
1.3.7. Wave Behaviour in Shallow Water
Waves moving out away from a storm eventually organize themselves into a swell, and
eventually, if they are not destroyed by interference, they reach the shore. The seafloor
shallows as the waves approach shore, and eventually the waves touch bottom (they
reach wave base). At this point we shift from "deep water" to "shallow water" (from the
wave's perspective).
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
11
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Figure 1.3.7.1: Near shore Wave Processes
The waves begin to slow down (celerity decreases) due to friction and wave celerity now
depends on water depth (remember, in deep water wave celerity depends on wavelength
and period: C= L/T). The shallower the water, the more the waves slow down.
Wavelengths shorten, but period remains the same.
The wave height increases and the trough flattens out. The wave gets so tall it can't
support itself, and the water crashes over the top. This is called a breaker, and breakers
form in an area called the surf zone.
The wave loses most of its energy by breaking (it actually gives off some light and heat),
and the remaining energy causes the water to rush up the shore. It loses the rest of its
energy to friction in this manner, then gravity pulls the water back out to sea. The surge
onshore is called swash; the slump back to sea is called backwash. Swash and
backwash occur in the swash zone. So as the waves come to shore from the sea,
 they change from deep-water waves to shallow water waves at wave base, where
water depth = 1/2 wavelength,
 they slow down (celerity decreases)
 wavelength decreases
 period stays the same
 height increases
 wave breaks and becomes
 swash, then backwash
For deep water conditions wave length and wave celerity can be written as
L0 
g 2
T (5.c)
2
C0 
g
T (5.d)
2
and
Co and Lo are independent of depth, as it has been shown that T does not vary with
depth d.
For the upper limit: x  2d / L  , gives d / L  1/ 2
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
12
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
For the lower limit: x  2d / L   /10 gives d / L  1/ 20
Thus one way of classifying waves is according to the relative depth ratio d/L
d/L>1/2 is called deep-water waves,
d/L<1/2 is called shallow water or long waves,
1/20<d/L<1/2is called intermediate or transitional waves.
1.3.8
Classification of Water Waves
Ocean waves may be classified according to several different criteria’s.
These are given below.
A. Classification of Water depth
From Table 1.3.6.1
Deep water
Intermediate depth
Shallow water
d/L>1/2
1/2<d/L<1/2
d/L>1/20
B. Classification of Wave Height
If the wave height is infinitely small, i.e.
H / L  0, andH / d  0 (6)
Thus in term in order of H 2 / L2 can be neglected in comparison with terms H/L. in fact,
for waves occurring in nature the wave steepness H/L is usually at most 0.05 to 0.08,
and due to this small value one could believe that the linearization represents a good
approximation for all practical purposes. The resulting waves are called
Small amplitude waves, small waves, infinitesimal waves,
Linear waves, sinusoidal waves, airy waves, simple harmonic waves,
First order stokes waves, or just first order waves.
If the wave height is of finite height (or finite amplitude), the effect of higher order terms is
taken into account in the mathematical formulation of wave equations. Finite amplitude
waves are also called nonlinear waves. Example of finite amplitude waves are stokes
waves and cnoidal waves.
If the wave height is comparable with the depth of water (or if H is in the order of depth
d), i.e.
H=O(d) order of depth,
The wave is called high wave!!
C. Classification of Height, Length and Depth
The wave height, length and water depth (H, L, d) are expressed by a dimensionless
number, which is decisive for the proper mathematical treatment of the wave motion.
Such a number is called Ursel parameter (or stokes parameter)
U 
HL2
d3
(7)
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
13
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
If this number is smaller than unity (i.e. U<<1,) the wave has small amplitude and is
described by linear wave theory.
Long and high waves in shallow water will have a large Ursel parameter (U>>1). This
leads to the so- called cnoidal waves. The solitary wave is the cnoidal wave limit for
U→∞. Cnoidal waves belong to the class of long waves, which are all characterized by d/
L<<1.
1.3.9
Wave Transformation
As the waves move from deep toward shallow water they are transformed by the
variation in the bottom configuration and the existence of obstacle such as islands and
breakwater. As a result, the height, length, celerity, and direction of waves may change.
Finally, waves become unstable and break at a certain water depth.
1.3.10 Wave Refraction
Imagine that a single wave train is coming towards shore. If the sea floor were a perfect
incline, the wave would reach wave base at the same distance offshore, no matter how
far left or right you looked. The wave would steepen at the same place offshore, and
break at the same place at the shore.
But in many places, the seafloor is not perfect incline. Even though it is under water, it
has hills and valleys as you walk along, parallel to the shore. That means that a given
wave train will reach wave base at different distances from shore. Some waves are
breaking a few feet farther offshore than others. It must be shallower out there! Waves
break where water depth is about 1/5 wave height. If you drew a line under all the
breaking waves, you would draw a line where water depth = 1/5 wave height.
When the area just offshore of the beach has topography, waves reach wave base
farther offshore.
At this point, orbits change to ellipses, and the wave slows down, just as discussed
above.
The part of the wave that hasn't touched bottom yet continues to move toward the beach,
now moving faster than the part of the wave that has slowed down.
This bending of waves around objects is called wave refraction. Wave refraction causes
waves to strike as near to parallel to shore as they can get.
Wave refraction helps erode land that juts out to sea, and deposit sediment in bays.
Waves bending toward shore converge on headlands (which are bits of rocky land jutting
out to sea). Converging waves concentrate energy, so the waves pound the headlands,
weathering them to sedimentary particles.
Waves bending toward shore diverge in the bays. This dilutes the energy of the waves,
and the bays are quiet places where sediment accumulates as a beach.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
14
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Wave Refraction in a bay
Top View of Refraction
Chapter 1
Wave Refraction at Headland
Phenomenon
1.3.11 Wave Diffraction
Diffraction usually happens when waves encounter surface-piercing obstacle, such as a
breakwater or an island. It would seem that on the land side of the island, the water
would be perfectly calm; however it is not. The waves, after passing the island in Figure,
turn into the region behind the island and carry wave energy and the wave crest into this
so-called 'shadow zone.' The turning of the waves into the sheltered region is due to the
changes in wave height (say along the crest) in the same wave.
If the sides of the island are sloping under the water, then refraction would also be
present.
1.3.12 Standing Wave
If the barriers are vertical, frictionless, rigid wall, and the incident wave direction is
perpendicular to the barrier, the wave will completely reflect, resulting in a standing wave.
All the waves discussed so far have been different types of progressive waves, where
the wave moves from one place to another. Standing waves are not progressive; they
just sit there and move up and down in the same place. Standing waves are easy to
make in your bathtub: just start sliding back and forth. The water goes up in the front of
the tub, then down. When it's up in the front of the tub, it's down in the back. So at one
point (the front of the tub, for example), you make a wave crest (water goes up), then a
wave trough (water goes down).
1.3.13 Wave Breaking
The separation of water particles from the wave under the action of gravity is known as
wave breaking. Wave breaking process causes energy dissipation by turbulence.
Wave breaking is one of the most commonly observed features of water waves.
Breaking is always a nonlinear phenomenon and is therefore extremely difficult to
describe analytically. Here I'll give a quick heuristic description of the main
physical mechanisms for the "plunging mode", i.e., the type experienced at the
beach and exploited by surfers.
Types of Wave Breaking
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
15
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Spilling breaker:
Plunging breaker:
Collapsing breaker:
Surging breaker:
Spilling Breaker
Plunging and Collapsing Breaker
Surging Breaker
Flat slopes for shore and shelf encourage spilling breakers. Somewhat steep bottoms
encourage plunging breakers. These waves curl magnificently before collapsing. Very
steep bottoms encourage surging breakers - these don't actually break, since the bottom
slopes too abruptly.
Note: Surfers prefer spilling and plunging waves-but they have to be sizeable. The east
coast has spilling waves, but they are only 1-2 feet high most of the time. There are a
number of reasons for this: the continental shelf there is broad and shallow; the fetch of
the Atlantic Ocean is less than that for the Pacific...to name a couple.
1.4. LONG WAVES and ABNORMAL WAVES in NATURE
1.4.1
Tidal Waves
Tides are the alternating rise and fall of the surface of the seas and oceans. They are
due mainly to the gravitational attraction (pull) of the moon and sun on the rotating earth.
Two high and two low tides occur daily and, with average weather conditions, their
movements can be predicted with considerable accuracy.
When the moon is new or full, the gravitational forces of the sun and moon are pulling at
the same side of the earth (See Figure 1.4.1 below). This occurrence creates the extra
large "spring" tides.
Figure 1.4.1: Diagram of Spring Tides
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
16
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
When the moon is at first and third quarter, the gravitational forces of the sun and moon
are pulling at 90 degrees from each other (See Figure 1.4.2 below). This occurrence
yields little net tides called neap tides.
Figure 1.4.2: Diagram of Spring Tides
The General characteristics of tides in different locations are shown below:
Wind waves: T<20sec, H<20 m
Tidal waves: T=24 hours, diurnal type
T=12 hours, semidiurnal type
H=1-6 m. in Pacific Ocean
Mombasa:Kenya 6 m.
Rio: 1.6 m.
Tokyo: 1.6 m.
England: 6 m. La Haye (Den Haag)=2.5 m.
Black Sea: H<0.20 m.
The Sea of Marmara: H<0.30 m.
The Mediterranean Sea: H<0.40 m.
1.4.2
Swell Waves
It is a wave system not raised by the local wind blowing at the time of observation, but
raised at some distance away due to winds blowing there, and which has moved to the
vicinity of the ship, or to waves raised nearby by winds that have since died away. Swell
waves travel out of a stormy or windy area and continue on in the direction of the winds
that originally formed them as sea waves. The swell may travel for thousands of miles
before dying away. As the swell wave advances, its crest becomes rounded and its
surface smooth. Its length increases until it is approximately from 35 to 200 or more
times its height.
Swell waves normally come from a direction different from the direction of the prevailing
wind and sea waves at the time of observation. However, sea and swell waves may
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
17
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
occasionally be seen coming from essentially the same direction, thus making it more
difficult to distinguish the two systems, especially if the sea waves are high.
Sea waves and one or more systems of swell waves are frequently present at the same
time, forming "cross seas." Again, sea waves may be absent (as would occur under
conditions of very light winds), but one or more systems of swell waves may still be
present.
What is the Difference Between Sea and Swell Waves?
"Sea 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. The height of sea
waves depends on three factors: 1. Wind speed 2. Length of time that the wind blows
over the water 3. Fetch of water over which the wind blows.
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.
1.4.3 Seiches
Seiches are periodic oscillations of water level set in motion by some atmospheric
disturbance passing over a Great Lake. The disturbances that cause seiches include the
rapid changes in atmospheric pressure with the passage of low or high pressure weather
systems, rapidly-moving weather fronts, and major shifts in the directions of strong
winds. Seiches exist on the Great Lakes, other large, confined water bodies, and on
partially-enclosed arms of the sea. The intervals (or periods) between seiche peaks on
the Great Lakes range from minutes to more than eight hours.
The term was first promoted by the Swiss hydrologist François-Alphonse Forel in 1890,
who had observed the effect in Lake Geneva, Switzerland. The word originates in a
Swiss French dialect word that means "to sway back and forth", which had apparently
long been used in the region to describe oscillations in alpine lakes.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
18
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Figure 1.4.3.1: Image courtesy of Professor Brennan, Geneseo State Univ. of New York)
The period of a seiche in an enclosed rectangular body of water is usually represented
by the formula:
in which L is the length, d the average depth of the body of water, and g the acceleration
of gravity.
Figure 1.4.3.2: Differences in water
level caused by a seiche on Lake
Erie, recorded between Buffalo, New
York (red) and Toledo, Ohio (blue) on
November 14, 2003
Larger seiches can be caused by wind, earthquakes, or underwater landslides. Seiches
caused by earthquakes are termed as seismic seiches, a term coined by Anders Kvale in
1955 to describe oscillations of lake levels in Norway and England caused by the M8.6
1950 Chayu earthquake. More recently the M7.9 Denali earthquake in 2002, caused
seiches as far as Louisiana and many other states in the continental United States. In
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
19
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
South Asia, the M7.6 Bhuj earthquake is also believed to have caused seiches in rivers
in Bangladesh and the M8.1 1934 Bihar earthquake caused a seiche in Lake Vembanad,
Kerala, capsizing many canoes. A great M8.5 earthquake in Indonesia in 1861 also
caused seiches that were noticed at many locations in Bengal and Orissa, including
Kolkata. The recent great M9.0 Sumatera-Andaman earthquake also generated
energetic seiches in eastern India, Bangladesh and Thailand.
Seismic seiches can be observed thousands of miles away from the epicentre of a
quake. Swimming pools are especially prone to seiches caused by earthquakes, as the
ground tremors often match the resonant frequencies of small bodies of water. The 1994
Northridge earthquake in California caused swimming pools to overflow across southern
California, and the massive Good Friday Earthquake that hit Alaska in 1964 caused
seiches in swimming pools as far away as Puerto Rico. Following the 2004 Indian Ocean
earthquake, it was postulated that the shock waves could have induced seiches as far
away as Oklahoma.
1.4.4. Freak Waves
Freak, rogue, or giant waves correspond to large-amplitude waves surprisingly appearing
on the sea surface (“wave from nowhere”). Such waves can be accompanied by deep
troughs (holes), which occur before and/or after the largest crest. Seafarers speak of
“walls of water”, or of “holes in the sea”, or of several successive high waves (“three
sisters”), which appear without warning. But since the 70s of the last century,
oceanographers have started to believe them. Observations gathered by the oil and
shipping industries suggest there really is something like a true monster of the deep that
devours ships and sailors without mercy or warning. There are several definitions for
such surprisingly huge waves. Very often the term “extreme waves” is used to specify the
tail of some typical statistical distribution of wave heights (generally a Rayleigh
distribution); meanwhile the term “freak waves” describes the large-amplitude waves
occurring more often than would be expected from the background probability
distribution. Sometimes, the definition of the freak waves includes that such waves are
too high, too asymmetric and too steep. More popular now is the amplitude criterion of
freak waves: its height should exceed the significant wave height in 2–2.2 times. Due to
the rare character of the rogue waves their prediction based on data analysis with use of
statistical methods is not too productive. During the last 30 years the various physical
models of the rogue wave phenomenon have been intensively developed and many
laboratory experiments conducted. The main goal of these investigations is to
understand the physics of the huge wave appearance and its relation to environmental
conditions (wind and atmospheric pressure, bathymetry and current field) and to provide
the “design” of freak wave needed for engineering purposes.
Recently, a large collection of freak wave observations from ships was given in the New
Scientist Magazine. In particular, twenty-two super-carriers were lost due to collisions
with rogue waves for 1969–1994 in the Pacific and Atlantic causing 525 fatalities. At
least, the twelve events of the ship collisions with freak waves were recorded after 1952
in the Indian Ocean, near the Agulhas Current, coast off South Africa. Probably, the last
event occurred in shallow water 4th November 2000 with the NOAA vessel; the text
below is an event description reproduced from Graham.
“At 11:30 a.m. last Saturday morning (November 4, 2000), the 56-foot research vessel
R/V Ballena capsized in a rogue wave south of Point Arguello, California. The Channel
Islands National Marine Sanctuary’s research vessel was engaged in a routine side-scan
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
20
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
sonar survey for the U.S. Geological Survey of the seafloor along the 30-foot-depth
contour approximately 1/4 nautical mile from the shore. The crew of the R/V Ballena, all
of whom survived, consisted of the captain, NOAA Corps officer LCdr. Mark Pickett,
USGS research scientist Dr. Guy Cochrane, and USGS research assistant, Mike Boyle.
According to National Oceanic & Atmospheric Administration spokesman Matthew Stout,
the weather was good, with clear skies and glassy swells. The forecasted swell was 7
feet and the actual swell appeared to be 5–7 feet. At approximately 11:30 a.m., Pickett
and Boyle said they observed a 15-foot swell begin to break 100 feet from the vessel.
The wave crested and broke above the vessel, caught the Ballena broadside, and quickly
overturned her. All crewmembers were able to escape the overturned vessel and deploy
the vessel’s liferaft. The crew attempted to paddle to the shore, but realized the
possibility of navigating the raft safely to shore was unlikely due to strong near-shore
currents. The crew abandoned the liferaft approximately 150 feet from shore and
attempted to swim to safety. After reaching shore, Pickett swam back out first to assist
Boyle to safety and again to assist Cochrane safely to shore. The crew climbed the rocky
cliffs along the shore. The R/V Ballena is a total loss.”
Various photos of freak wave are displayed in Fig. 1.4.4.1. The description of the
conditions when one of the photos (left upper) was taken is given below.
“A substantial gale was moving across Long Island, sending a very long swell down our
way, meeting the Gulf Stream. We saw several rogue waves during the late morning on
the horizon, but thought they were whales jumping. It was actually a nice day with light
breezes and no significant sea. Only the very long swell, of about 15 feet high and
probably 600 to 1000 feet long. This one hit us at the change of the watch at about noon.
The photographer was an engineer, and this was the last photoon his roll of film. We
were on the wing of the bridge, with a height of eye of 56 feet, and this wave broke over
our heads. This shot was taken as we were diving down off the face of the second of a
set of three waves, so the ship just kept falling into the trough, which just kept opening up
under us. It bent the foremast (shown) back about 20 degrees, tore the foreword
firefighting station (also shown) off the deck (rails, monitor, platform and all) and threw it
against the face of the house. It also bent all the catwalks back severely. Later that night,
about 19:30, another wave hit the after house, hitting the stack and sending solid water
down into the engine room through the forced draft blower intakes.”
These photos and descriptions show the main features of the freak wave phenomenon:
the rapid appearance of large amplitude solitary pulses or a group of large amplitude
waves on the almost still water in shallow as well as in deep water. They highlight also
the nonlinear character of the rogue wave shapes: steep front or crest beard, and also
two- and three-dimensional aspects of the wave field.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
21
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Figure 1.4.4.1: Various Photos of Freak Waves
1.4.5 Resonance of Basins
Resonance may be described as the coincidence of natural period of oscillatory motion
of a system with the period of external effect on this motion and the resultant increase in
the magnitude of motion. This oscillatory motion-not exactly motion actually- may be
caused by sound, magnetic effects, waves etc. For instance, considering a woman on a
swing is a good and simple example. Since the swing is making an oscillation, it certainly
has a free oscillation period, say Ti. Besides, if a man is pushing the swing from back,
thus putting the external effect on this motion, another oscillation period , say Text, is
added on the system. If free oscillation period (Ti) is equal to period of external effect,
say Text, speed of this system gains an unexpected increase in its magnitude. This
concept is called resonance, and the corresponding period is called resonance period
(free oscillation period). Trying to swing the woman faster by giving an external force
with higher or smaller oscillation periods definitely does not give expected results.
From coastal engineering point of view, every closed basin also has its own free
oscillation period. Determination of these oscillation periods is considerable since
incoming waves with this oscillation periods make effect within basin higher than
expected.
When a wave enters a closed basin (harbor, bay etc.), surface fluctuations occur within
the basin which affects coastal infrastructure, navigation of marine vehicles, public safety
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
22
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
etc. Thus, magnitudes of these fluctuations are significantly important and they depend
on 2 basic parameters: Boundary conditions of the system and incoming wave
properties. Determination of incoming wave properties is the very first step in coastal
engineering projects. However, system properties are never less important than the
wave properties. The major properties and boundary conditions of a system are:
-
the geometry
depth profile
energy reflection and dissipation characteristics
These boundary conditions give the resonance period of basin. Resonance period of a
coastal basin is the critical value which the incoming waves with this period cause
unexpectedly high fluctuations in the system. A basin has more than one resonance
periods. The larger a basin is, the greater its free oscillation period is. This means that
longer waves (with longer periods ) become more important for larger basins.
The periods of free oscillations (Tn) inside a closed basin (the boundaries are vertical,
solid, smooth and impermeable) is in general
n=1, 2, 3…
(3)
where l is the length in the direction of wave and d is the depth of the basin, and n is
integer number represents each mode.
If the basin is semi enclosed (one of the boundaries is open boundary) then the periods
of free oscillations become
n= 1, 2, 3…
(2)
The waves propagating inside the basins may cause long wave generation with higher
amplitudes if their period coincides with any period of free oscillations of the basin. There
are theoretical methods for computing the periods of free oscillations for the regular
shaped basins, but there is not any specific theoretical approach for determination of the
periods of free oscillations for irregular shaped basins and thus numerical or semi
numerical approaches has been developed for problem. Classically, several numerical
computations can be performed to calculate the long wave amplitudes inside the basins
in relation to the inputted waves with certain periods. When this test is repeated by
inputting waves with different periods separately and the computed amplitudes can be
compared and the period causing highest amplitude can be selected as one of the
periods of free oscillations of the basin. This method needs many test runs with many
different input wave periods and takes extremely long time efforts for determination of a
single period of free oscillations.
A method for computing periods of free oscillations of the irregular shaped basins is
presented in Yalciner et. al. (1993, 1995) and used in Çakıroğlu (1997) and Karatoprak
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
23
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
(2000) is an alternative short cut numerical method to the classical method, in which the
periods of oscillation were obtained by the output data of a single run.
The computer programs used were developed in the Disaster Control Center, Tohoku
University, Japan, and modified for the seas around some bays around Turkey by
Yalçıner et. al. (1993, 1995). The equations used for the determination of the periods are
presented in the following sections.
Long waves have small relative depth (the ratio of water depth to wave length) for which
the vertical acceleration of water particles is negligible compared to the gravitational
acceleration and the curvature of trajectories of water particles are relatively small. As a
result, the pressure can be approximated as hydrostatic since the vertical motion of the
water particles has no effect. It can be assumed that horizontal velocities of water
particles are vertically uniform.
Based upon above approximations the motion of long waves is expressed by the
following equations (Shuto, Goto, Imamura 1990).
  uh +    vh +  


0
t
x
y
u
u
u
  x
u
v
g

0
t
x
y
x 
v
v
v
  y
u
v
g

0
t
x
y
y 
(1)
(2)
(3)
Where x and y are horizontal coordinates, t time, h the still water depth,  the vertical
displacement of water level, u and v water particle velocities in the x- and y-directions, g
gravitational acceleration,  x /  and  y /  bottom friction in the x and y directions. For
uniform flow, the bottom friction is expressed as:
x  1

f
u u2  v 2 ,
2D
y  1

f
v u2  v 2
2D
(4)
where D is the total water depth defined by +h and f is the friction coefficient. f is
expressed by using the Manning’s roughness n as follows,
n
f D1/ 3
2g
(5)
By using D, total depth as defined h + , the discharge fluxes M and N in x, y directions
respectively can be described in relation with to u and v by the following expressions:
M  u (h   )  uD , N  v(h   )  vD
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
(4)
24
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
If we define shear stress in terms of Manning’s coefficient n and if the equations (1-3) are
integrated from sea bottom to water surface, the following equations are obtained for
discharge fluxes M and N.
 M  N


0
t x y
(5)
 M   M 2    MN 
 gn 2



  gD


M M 2  N2  0
 t  x D   y D 
 x D7/3
(6)
 N   MN    N 2 
 gn 2

  gD

 7/3 N M 2  N 2  0


 t  x D   y D 
 y D
(7)
The numerical model TUNAMI-N2, used for the simulation of the propagation and coastal
amplification of long waves. The model was originally created by Prof. Imamura in
Disaster Control Research Center in Tohoku University (Japan) through the Tsunami
Inundation Modeling Exchange (TIME) program. TUNAMI-N2 is one of the key tools for
developing studies for propagation and coastal amplification of tsunamis in relation to
different initial conditions. It solves the modified form of Eqs. 5-7 using depth averaged
velocities and discharge fluxes with the leap-frog scheme in finite difference technique for
the basins of irregular shape and topography. The program can compute the wave
propagation at all locations, even at shallow and land regions (Imamura, 1996) within the
limitations of grid size.
The computer program TUNAMI-N2 developed in the Disaster Control Center, Tohoku
University, Japan are used for the simulation of long waves. and it is based on the
nonlinear form of the long wave equations The models have been applied to several
case studies for the propagation of the tsunamis in the oceans, seas and the bays.
(Imamura and Goto, 1988; Imamura and Shuto, 1989; Yalciner et. al., 1995). The run-up
can also be outputted in the runs of TUNAMI-N2. One of the applications of these models
is to investigate the modes of free oscillations of the basins with any geometry and
bathymetry.
In the method the forcing function is inputted as a single wave with a period that is
shorter than the resonance periods. The equation of motion can be solved for unknown
(t) to analyze it by Fourier Transformation Technique. This analysis gives the spectrum
curve for (t) and the peaks of this curve are the periods of free oscillation of the system.
The procedure explained is advantageous in the way that the periods of oscillation can
be determined by a single numerical computer run. Details on determination of the
periods of oscillation, test of the method and limitations will be discussed further in the
following sections.
The application of the method should be carried out by taking care of the following
limitations and points.
The external force, i.e. the initial wave given to the basin should have a wave period that
is in the limits of long wave. On the other hand, it should not be forgotten that the period
of the initial wave is the lower limit of the periods of oscillation computed in the study.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
25
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
The duration of analysis should be long enough to completely investigate the water
surface fluctuations caused by the disturbing wave. Otherwise, some of the longest
periods of oscillation may not be found. Smaller time steps increase the accuracy and the
data stored for the Fourier Transformation. This makes the frequency spectrum more
precise to find the correct periods located at the peaks.
The basin should be described by regularly spaced grid points, which have a grid
spacing small enough to properly describe the boundary conditions and the bathymetry
of the basin. Smaller grid spacing will result in less numerical errors.
Quantity and quality of the locations chosen should be enough to describe the
mechanism of the basin. Locations chosen should be well distributed along the basin, as
the periods of oscillation will be chosen from the periods of oscillation for the selected
locations. Number of locations should be enough to prevent wrong judgment due to
special conditions of some locations.
The method used in this study was tested by Çakıroğlu (1997), for regular shaped basins
whose resonance periods are known analytically.
The results obtained in those tests are in close agreement with the results given in Bruun
(1981) for regular shaped basins. It was also determined that the shape of initial impulse
in this study has no effect on the results.
The results have also been confirmed by testing the regular shaped basins with classical
m.
Test of the Method
For verification of the proposed method, rectangular shaped basin is chosen, (Table
1.4.5.1). In the table the plan form and depth profile with the first four periods of free
oscillations are shown, (Per Bruun, 1981). Agitation is triggered by giving a triangular
shaped impulse with 10 m. height and 100 sec. period, from the open boundary, (Figure
1). The grid map of the basin is prepared by using 500 m. grid size, (Figure 2). The time
step is chosen 0.3 seconds. The water surface fluctuations at 7 points located in different
parts of the basin, are stored at every time step. By using the Fast Fourier
Transformation technique the energy spectrum analysis are applied to the data of water
surface fluctuations. The modes of free oscillations are chosen by detecting the periods
(or frequencies) which correspond to the peak values in the energy spectrum curve.
Water surface fluctuation and spectrum curve of a chosen point on the basin are given, in
Figure 3-4. The results are coincides with the theoretical values, (Table 2).
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
26
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Table 1.4.5.1: Modes of Free Oscillations in Semi-enclosed Rectangular Basin with Horizontal
Sea Bottom
67 km.
open boundary
Figure 1.4.5.1: The Shapes of Two Different Impulse Waves
d = 500 m.
105 km.
Figure 1.4.5.2: Map of the Rectangular Basin.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
27
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
Table 1.4.5.2: Periods of Free Oscillations of Rectangular Basin (in minutes)
MODE
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
9
10
93.6
31.2
18.7
13.7
10.6
9.5
8.6
7.9
6.4
5.6
93.6
31.2
19.3
13.7
10.6
8.6
7.4
6.4
5.6
93.6
31.2
19.3
13.7
10.6
8.6
7.3
6.4
5.6
93.6
31.2
19.3
13.7
10.6
8.6
7.3
6.4
5.6
93.6
31.2
18.7
13.7
10.6
8.6
7.3
6.4
5.6
93.6
31.2
18.7
13.7
10.6
9.5
8.6
7.3
6.4
5.6
93.6
31.2
19.3
13.7
10.6
8.6
7.3
6.4
5.6
Bruun
(1981)
99.9
33.3
20.0
14.3
11.1
9.1
7.7
6.7
5.9
125.0
150.0
10.00
7.50
Point 1
Elevation (m.)
5.00
2.50
0.00
-2.50
-5.00
-7.50
-10.00
0.0
25.0
50.0
75.0
100.0
Time (min.)
Figure 1.4.5.3: Water Surface Fluctuations of the Rectangular Basin for Point 1
450.0
375.0
Point 1
S(w)
300.0
225.0
150.0
75.0
0.0
0.000
0.005
0.010
0.015
0.020
Frequency (1/sec.)
Figure 1.4.5.4: Spectrum Curve of the Rectangular Basin for Point 1.
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
28
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 1
For further information, see:
- http://earthobservatory.nasa.gov/NaturalHazards/natural_hazards_v2.php3?topic=flood
- http://www.usgs.gov/themes/flood.html
- http://www.fema.gov/hazards/floods
- http://earthobservatory.nasa.gov/NaturalHazards/natural_hazards_v2.php3?topic=volcano
- http://www.jpl.nasa.gov/earth/natural_hazards/volcanoes_index.cfm
- http://earthobservatory.nasa.gov/Library/Hurricanes/
- http://www.fema.gov/hazards/hurricanes/
- http://www.gesource.ac.uk/hazards/Storms.html
- http://earthobservatory.nasa.gov/Library/DroughtFacts/
- http://earthobservatory.nasa.gov/Library/DroughtFacts/
- http://pubs.usgs.gov/gip/earthq1/intro.html
- http://www.enchantedlearning.com/subjects/tsunami/
- http://njnie.dl.stevens-tech.edu/curriculum/tide.html
- http://www.mid-c.com/manmar/Swell.htm
- http://lighthouse.tamucc.edu/Waves/SeaSwellDefinition
- http://www.seagrant.wisc.edu/CoastalHazards/Default.aspx?tabid=426
- http://en.wikipedia.org/wiki/Seiche
- http://asc-india.org/menu/waves.htm
- http://en.wikipedia.org/wiki/Seiche
- Pelinovsky, Kharif, “Physical mechanisms of the rogue wave phenomenon”, European
Journal of Mechanics B/Fluids 22 (2003) 603–634,September 2003
© A.C.Yalçıner, H.Karakuş, C.Özer, G.Özyurt
29