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Team #2311 Problem B Page 1 of 31
For office use only
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2009
12th Annual High School Mathematical Contest in Modeling (HiMCM) Summary Sheet (Please attach a copy of this page to each copy of your Solution Paper.)
Team Control Number: 2311 Problem Chosen: B
Please type a summary of your results on this page. Please remember not to include the name of your school, advisor, or team members on this page.
In order to best measure the effect of a tsunami on a coastal city, we tested the severity of damage to the infrastructure
in several American cities using several factors such as the slope of the city, the density of buildings in the city, the
length of coastline, and a likely epicenter for the earthquake. By compiling this data we were able to estimate the
devastation of earthquakes that would register 5.0, 7.0, and 9.0 on the Richter Scale based on the number of buildings
the tsunami would reach and the mean price of buildings in each city.
The first segment of our model measures the wave from the epicenter to cresting. To ascertain this, we gathered
relevant data on each city, for example, statistics included the dimensions of epicenter (meaning the depth and distance
from shore) and the terrestrial features off the coast. With this, we determined the average energy of the wave leading
up to the impact and how the protruding features, like the harbors and islands, would slow the tsunami. The aim of this
portion of the model was to find the mass, velocity, amplitude, and kinetic energy of the wave just as it begins to fall on
the city, thereby causing the destruction. We developed an equation that uses the provided parameters to find these
desired quantities and describes the characteristics of the wave.
The second step of our model aimed to measure the destruction a wave would cause once it reached the shore and
subsequently broke. This section relied on the first segment’s results in addition to the dimensional size of each city and
the demographics of buildings in each city. We determined which buildings would be affected by finding the highest
point the wave would climb and figured the number of buildings that were in that swath of land. We derived an
equation that outlines the work needed to seriously harm each building, assuming generalizations on the traits of all the
buildings. Next, total monetary cost was compiled, stemming from ratio of commercial to residential property and the
price differences of each in all the cities.
The fiscal cost of tsunamis fluctuated immensely with respect to Richter Scale value and the city. The most damaging
example is a 9.0-magnitude tsunami in New York City resulting in $3.98 billion worth of reconstruction and repair. On
the other end of the spectrum, a 5.0-magnitude landing in Hilo, Hawaii totals to $21.5 million in infrastructure
devastation. The amounts appear daunting in scope, more notably on the Atlantic and Gulf of Mexico coasts; however,
the likelihood of any sized tsunami in these areas is minuscule to insignificant. In summation, our model successfully and
precisely estimates the cost of variously sized tsunamis that reach shore in American cities.
Team #2311 Problem B Page 2 of 31
Table of Contents
Restatement of the Problem 3
Assumptions with Justifications 3
The Model 5
Part I: Epicenter to Shore 5
Part II: Devastation of Cities 7
Part III: Putting the Pieces Together 9
Part IV: Application of Model to Cities 9
Hilo, HI 10
San Francisco, CA 10
New Orleans, LA 11
Charleston, SC 11
Boston, MA 12
New York, NY 12
Corpus Christi, TX 13
Strengths and Weaknesses 14
Extensions 14
Appendices 15
Appendix A: Equation Derivations 15
Appendix B: Data 21
Appendix C: The Cities 26
Bibliography 29
Article to the Local Newspaper 31
Team #2311 Problem B Page 3 of 31
Restatement of the Problem
Through building a mathematical model, measure the effect of earthquake-generated tsunamis impacting several
different coastal cities. Use the following cities for analysis: Hilo, HI; San Francisco, CA; New Orleans, LA; Charleston, SC;
New York City, NY; Boston, MA; and Corpus Christi, TX. Using property damage, losses of life, or any other reasonable
indicator(s), calculate the devastation caused by tsunamis that vary in magnitude. Write a letter to the editor of one of
the cities’ periodicals describing the results.
Assumptions and Justifications
A1. Weather variables, such as wind, precipitation, and air and water temperature, have a negligible effect on the
intensity of a tsunami.
-The aforementioned factors are very diverse in the given cities and vary drastically from season to season, meaning
no unified approach can be applied to them.
A2. Water is an ideal fluid, so viscosity is negligible.
-We found that this occurs under ideal conditions through our research, and we assume that we are dealing with
ideal conditions.
A3. The shape of the wave is sinusoidal.
-A sinusoidal shape is the model closest to the actual shape of the wave. Also, making the shape sinusoidal makes
finding the volume cross-sectional area of the wave less arduous.
A4. During a tsunami, there is only one wave that causes damage.
-Depending on the location and earthquake, the difference in time and magnitude of multiple waves would create
several different cases for the interactions of multiple waves. These extra variables made the math very difficult, so we
were forced to eliminate them.
A5. 1% of the energy produced from earthquake is transmitted to the tsunami.
-This simplifies the problem and is approximately equal for all earthquakes. Most of the energy goes into P and S
waves through the earth, heat, sound, and movement of water that doesn't support a tsunami.
A6. An earthquake that registers 5.0 on the Richter Scale will produce a wave with an initial amplitude of 1 meter. Each
step up on the Richter Scale will increase the amplitude by a factor of 1.32.
-Past research has no decisive conclusion on a relationship between earthquake magnitude and wave amplitude,
instead the consensus is that a typical base value is 1 meter and each progressive step marginally increases the wave
amplitude.
A7. While the wave is traveling through deep water, it does not lose any energy.
-This is approximately true. The energy lost is mainly through viscosity and air resistance. Because viscosity is
negligible and the amplitude is relatively small, both of these factors are negligible.
A8. All points on the ring of waves produced by the tsunami have the same energy (up until the waves encounter the
shore of the target city).
-This allows us to look specifically at the features of the target city and calculate energy without considering
Team #2311 Problem B Page 4 of 31
geological figures elsewhere.
A9. Before the wave crests, the area in the cross-section of the wave remains constant.
-This is approximately true, and assuming so allows us to calculate the effects of changing depth on changing height
of the wave.
A10. The presence of a harbor reduces the kinetic energy of the incoming wave by 5%.
-The shapes of all harbors are very similar. Thus, all harbors will have almost exactly the same effect on the tsunami.
This number was developed through research.
A11. The presence of barrier islands, breakwaters, or other islands will reduce the kinetic energy of the incoming
tsunami according to the equation 4
4f i
KE KEW
=+
-All obstacles of this sort are the same in that they all inhibit the wave's motion and reduce its kinetic energy as it
passes over/around it. This drop is proportional to the width of the obstacle. In addition, all islands of this type are
relatively flat and at sea level.
A12. The bayous surrounding New Orleans affect the loss of energy of the tsunami 25% as much as an island of the same
size.
-New Orleans is surrounded by bayous, and the tsunami would have to pass through a bayou in order to get to New
Orleans. The bayous would cause some loss of energy because they are not open water, but at the same time, they do
have some water, so they would not cause as much of a loss of energy. In addition, the rivers and streams connecting
these bayous to New Orleans would transmit the tsunami more easily. This assumption allows us to calculate the energy
loss of the tsunami as it travels through the bayous.
A13. Once the wave crests, it is no longer a wave; it is a mass of water moving towards the city.
-Again, this is nearly true, and assuming so allows us to apply conservation of energy to calculate the damage done
by the tsunami.
A14. The wave crests completely before encountering shore.
-This causes the water to be approximately level when it reaches shore. This allows us to accurately consider the
wave cresting and then hitting shore as a level mass of water rather than having to include a complicated expression
that explained cresting and destroying and the same time.
A15. The land in all of the cities is linearly sloped, except the bayous in New Orleans, which are flat (see assumption 12),
and the mountain in San Francisco (Mt. Sutro), which is impassable.
-By looking at topographic maps, this is true. The slopes are not exactly linear, but by approximating it as linear, we
can greatly simplify the problem. Because the mountain in San Francisco is so high relative to the rest of the city (925ft
compared to 52ft) and the mountain is so far inland, no feasible tsunami can pass over the mountain.
A16. The distribution of buildings and the population density will be uniform throughout.
-Obtaining the block by block layout of several cities would be difficult, time-consuming, and superfluous; and add
little to our generalized model.
Team #2311 Problem B Page 5 of 31
A17. All affected buildings in cities under consideration can be considered brittle, thus experiencing minimal
deformation before breaking.
-Many building materials, such as wood and concrete, are brittle. Brittle materials have a linear stress vs strain curve,
which makes modeling collisions between the tsunami and building much simpler.
A18. The average building in any city is two stories tall.
-The average building height near the coast will not be drastically different for different cities. Assuming constant
height allows us to simplify our model and eliminate one variable.
A19. All friction, heat, noise loss, and other sources of energy loss as the tsunami advances inland (besides change in
gravitational potential energy and building destruction) are negligible.
-These diverse causes of energy loss would be very difficult to account for, and because they are relatively small
compared to the total energy of the tsunami, they can be ignored.
The Model
From the time the earthquake occurs until after the tsunami has passed, the tsunami acts like two different objects.
Before the wave crests, it acts like a wave: a sinusoidal disturbance passing through the water. When it leaves the ocean
and moves onto land, it acts like a moving mass of water. Instead of trying to design a model that encompasses both of
these facets, we developed a model comprised of two parts: one to deal with how the wave reaches shore, and one to
explain how the wave interacts with the land and destroys buildings.
Part I: Epicenter to Shore
Before a tsunami occurs, tension in the Earth’s tectonic plates builds up. The epicenters of earthquakes that cause
tsunamis are usually at a site where an oceanic plate is subducting underneath a continental one. Because of friction,
the continental plate is compressed and bent backwards. When this tension reaches a critical level, it releases in a very
rapid movement, pushing upwards and forwards. The upward part of this movement displaces water. This causes a
ripple effect to emanate in a circular ring from the epicenter.
(For all of the following equations, see Appendix A for complete derivations)
The magnitude of an earthquake can be defined by the Richter Scale: a base ten logarithmic scale. Because the end goal
of the model is to calculate the amount of destruction done by a tsunami in a set of given conditions, we want to know
the energy of the incoming wave. To find this, we need to know the energy of the initial earthquake. According to our
research, the relationship is
34.8
210R
E+
= [Equation 1.1]
Where E is energy in Joules and R is the Richter Scale magnitude. In addition, we need to find the initial amplitude of this
wave. This is helpful because it allows us to find when the wave will begin cresting. Because the Richter Scale is
logarithmic, the desired relationship should be exponential. The lowest magnitude of earthquake that can produce a
tsunami is 5.0 on the Richter Scale. In addition, the smallest initial amplitude of a tsunami is 1 meter. The highest
recorded magnitude of earthquake that produced a tsunami is approximately 9.0 on the Richter Scale and the initial
amplitude was 3 meters.
Team #2311 Problem B Page 6 of 31
Using these two data points, we can derive the relationship
Ai = [Equation 1.2]
Where Ai is the initial amplitude in meters and R is the Richter Scale magnitude. Finally, the velocity of the created wave
is needed because it allows us to calculate the final speed of the wave as it approaches shore. The velocity of this wave
is defined as
i iv gd= [Equation 1.3]
Where vi is the initial velocity in m/s, g is the gravitational acceleration (9.81 m/s2), and di is the initial depth of the
water, or the depth of the epicenter.
The wave loses no energy as it travels towards the shore, and because the water is so deep, it doesn’t lose any
speed either. Once the wave reaches significantly shallow water, the ground forces the wave to slow down. However,
the frequency of the wave cannot change, so in order to keep the frequency constant when velocity decreases is to
decrease the wavelength as well. This means that the wave bunches up. Because the same amount of water needs to fit
in a smaller space, the amplitude of the wave increases significantly. At a certain point, the amplitude will grow so tall
relative to the depth of the water that it becomes unstable. At this point, the wave begins to crest. We need to find this
point because then we can find the final velocity, depth, and amplitude before the wave crests. The wave becomes
unstable once the amplitude is at least .88 times the depth of the water because of Assumption A2. So, if we can find
the point when the amplitude is equal to .88 times the depth of the water we can find the cresting depth. We developed
a relationship using calculus to relate the cresting depth and amplitude to the initial depth and amplitude. We can solve
this for the cresting depth and get
231.089f i id A d= [Equation 1.4]
Where Ai is the initial amplitude and di and df are the initial and cresting depths respectively. Using this, we can also find
Af by multiplying by .88. Using equation 1.2, we can find the cresting velocity. Finally, we want to find the energy of the
incoming wave. Because Etsunami = 1% x Eearthquake,
32.8
210R
tsunamiE
+
= [Equation 1.5]
However, not all of the energy goes to the specified location. The fraction χ of energy that goes to the location is defined
as
2 rχ
π=
l [Equation 1.6]
Where l is the coastal length in meters and r is the distance from the location to the epicenter. Thus, the energy
brought to the specific area of the city in the tsunami is
32.8
210city tsunam
R
iE Eχ χ
+
= = [Equation 1.7]
Thus, we now have the amplitude, energy, depth, and velocity of the wave when it begins to crest.
Team #2311 Problem B Page 7 of 31
Because the wave will crest and collapse before it reaches the shore, it will level out at the average height of the wave.
Using the mean value theorem of integrals, we can find
231.917i i
A dAV
π= [Equation 1.8]
Using this, we can calculate the gravitational potential energy of the wave.
23
2
1.917f f i i
g
A g A dPE
ρ λ
π=
l [Equation 1.9]
Because the center of mass was at the height of the average value, but now it is at one-half of the average value, so one-
half of its gravitational energy is transformed to kinetic energy. Because of conservation of energy, the wave’s final
kinetic energy after it falls is equal to the initial kinetic energy before it fell plus the gravitational potential energy
converted to kinetic energy. Thus,
2 2 2 23 3
2
( ) .8433 ( )
2
i i i i
f
g A d g A dKE
ρ λ ρ
π π= +
l l [Equation 1.10]
This value can then be used in the second part of the model where devastation is calculated.
Part II: Devastation of Cities
The second part of our model regards the wave after it hits the shore. Our goal is to calculate the devastation in terms of
cost of damage. To do this, we found the total area affected for each city, multiplied by the average number of buildings
per area, and multiplied this by the average cost of a building. The latter two items were found through research, thus,
this section of our model was mainly concerned with determining the total area of each city affected. Rather than
attempt to account for the complex fluid dynamics of a water wave climbing a shore, after the wave crested, we treated
it as a single mass and used conservation of energy to find the maximum height.
An expression for kinetic energy was found in the previous section of the model. Applying the work-energy theorem,
f i
W E
W PE KE
= ∆
= −
∑∑
[Equation 2.1]
In this case, the only work done on the tsunami is the work of collisions with buildings and other objects. Using the
formula for shear stress,
F xS
A h
∆= • [Equation 2.2]
We derived a formula for the work required to destroy a building:
2
2
VUW
S= [Equation 2.3]
Team #2311 Problem B Page 8 of 31
Where W is the work required, V is the volume of the building, U is the ultimate strength of the building, and S is the
shear modulus.
Initially we used this equation to calculate the work done whenever the tsunami strikes a building. However, we later
realized that not every building struck by the tsunami will be completely demolished. To account for this, we multiplied
the work done by each building by a factorv
v k+, where v is the velocity of the wave and k is a constant. We chose to
make the work proportional to velocity because we reasoned that the faster the wave travels, the more each object
struck will be demolished, and therefore the more work will be done on the wave by each building. The constant k was
determined based on data gathered from tsunamis that struck Hilo, Hawaii.
We then used this value for work to determine the height at which the kinetic energy is zero, which is the maximum
height reached by the wave.
2
2
f i
i
W PE KE
VU vbA Mgy KE
S v k
= −
− • = −+
[Equation 2.4]
Where b is the number of buildings per square meter and A is the total area affected.
Using m as the slope of the bank up which the tsunami climes,
2
11A y
m= +l [Equation 2.5]
Where ℓ is the length of the coastline in meters.
Substituting and solving for y,
2
2
11
2
iKE
yVU v
Mg bS v k m
=
+ • • ++
l
[Equation 2.6]
y = maximum height reached by wave (m)
M = mass of wave (kg)
g = acceleration due to gravity (m/s^2)
V = average volume of a building (cu. m)
U = ultimate strength of average building (Pa)
S = shear modulus of building material (Pa)
v = velocity of wave (m/s)
b = # of buildings per square meter
Team #2311 Problem B Page 9 of 31
ℓ = length of coastline (m)
m = slope of ground in the city
To simplify calculations, we used average values for U, V, and S. U and S were found through research, and the average
volume of a building was found for each city. The average volume was calculated by multiplying the percent of buildings
that are residential in a city by the average volume of a residential building, and adding the result to the percent of
buildings used commercially multiplied by the average volume of a commercial building. These data are all listed in
Appendix B.
Using equation 2.5, area affected was calculated from y. Multiplying this result by b (buildings/sq. meter) and the
average cost of a building for each city (C), the total devastation was found:
0 2
2
2
11
11
2
Cb KEm
DVU v
Mg bS v k m
• +
=
+ • • ++
l
[Equation 2.7]
Part III: Putting the Pieces Together
Both of these pieces of our model contain a large number of equations. By combining all of the equations in the first part
of the model, we obtain equation 1.10. By combining all of the equation in the second part of the model, we find
equation 2.7. However, we can consolidate our model even further. Although the equation is very large, we can
substitute equation 1.10 (kinetic energy) into equation 2.7 and find one expression for the destruction a tsunami causes
a city.
2 2 2 23 3
2 2
2
2
( ) .8433 ( ) 1( ) 1
2
11
2
i i i ig A d g A d
Cbm
VU vMg b
S v k m
D
ρ λ ρ
π π+ • +
+ • •+
=
+
l l
l
[Equation 3.1]
Part IV: Application of Model to Cities
For each city under consideration, eight factors were used to calculate devastation:
1. Distance of city from fault line.
2. Depth of fault line.
3. Length of coastline.
4. Features around coast (e.g. harbor/bay, island, bayous, breakwater).
5. Slope of city land.
6. # of commercial and residential buildings in the city.
7. Area of the city
8. Average building cost in the city.
Team #2311 Problem B Page 10 of 31
Factors 1 through 4 are used to determine the mass, velocity, height, and energy of a tsunami wave as it hits the land.
Factors 3, 5, 6, and 7 are used to calculate the distance inland the tsunami reaches and the number of buildings affected.
Factor 8 is used to calculate the cost of damage.
For each city, all data used and figures calculated are presented in Appendix B. Here, the horizontal distance inland
reached by the tsunami, the number of buildings affected, and the total cost of damages are given for tsunamis
generated by earthquakes of magnitude 5, 7, and 9 on the Richter scale.
(See Appendix C for visual representations of each city)
Hilo, HI
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 85.563 57 21.503
7 161.712 108 40.640
9 283.961 190 71.362
Hilo is one of the few cities that regularly experience tsunamis because of its location in the Pacific. Hilo is characterized
by a very low population and a high land slope. It is also located on a funnel-shaped bay called the Hilo Bay, which
increases its chances of encountering tsunamis. However, due to demographic and topographic eccentricities, Hilo has
the lowest devastation out of all the cities we tested with our model.
San Francisco, CA
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 3.929 51 42.566
7 9.619 124 104.192
9 23.195 298 251.236
San Francisco proves difficult to model for several reasons. First is the elevation. Most of the city is fairly flat, with the
exception of the central mountains. One of our assumptions is that all cities have constant linear slopes, but a linear
model clearly could not account for mountains. Eventually, we decided that because of the mountains' height and
distance from the coast, no tsunami wave could feasibly get over the mountain, so we disregarded them in our model.
The second difficulty with San Francisco is the San Francisco Bay. The front of the city will be hit normally by a tsunami
wave, but a portion of the wave will also curve around into the bay and hit the city from behind. Half of the wave energy
will hit the city normally, at full strength, and the other half will be diminished by an assumed value of 5% in accordance
with assumption #22. Thus, the average energy of a wave hitting San Francisco is 97.5% of the energy of the wave.
Team #2311 Problem B Page 11 of 31
New Orleans, LA
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 30.763 1043 159.609
7 76.388 2591 396.323
9 188.453 6392 977.752
New Orleans is one of the cities most at risk from tsunami waves, having the greatest number of buildings affected and
total cost of damage after New York City. The flatness of the terrain of New Orleans contributes to its vulnerability, as
the wave can penetrate farther horizontally into the city without losing kinetic energy by rising vertically. The long coast
of New Orleans means that the city has more area exposed to a tsunami. These factors apply to hurricanes as well as
tsunamis, and contributed to the massive devastation caused by Hurricane Katrina in 2005.
New Orleans has several unique features that affect the impact of a tsunami. First is a 1 mile wide barrier island. We
modeled the affect of barrier islands an breakwaters by multiplying the tsunami's energy by , where w is the
width of the barrier and n is a constant, determined from data to be approximately 4. Thus, a tsunami crossing a 1 mile
wide barrier island loses 1/5 of its total energy. After the island, tsunamis must traverse about 4 miles of bayous. We
estimated the energy lost in crossing the bayous is equal to 25% of the energy lost crossing an equivalent solid barrier
island. Thus, for 4 miles of bayous, the tsunami loses of its initial energy, after losing 1/5 of
its energy from the New Orleans barrier island. Thus, the energy of a tsunami striking New Orleans is its
energy on the open ocean.
Charleston, SC
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 145.953 98 25.998
7 262.613 175 46.778
9 441.690 295 78.676
The city of Charleston, South Carolina is a sharp departure from its eastern counterparts. It acts as a tsunami-foil in that
it has a low building price per unit, low building density, small coastline, and low population density. The combination of
these four factors indicates a strong resistance to tsunami damage. Charleston also has a harbor, which changes the
energy of the wave to 95% of its original. The possible damage caused by the three levels of tsunamis is minuscule when
contrasted with the totals of New York or Boston, amounting to less than a tenth of the cost of reconstruction.
Additionally, Charleston has a fairly high land slope so the sparse building distribution will be even less impacted by a
tsunami of any magnitude. The above characteristics of Charleston protect it from massive damage, and consequently
the death toll as well.
Team #2311 Problem B Page 12 of 31
Boston, MA
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 8.660 642 274.010
7 21.362 1584 675.912
9 52.113 3865 1648.939
The city of Boston, Massachusetts has two identifying traits, in reference to a tsunami's potential effect. The composite
price per building is comparatively high to the other cities and, for Boston; this is combined with an exceptionally high
ratio of buildings to square meter. What this does is intensify the tsunami's infrastructural damage. Since the buildings
are placed closely together and are expensive, the final cost of reconstruction will be increased relative to the other six
cities. Boston also has a large population density, which also fuels the tsunami's damage since building loss and death
toll will be highly correlated. The numerous islands scattered in Boston's harbor do act as a shield from tsunami and
decrease the energy of an oncoming tsunami by about 4/(1.5+4) = 0.73. Like other Eastern Seaboard cities, Boston has a
limited history of sea-bound earthquakes, which lessens the potential threat of a tsunami, but existence of a tsunami
would cost a radically vast amount of money.
New York, NY
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 24.461 1165 635.927
7 61.227 2916 1591.727
9 153.115 7293 3980.546
New York City, the most populous municipality of the given cities, is similar to Boston in many respects. Both cities have
a high building density coupled with a high cost per building. A factor that separates Boston and New York and adds to
New York's vulnerability is its immense coastline. Like its New England neighbor, the high density and cost of buildings
means that a natural disaster would be very costly to the city, and by extension, increase the death toll. The coastline
factor would mean a larger swath of land is affected and hence more buildings are harmed. New York City has a harbor,
which dampens the hurricanes energy by 0.95, but no major islands. New York is largely ill-equipped and poorly laid-out
for an earthquake-generated tsunami wave. Its large coast exposes it to a larger section of the wave and its building
density coupled with its high cost per building further increase the prospective damage of any sized tsunami.
Team #2311 Problem B Page 13 of 31
Corpus Christi, TX
Richter scale value Horizontal Distance (m) # of buildings affected Total cost (millions of $)
5 105.088 453 48.005
7 260.824 1125 119.146
9 642.964 2773 293.711
Corpus Christi, Texas is unique from the other cities in that its land slope is significantly less, meaning it’s flatter than the
rest and has a stretch of land that covers the entire coastline. This adds to the steps in our calculations, because we have
to use the limiting factor for islands, which is defined as 4/(w+4), where w is the width of the island. The kinetic energy
for the wave will be four-fifths of the approaching wave since it passes over 1 mile of land prior to reaching the city.
Furthermore, the Corpus Christi Bay will again diminish the wave's kinetic energy by a factor of 0.95. Relative to the
other cities, Corpus Christi has a low building per meter ratio and on a building-to-building ratio are noticeably cheaper.
This is offset; however, by the low land slope, which adds to the tsunami's destructive power. All in all, Corpus Christi is
pretty severely damaged by a tsunami of any magnitude. That said, only one earthquake has ever been recorded to have
appreciably affect Corpus Christi, and it was pegged at 3.8 on the Richter Scale, so, in summation, the likelihood of a
damaging tsunami is insignificant.
This is a bar graph illustrating the differences in devastation for each city:
Team #2311 Problem B Page 14 of 31
Strengths and Weaknesses
Strengths Weaknesses
⇒ Our model is very adaptable. Given any location, we can
find the most likely location of the epicenter, the terrain of
the city, any protective land features, population density,
and average building cost and strength. Using these inputs,
we can find the monetary damages from tsunamis of
various magnitudes for any location.
⇒ Our model greatly simplifies some of the extremely
complex aspects of a tsunami such as cresting, interaction
with obstacles, formation of a wave destruction of
buildings, and effects of topography.
⇒ If another measurement of devastation, such as loss of
life, amount of commercial property destroyed, proportion
of the city that is submerged, etc., then the equation can
easily be used to calculate these values. This is because
there are many separate intermediate equations in
addition to the final equation that can be combined in
different ways to yield different measurements.
⇒ We only base our model off of the most likely epicenter
location. Thus, our results pertain to the average case, not
the range of possible results.
⇒ Although the equations we derived are cumbersome
and a composite of many smaller equations, they are
easily adaptable and adjustable. Simple addition of data,
or data from a new city, can be straightforwardly entered
into the equation to produce new results.
⇒ Our model makes numerous assumptions concerning
constants in our derived equations. Although the constants
are reasonable, they are based off of limited data and
research.
⇒ Our model utilizes very complex aspects of tsunami
waves, such as the cresting point, interaction with
infrastructure, and the relationship between initial
epicenter and final kinetic energy.
⇒ Our data, such as cost of buildings and building density,
are significantly individualized, meaning our effects are
unique to each city.
Extensions
1. We would have liked to further delve into the effects of the bayous around New Orleans (and bayous/estuaries in
general) on tsunamis. We assumed its effects we 25% of that of dry land, but this is not actually the case. We would
have liked to calculate a more realistic effect.
2. We would have liked to further study the effects of varying topography on tsunamis, rather than assuming the land is
uniform. We could have used other models than linear models to describe the land, but the land resembled a linear
model. Using non-uniform topography would increase the accuracy of our model.
Team #2311 Problem B Page 15 of 31
3. We would have liked to further investigate the effects of bays, harbors, and protective islands on the strength of the
tsunamis. We simply assumed that harbors and bays reduced the energy of the tsunami by 5% and the protective
islands reduced the energy of the tsunami by the factor in equation in Assumption A11.
4. We would have liked to take into account the possibility of multiple waves, but it made the problem much more
complex, and we did not have enough time to pursue this option. It would have increased the accuracy of our model.
Appendices
Appendix A: Equation Derivations
Calculation of the cresting depth (d2) given the initial amplitude and depth
1 2
1 2
2 2
1 2
1 20 0
2 2
1 2
1 20 0
Because the area under the curve of the cross-section of the wave remains constant,
2 2sin( ) sin( ) = so,
2 2sin( ) sin( ) so,
v v
f f
vA x dx A x dx
f
f fA x dx A x dx v gd
v v
A
λ λ
π πλ
λ λ
π π
=
= =
∫ ∫
∫ ∫
1
2 2
0 0
2 2
0 0
2 2
2 2sin( ) sin( )
2 2[ cos( )] [ cos( )]
2 2
[1 cos( )] [1 cos( )]2 2
Because .88 ,
.88
fi
f
gdgd
f f
i f
i f
gdgd
f fi i f f
i f
f fi i
f fi i
i i f f
i i
f fx dx A x dx
gd gd
A gdA gd f fx x
f fgd gd
A gdA gd
f f
A gdA gd
f f
A gd A gd A d
A gd
π π
π π
π π
π ππ π
π π
=
− = −
− = −
=
= =
=
∫ ∫
2 2
23 2
2 23 3
.7744
1.291.774
1.291 1.089
f f
i i f f
i if i i
f i i i i
d gd
A gd d gd
A dd A d
d A d A d
=
= =
= =
A1 = Initial amplitude (m)
d1 = Initial depth (m)
Team #2311 Problem B Page 16 of 31
λ1 = Initial Wavelength(m)
A2 = Cresting amplitude (m)
d2 = Cresting depth (m)
λ2 = Cresting Wavelength(m)
f = Frequency (Hz)
Calculation of mass of the incoming wave
2
0
20
2sin( )
2[ cos( )]
2
2[1 cos( )]
2 2
22
m A x dx
Ax
A
A
A
λ
λ
πρ
λ
λ πρ
π λ
ρ λ π λ
π λ
ρ λ
π
ρ λ
π
=
= −
= −
=
=
∫l
l
l
l
l
A = Amplitude (m)
ρ = Density of Seawater = 1027 kg/m3
λ = Wavelength (m)
l = Shoreline length (m)
Average height of a wave/its height after collapsing
2
0
2
0
20
23
23
1 2sin( )
2
2 2sin( )
2 2[ cos( )]
2
.882 2[1 cos( )]
1 2 2
.958322
1 2
1.917
f
f
f
i
i i
i i
AV A x dx
A x dx
A x
d
A d
A d
λ
λ
λ
π
λ λ
π
λ λ
λ π
λ π λ
π λ
π λ
π
π
=
=
= −
= −
=
=
∫
∫
Team #2311 Problem B Page 17 of 31
AV = Average value (m)
Ai = Initial amplitude (m)
di = Initial depth (m)
Af = Cresting amplitude (m)
λ = Cresting wavelength (m)
Calculation of Gravitational Potential Energy Right at Cresting
23
23
2
1 .9 1 7
1 .9 1 7
g
f f i i
f f i i
P E m g h
A A dg
A g A d
ρ λ
π π
ρ λ
π
=
=
=
l
l
Af = Cresting amplitude (m)
Ai = Initial amplitude (m)
di = Cresting depth
ρ = Density of Seawater = 1027 kg/m3
λf = Cresting wavelength (m)
l = Shoreline length (m)
g = Gravitational acceleration = 9.81 m/s2
Development of the model to connect Richter Scale Magnitude and initial amplitude.
5
4
5
The equation must have A=1 when R=5 and A=3 when R=9.
Because the Richter Scale is logrithmic, the desired equation is exponential.
A =m
1
5
3
1.3161
(1.3161)
R n
i
n
R
i
m
n
k
k
A
−
−
−
=
=
=
=
=
Ai = Initial Amplitude (m)
R = Richter Scale Magnitude
Team #2311 Problem B Page 18 of 31
Calculation of Kinetic Energy of the wave before it moves inland (PE zero is the AV)
2 23 23
2
2 2 2 23 3
2
1
2
( 1.089 ) 1.917
2 2
( ) .8433 ( )
2
f i lost i
f
i if i i
i i i i
KE KE PE KE PE
Ag A d A g A d
g A d g A d
ρ λρπ
π
ρ λ ρ
π π
= + = +
= +
= +
l
l
l l
Ai = Initial amplitude (m)
Af = Cresting amplitude (m)
di = Cresting depth (m)
ρ = Density of Seawater = 1027 kg/m3
λf = Cresting wavelength (m)
l = Shoreline length (m)
g = Gravitational acceleration = 9.81 m/s2
Derivation of Equation Relating KEi and Other factors to Destruction
f i
W E
W PE KE
= ∆
= −
∑∑
At maximum height,
iW Mgy KE= −∑
By the Pythagorean Theorem and definition of slope, 2
2 2 2
2
2
( )
11
yAreaAffected y
m
AreaAffected ym
= +
= +
l
l
The total work is the work required to destroy one building times the buildings per area times the affected area, so
2
11 iW Wb y Mgy KE
m= − + = −∑ l
W is the work the wave exerts on one building, so the work exerted on the wave by the building is -W.
Team #2311 Problem B Page 19 of 31
Solving for y,
2
11
iKE
y
Wbm
=
− +l
To calculate work done when a building is destroyed, we used
F xS
A h
∆= •
For brittle substances, breaking occurs at the ultimate strength;
FU
A=
xU S
h
∆= •
Uhx
S∆ =
Using the definition of work,
W Fd F x= = ∆
Substituting,
2Uh VU
W UAS S
= =
Since V = Ah.
This expression would give the work done in the destruction of one building if all of the force were applied at the top, at
height = h. When a wave strikes a building, the wave force is distributed over every height from 0 to h, thus, the work
done is one half of the work if all force were applied at the top, so
2
2
VUW
S=
This equation gives the work done on a building by the tsunami when the tsunami completely destroys the building.
However, when a tsunami collides with a building, the building is not always completely destroyed, instead being
partially damaged by the wave's force and partially damaged by the water. The faster the wave travels, the more work it
will exert on buildings it collides with, so we modified the work exerted equation by multiplying it by a factor of v
v k+.
Team #2311 Problem B Page 20 of 31
Using data from the 1960 tsunami in Hilo, Hawaii, we estimated k to be 500 m/s. Substituting this into the equation for y,
2
2
11
2
iKE
yVU v
Mg bS v k m
=
+ • • ++
l
Using our definition of devastation,
D Cb AreaAffected= •
2
11D Cby
m= +
0 2
2
2
11
11
2
Cb KEm
DVU v
Mg bS v k m
• +
=
+ • • ++
l
Team #2311 Problem B Page 21 of 31
Appendix B: Data
Phase 1: Epicenter to Cresting
City
Shoreline
Length
(m)
Population
Density
(people/m2)
Land
Slope (%
grade)
Land Area
(m2)
Fault Line
Depth (m)
Distance to
Fault (m)
% of E @ City
(χ)
Hilo, HI 5486 0.000289886 3.077
14063635
4.4 1960 50000 1.746249511
San
Francisco,
CA 19022.25 0.006688448 2.223
12095244
4.7 200 12500 24.21990139
New
Orleans, LA 70960 0.000972205 0.1894
46775185
2.7 2000 160000 7.058527688
Charleston,
SC 12870 0.000384751 2.051
38072825
2.2 1650 160000 1.280203655
NY City, NY 53220 0.010594643 0.08463
78942837
5.9 2370 300000 2.823411075
Boston, MA 35480 0.004849829 1.253
12543312
4.2 2480 320000 1.764631922
Corpus
Christi, TX 30160 0.000692744 0.06629
40041216
1.8 2000 100000 4.800117138
SanF. IS
ADJUST-
ED
.5 is
full, .5
gets
.95 of it
(harbor)=.
975
Team #2311 Problem B Page 22 of 31
Phase 2: Cresting to Destruction
City
Richter
Scale
Energy of
Tsunami (J)
Initial
Amplitude (m)
Cresting Depth
(m)
Cresting
Amplitude (m)
Mass of wave
(kg) Potential E @
cresting (J)
Hilo, HI 5 19952623150 1 12.51464949 11.01289155 3782599839 264178316.4
Hilo, HI 7 1.99526E+13 1.732050808 18.04924785 15.88333811 7868124735 659930519.8
Hilo, HI 9 2.00E+16 3 26.03151996 22.90773756 16366358979 1648539127
San Francisco, CA 5 19952623150 1 5.848035476 5.146271219 2864040123 428050055.8
San Francisco, CA 7 1.99526E+13 1.732050808 8.434326653 7.422207455 5957443529 1069290242
San Francisco, CA 9 2.00E+16 3 12.16440399 10.70467551 12391981911 2671139989
New Orleans, LA 5 19952623150 1 12.5992105 11.08730524 49590385711 3440167693
New Orleans, LA 7 1.99526E+13 1.732050808 18.17120593 15.99066122 1.03152E+11 8593709299
New Orleans, LA 9 2.00E+16 3 26.20741394 23.06252427 2.14565E+11 21467511501
Charleston, SC 5 19952623150 1 11.8166575 10.3986586 7911613424 585188616.5
Charleston, SC 7 1.99526E+13 1.732050808 17.04256921 14.9974609 16456819097 1461830150
Charleston, SC 9 2.00E+16 3 24.57963812 21.63008154 34231563183 3651724124
NY City, NY 5 19952623150 1 13.33263885 11.73272219 41648974887 2730320688
NY City, NY 7 1.99526E+13 1.732050808 19.22899266 16.92151354 86633358908 6820476321
NY City, NY 9 2.00E+16 3 27.7330064 24.40504563 1.80205E+11 17037887687
Boston, MA 5 19952623150 1 13.5357989 11.91150304 28618615117 1847949842
Boston, MA 7 1.99526E+13 1.732050808 19.52200015 17.17936013 59529118344 4616270239
Boston, MA 9 2.00E+16 3 28.15559633 24.77692477 1.23826E+11 11531671714
Corpus Christi, TX 5 19952623150 1 12.5992105 11.08730524 21077311627 1462168230
Corpus Christi, TX 7 1.99526E+13 1.732050808 18.17120593 15.99066122 43842574949 3652568665
Corpus Christi, TX 9 2.00E+16 3 26.20741394 23.06252427 91196230911 9124297447
Team #2311 Problem B Page 23 of 31
City
Richter
Scale
Initial Kinetic
Energy (J)
Total Kinetic
Energy (J) Velocity (m/s)
Height reached by
tsunami (m)
Area covered by
tsunami (m2)
Hilo, HI 5 2.20583E+11 2.20715E+11 11.08010431 2.632781454 469622.194
Hilo, HI 7 6.61748E+11 6.62078E+11 13.30650673 4.975865066 887569.5573
Hilo, HI 9 1.98525E+12 1.98607E+12 15.98027568 8.737480749 1558547.472
San Francisco, CA 5 78046191173 78260216201 7.574247687 0.087361368 74773.74433
San Francisco, CA 7 2.34139E+11 2.34673E+11 9.096193955 0.213842620 183030.7126
San Francisco, CA 9 7.02416E+11 7.03751E+11 10.92395547 0.515630194 441334.6686
New Orleans, LA 5 2.03799E+12 2.03971E+12 11.11747521 0.058265412 2182957.254
New Orleans, LA 7 6.11396E+12 6.11826E+12 13.35138683 0.144678366 55420483.244
New Orleans, LA 9 1.83419E+13 1.83526E+13 16.03417384 0.356930071 13372652.191
Charleston, SC 5 4.35635E+11 4.35927E+11 10.76668055 2.993493389 1878808.498
Charleston, SC 7 1.3069E+12 1.30763E+12 12.93010456 5.386200026 3380544.757
Charleston, SC 9 3.92071E+12 3.92254E+12 15.5282404 9.059062969 5685746.477
NY City, NY 5 2.58752E+12 2.58888E+12 11.43648491 0.020701685 1301836.3
NY City, NY 7 7.76255E+12 7.76596E+12 13.73449737 0.051816367 3258499.4
NY City, NY 9 2.32876E+13 2.32962E+13 16.49426545 0.129580892 8148762.325
Boston, MA 5 1.31278E+12 1.31371E+12 11.52328891 0.108507808 307275.2765
Boston, MA 7 3.93834E+12 3.94065E+12 13.83874349 0.267660477 757968.0112
Boston, MA 9 1.1815E+13 1.18208E+13 16.61945848 0.652978156 1849120.796
Corpus Christi, TX 5 9.89945E+11 9.90676E+11 11.11747521 0.069662536 3169441.179
Corpus Christi, TX 7 2.96984E+12 2.97166E+12 13.35138683 0.172900137 7866449.374
Corpus Christi, TX 9 8.90951E+12 8.91407E+12 16.03417384 0.426221153 19391812.95
Team #2311 Problem B Page 24 of 31
City
Richter
Scale
Building Density
(bldg / sq. m)
Work to destroy
per building (J)
Horizontal Distance Water
Travels (m)
Hilo, HI 5 0.000122 2147165428 85.56325817
Hilo, HI 7 0.000122 2567425913 161.7115718
Hilo, HI 9 0.000122 3067339156 283.9610253
San Francisco, CA 5 0.000676 1499705503 3.929886086
San Francisco, CA 7 0.000676 1795667505 9.619551048
San Francisco, CA 9 0.000676 2148768937 23.19524037
New Orleans, LA 5 0.000478 1927601189 30.76315302
New Orleans, LA 7 0.000478 2304853811 76.38773284
New Orleans, LA 9 0.000478 2753593769 188.4530469
Charleston, SC 5 0.0000519 2087919493 145.9528712
Charleston, SC 7 0.0000519 2496883777 262.6133606
Charleston, SC 9 0.0000519 2983487792 441.6900521
NY City, NY 5 0.000895 2214682564 24.46140237
NY City, NY 7 0.000895 2647796919 61.22694914
NY City, NY 9 0.000895 3162846443 153.1146074
Boston, MA 5 0.00209 1998180236 8.659840982
Boston, MA 7 0.00209 2388875038 21.36157039
Boston, MA 9 0.00209 2853446464 52.11318085
Corpus Christi, TX 5 0.000143 2154032337 105.0875492
Corpus Christi, TX 7 0.000143 2575600011 260.8238609
Corpus Christi, TX 9 0.000143 3077052482 642.964479
Team #2311 Problem B Page 25 of 31
Results:
City
Richter
Scale
Number of Buildings
Destroyed
Cost per
Building ($)
Total Cost of Tsunami
(millions of $)
Hilo, HI 5 57.29390767 375309 21.5029192
Hilo, HI 7 108.283486 375309 40.63976684
Hilo, HI 9 190.1427916 375309 71.36230098
San Francisco, CA 5 50.54705117 842108 42.56607616
San Francisco, CA 7 123.7287617 842108 104.1929801
San Francisco, CA 9 298.342236 842108 251.2363837
New Orleans, LA 5 1043.45357 152962 159.6087445
New Orleans, LA 7 2590.99099 152962 396.3231639
New Orleans, LA 9 6392.12775 152962 977.7526445
Charleston, SC 5 97.51016104 266617 25.99786661
Charleston, SC 7 175.4502729 266617 46.77802541
Charleston, SC 9 295.0902421 266617 78.67607509
NY City, NY 5 1165.143489 545793 635.9271601
NY City, NY 7 2916.356963 545793 1591.727216
NY City, NY 9 7293.142281 545793 3980.546005
Boston, MA 5 642.2053278 426671 274.0103894
Boston, MA 7 1584.153143 426671 675.9122058
Boston, MA 9 3864.662464 426671 1648.939398
Corpus Christi, TX 5 453.2300886 105917 48.00477129
Corpus Christi, TX 7 1124.90226 105917 119.1462727
Corpus Christi, TX 9 2773.029251 105917 293.7109392
Team #2311 Problem B Page 26 of 31
Appendix C: The Cities
Corpus Christi, TX:
Hilo, HI:
Team #2311 Problem B Page 27 of 31
San Francisco, CA:
New Orleans, LA:
Team #2311 Problem B Page 28 of 31
New York City, NY:
Boston, MA
Team #2311 Problem B Page 29 of 31
Charleston, SC:
Bibliography
"Atlantic Ocean Floor Topography Lab." UCLA. Web. 8 Nov. 2009.
<http://www.msc.ucla.edu/oceanglobe/pdf/atlantic_topo.pdf>.
"The Formation of a Tsunami." Tsunami Institute. Web. 8 Nov. 2009. <http://www.tsunami-alarm-
system.com/en/phenomenon-tsunami/phenomenon-tsunami-formation.html>.
Google Maps. Web. 08 Nov. 2009. <http://maps.google.com>.
Gulick, Sean. "Tsunamis - Walls of Water." UTexas. Web. 8 Nov. 2009.
<http://www.ig.utexas.edu/outreach/cataclysms/tsunamis/overview_010305.pdf>.
Hills, Paul. "Materials." PWS Tutorial. Web. 08 Nov. 2009.
<http://homepages.which.net/~paul.hills/Materials/MaterialsBody.html>.
"How Tsunamis Form as a Result of Earthquakes." Helium. Web. 08 Nov. 2009.
<http://www.helium.com/knowledge/231528-how-tsunamis-form-as-a-result-of-earthquakes>.
Nelson, Stephen. "Tsunamis." Earth Science Australia. Web. 08 Nov. 2009.
<http://earthsci.org/education/teacher/basicgeol/tsumami/tsunami.html>.
Team #2311 Problem B Page 30 of 31
Nelson, T.J. "Tsunami Myths." Entropy. Web. 08 Nov. 2009. <http://brneurosci.org/tsunami.html>.
"Ocean Waves." Ocean World. Web. 08 Nov. 2009.
<http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_01.htm>
Pararas-Carayannis, George. "Chile - The Earthquake and Tsunami of 22 May 1960 in Chile." Disaster Pages of Dr. George
PC. Web. 08 Nov. 2009. <http://www.drgeorgepc.com/Tsunami1960.html>.
"Properties of Common Solid Materials." EFunda. Web. 08 Nov. 2009. <http://www.efunda.com/materials/common_
matl/common_matl.cfm?MatlPhase=Solid&MatlProp=Mechanical>.
R.D., Catchings. "San Andreas Fault Geometry at Desert Hot Springs, California, and Its Effects on Earthquake Hazards
and Groundwater." Bulletin of the Seismological Society of America. Web. 08 Nov. 2009.
<http://www.bssaonline.org/cgi/content/abstract/99/4/2190?rss=1>.
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Smith, Jane. "Wave Breaking on an Opposing Current." US Army Corps of Engineers. Web. 8 Nov. 2009.
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<http://www.observernews.net/artman/publish/article_001726.shtml>.
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en.wikipedia.org
www.city-data.com
Team #2311 Problem B Page 31 of 31
To the Editors at the Hawaii Tribune-Herald: November 8, 2009
Written by Team 2311
The Tsunami Threat in Hilo
The city of Hilo is not unfamiliar with tsunamis, and we, through analyzing its past, have calculated myriad traits
associated with the potential threat of future tsunamis. Between 1962 and 1985, Hilo witnessed three notable below-
sea earthquakes that were recorded to be 4.0, 5.0, and 5.5 on the Richter Scale. None caused major tsunamis, but they
did remind Hawaiians of the more extreme example just 2 years prior to that range. That wave was created just off the
coast of South America by an 8.5 Richter-value earthquake. The 1960 tsunami left most of the Hawaiian Islands
unaffected; in contrast, Hilo incurred waves of up to 35 feet (10.7 meters), 540 demolished buildings, and 61 deaths.
To produce a likely tsunami, we centered our research on the three earthquakes between 1962 and 1985. These
three earthquakes occurred around 50 km off the shore of Hilo and about 1960 meters below sea-level. Next, we
gathered data about the city of Hilo itself. Hilo has a mean percent-grade of 3.077 and a mean elevation of 38 feet (11.6
m). The building density (buildings per sq. meter) is 0.000122 and an average cost of $375,309 per building (that’s a
composite of residential and commercial buildings).
These facts allow us to calculate the potential loss of infrastructure in Hilo, which would be highly correlated with
a death toll. To estimate Hilo’s response to a tsunami, we tested three earthquakes of 5.0, 7.0, and 9.0 Richter Scale
units. For a 5.0 earthquake 50 km from Hilo’s coast, a 3,780,000,000 kg wave would hit the shore travelling at 11.1
meters per second. This was would affect the buildings of Hilo 86 meters inland. A 5.0 earthquake-generated tsunami
would amount to $21.5 million in damages. As the initial magnitude of the earthquake increases, so will the mass and
velocity of the wave and the devastation it causes. For a 7.0 initial Richter value, the wave would accumulate $40.6
million in reconstruction and repair costs, and infrastructure loss from a 9.0 earthquake-tsunami would reach $71.4
million.
These numbers are undoubtedly large, and the death toll only adds to the prospective harm. However, under-sea
earthquakes are exceedingly rare and a subsequent tsunami is even less so. Tsunamis represent a menacing natural
disaster for Hilo, but, as with every climate and region, these disasters are minimally likely.