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Created by:
Eric Riedner
University of
Wisconsin-Madison
Department of Civil and
Environmental Engineering
Introduction
Shoreline erosion is a
major concern to many landowners with lake property
because it causes a decrease in property value and
sometimes endangers structures that were built too
close to the water. The main cause of shoreline
erosion is waves that through continuous battering
of the shore slowly erode away rock and soil. These
waves are often caused by wind, but in some areas
boat generated waves are also a major contributor to
shoreline erosion. This report investigates the
effects of boat generated waves on shoreline erosion
on one such lake in Wisconsin.
Crystal Lake is a fairly
small lake in South-Central Wisconsin located
halfway between Fond du Lac and Sheboygan. The lake
has an area of 152 acres with a mean depth of 25
feet and a maximum depth of 61 feet [www.lake-link.com].
A topographic map of the lake can be seen in the
figure below. The lake is surrounded completely by
houses and cabins that are squeezed close together
around the lake. The lake is one of the most
beautiful in the area with its clear water and
surrounding scenery and because of this it is
increasingly becoming more crowded. Along with the
increased popularity of the lake is an increased
number of recreational boats on the lake, which are
used primarily for waterskiing and tubing. These
activities have become an important part of lake,
but it is possible
that the waves created by these boats are damaging
the shoreline of Crystal Lake.
The objective of this study is to evaluate the
effects of wind generated waves and boat generated
waves on shoreline using some empirical models and
laboratory studies.
Crystal Lake
Calculations
Overview
The approach
to quantify the effects of boat generated waves is
based upon the draft report by William Fitzpatrick
at the Wisconsin Department of Natural Resources.
The paper was in rough draft form and was created in
order to determine the effects of boat waves on the
shoreline of Lake Mendota. The model used a
comparison of boat generated waves to wind generated
waves in order to determine if the energy delivered
to the shoreline by boat waves is significant with
relation to the energy delivered by wind waves over
the course of one day. The main steps in
determining these two values are calculating wave
heights, converting the wave heights to wave energy,
and determining the number of times these waves
strike the shore.
Our approach will
combine CEE 514 notes, Fitzpatrick’s paper, and
Robert Sorensen report that looks into the effects
of boat generated waves on the Upper Mississippi
River. Sorensen’s report explains the physics of
boat generated waves and compares a number of
theories that are used to determine wave heights
created by boats. For this report all of these
theories were considered. We found that the method
created by N.G. Bhowmik fits an analysis on Crystal
Lake the best. Bhowmik uses an empirical equation
derived from experiments involving boats of similar
size, speed, and type that are commonly seen on
Crystal Lake, using visual observation records.
The most difficult
part of using these methods and equations is
attempting to put a single value on many of the
different parameters needed to create two numbers in
the end that are used to compare the energies of the
two kinds of waves. This report recognizes the
shortcomings of singularizing these values by
averaging and making undocumented assumptions and it
should be stressed that the final values are meant
to only give a rough comparison between the effects
of wind and boat waves.
Wind Waves
Wind wave
height can be determined using the following
equation [Fitzpatrick].
Hs = (0.283U2)/g)*(tanh(0.530(gd/U2)0.75))*(tanh((0.0125(gF/U2)0.42)/(tanh(0.530(gd/U2)0.75))))
Hs=
significant wave height in feet, the value that is
solved for
U = wind speed
in ft/s = 14.7 ft/s (10mph) which is the average
value for the state of Wisconsin [NOAA]
F = fetch
length in 2650 ft
d = depth in
feet, average lake value determined from a
topographic map = 25 ft
g = 32.2
ft/s2
Using the parameters in the
given equation, Hs was determined to be
0.29 ft or 0.09
m.
As a check this value
can compared to one created by the SMB (Sverdrup,
Munk, and Bretschneider) method. This value is
determined using the fetch-limited approach instead
of the duration-limited approach due to the small
size of the lake and the average wind value that is
used, which is sustained for a relatively long
period of time.
(gF)/U2=
(9.81 m/s2
* 808 m)/(4.48 m/s)2
= 394
Using the graph associated with the
SMB method, gHs/U2
= 0.042, and
Hs = 0.086 m = 0.28
ft. This value is
consistent with the estimated one using
Fitzpatrick’s equation.
In addition, the SMB
method can also calculate the period, T, of the wind
waves.
Using the graph associated with the
SMB method, gT/(2*pi*U2)
= 0.4, and T = 1.1 seconds.
Following the CEE 514
Notes, The following equation can be used to
determine the weighted average energy, Ew, of a wave
given the significant wave height, i.e.
Ew = 1/16 * r
*g*Hs2
=0.613*Hs2
= 0.613*0.086^2 = 0.0045 kN/m2
Dividing Ew
by a wave period T, the wave power, Ep, of wind
waves can be computed, i.e.
Ep = Ew /T =
0.0045/1.1 = 0.004 kN/m2/sec
Eday
= Ep*86400 = 0.0045*86400 = 388 kN/m2
Boat Waves
Boat generated
wave heights can be calculated using the following
equation that was created by Bhomik et.al. in 1991
and was summarized and evaluated by Sorensen. This
equation was created using a regression analysis
from a selection of boats that are very similar to
those found on Crystal Lake.
Hm = 0.537*V-0.346*x-0.345*Lv0.56D0.355
Note that this
equation implies that wave height would decrease if
vessel speed increases!
(This only works
for planing hull boat)
Hm
= maximum wave height, the value that is solved for
V = vessel
speed is 9 m/s. This speed was determined by visual
inspection and by suggested water skiing speeds (www.waterski.about.com),
since water skiing and tubing are the main source of
boat traffic on the lake.
x = distance
to the measurement point in meters = 100 m. This
distance was estimated by observing that boats on
the lake tend to travel in a circular direction
around the lake traveling close to shore so that
more boats can operate at one time with skiers in
tow.
Lv = length of
the boat in meters = 5.5 m. This length was
determined through visual observation and
researching standard sizes of recreational boats.
Almost every boat on the lake that created
significant waves is near the standard recreational
boat size of 5.5 m.
D = draft of
the boat in meters = 0.4 m. This value was
estimated by interpolating Bhowmik’s data related to
boat speed, size, and type.
Using the parameters in the
equation, Hm was determined to be
0.10 m or 0.32 ft.
While standing on
shore there are approximately 13 measurable waves
created by a passing vessel [Sorensen]. The figure
below shows these waves and the relative height of
each compared to the Hm. The equation E
= 9.80*(H/2)2/2, which determines the energy of the
wave given the wave height, was incorporated with
this typical wave train in order to determine the
total energy delivered to the shore by a passing
boat.
Typical vessel
generated wave record with wave size relation to Hm
denoted
E=0.1138*(0.075*Hm)2+0.1138*(0.3*Hm)2+0.1138*(0.725*Hm)2+0.1138*(Hm)2+0.1138*(0.775*Hm)2+0.1138*(0.45*Hm)2+
0.1138*(0.375*Hm)2+0.1138*(0.325*Hm)2+0.1138*(0.275*Hm)2+0.1138*(0.25*Hm)2+0.1138*(0.225*Hm)2+
0.1138*(0.2*Hm)2+0.1138*(0.175*Hm)2
E = energy exerted by a passing boat
on a point on shore in kN/m2
Hm
= maximum wave height in meters = 0.10 m
Solving the equation, E = 0.0033 kN/m2.
In order to convert
the boat wave energy from one pass into the energy
created per day, estimates were made concerning boat
traffic on the lake. Through visual observations it
was estimated that a boat passes a point on the
shore every 3 minutes during the peak 8 hours on the
weekend. During four of the weekdays it was
estimated that a boat passes once every 10 minutes
during the peak 4 hour period and on one weekday no
motorboats are allowed on the lake. Using these
assumptions it was determined that there are 60 boat
passes on the lake during the average day.
The energy
created by boat waves per day can then be computed
by using the following equation:
Eday
= E*(boat passes in a day) = 0.0033*60 = 0.20 kN/m2
Results
(1) Energy Viewpoint:
These calculations show that the energy created by
wind waves in one day, 388 kN/m2,
is far greater than the energy created by boat
generated waves in one day, 0.20 kN/m2.
The energy approach assumes wind continues generates
waves for 24 hours, which is not true. Also, the
approach adopted here does not take into account of
shoaling and bottom effects. In addition, the boat
generated waves 0.20 kN/m2
also neglect the long resistance of wakes generated
by boats. Therefore, the energy approach usually
may mislead the results.
(2) Maximum Wave
Height Viewpoint:
The wave heights
generated by both wind and boat are
comparable to each other, i.e 10cm. If boats get
closer to the shore (say 100ft), the boat generated
waves can reach to 15cm wave height. Normally, the
extreme wave height plays an important role in
damaging shoreline or structures. Thus one may
easily conclude that boat generated waves is the
main factor to cause shoreline erosion.
Discussion:What
is missing?
As we know that
recreational boating is important to the people who
use the lake. The results shown in the above study
may contradict to each other! Therefore, a
detail study of monitoring shoreliner response to
boat generated waves and wind waves is definitely
needed.
References
Bhowmik, N.G., Soong,
T.W., Reichelt, W.F., and Seddik, N.L. (1991). Waves
generated by recreational traffic on the Upper
Mississippi River system, Research Report 117,
Department of Energy and Natural Resources, Illinois
State Water Survey, Champaign, IL.
Crystal Lake,
Sheboygan County. www.lake-link.com. (2001).
Fitzpatrick, William.
(2000). Boat and Wind Energy Screening Model,
Review Draft. Wisconsin Department of Natural
Resources.
How Fast Should You
Go. http://waterski.about.com/library/weekly/aa091500.htm.
(2001)
NOAA Web Site for
Plymouth Wisconsin. www4.ncdc.noaa.gov. (2001)
Sorensen, Robert M.
(1997). Prediction of Vessel-Generated Waves with
Reference to Vessels Common to the Upper Mississippi
River System, Upper Mississippi – Illinois Waterway
System Navigation Study, ENV Report 4.
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