Full text of "Hydraulics of Great Lakes inlets"

See other formats

US. Ar, Co@ack. Cre, Cea ern IT
eS

Hydraulics of Great Lakes Inlets

by
William N. Seelig and Robert M. Sorensen

TECHNICAL PAPER NO. 77-8
JULY 1977

Approved for public release;
distribution unlimited.

U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER

Kingman Building
Fort Belvoir, Va. 22060

Reprint or republication of any of this material shall give appropriate

credit to the U.S. Army Coastal Engineering Research Center.

Limited free distribution within the United States of single copies of

this publication has been made by this Center. Additional copies are
available from:

National Technical Information Service
ATTN: Operations Division
5285 Port Royal Road

Springfield, Virginia 22151

Contents of this report are not to be used for advertising,
publication, or promotional purposes. Citation of trade names does not

constitute an official endorsement or approval of the use of such
commercial products.

The findings in this report are not to be construed as an official

Department of the Army position unless so designated by other
authorized documents.

MBL/WHOI

MUM

(QUAI

0 0301 0089976 1

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

REPORT DOCUMENTATION PAGE BE EOREICONEH EN CIEOR

1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER
TP 77-8

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

HYDRAULICS OF GREAT LAKES INLETS Technical Paper

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(a)

William N. Seelig
Robert M. Sorensen

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK
Department of the Army UN
Coastal Engineering Research Center (CERRE-CS) A31220

Kingman Building, Fort Belvoir, Virginia 22060

AREA & WORK IT NUMBERS

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Department of the Army July 1977
Coastal Engineering Research Center 13. NUMBER OF PAGES

Kingman Building, Fort Belvoir, Virginia 22060 69
14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report)

UNCLASSIFIED

1Se. DECLASSIFICATION/ DOWNGRADING
SCHEDULE

Approved for public release; distribution unlimited

16. DISTRIBUTION STATEMENT (of thie Report)

. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

- SUPPLEMENTARY NOTES

- KEY WORDS (Continue on reverse side if necessary and identify by block number)

Great Lakes Lake seiching
Inlet-harbor resonance Nontidal inlet
Inlet hydraulics

20. ABSTRACT (Continue em reverse side if rreceacary and identify by block number) } :
Reversing currents in inlets on the Great Lakes are generated primarily by

long wave seiching modes of the lakes rather than by the astronomical tide.
Field measurements were conducted in 1974-75 at nine harbors on the Great
Lakes to: (a) Investigate the nature of long wave excitation and the gener-
ating mechanism for significant inlet velocities, (b) establish techniques for
predicting inlet-bay system response, and (c) develop base data for future
planning and design studies. Data collected include continuous harbor water
(continued)

DD , fie 1473 ~—sC EDITION OF ? NOV 65 IS OBSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION GF THIS PAGE (Wren Data Entered)

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)

level measurements at all sites, inlet velocity measurements at the primary site
(Pentwater, Michigan), and channel hydrographic surveys at the sites where more
recent data were needed. Available historic water level and velocity data for
some of the harbor sites were also used.

Amplified harbor oscillations and generation of the highest inlet veloc-
ities are caused by the Helmholtz resonance mode which has a period of 0.6 to 5
hours for the inlet-bay systems studied. A recently developed, simple numerical
model is shown to be effective in predicting inlet-bay response over the range
of excitation periods encountered. A finite-difference form of the continuity
equation is shown to adequately predict inlet velocities if high-quality bay
water level records are available. Selected data from the study sites are
presented to demonstrate the hydraulic response of the inlet-bay systems and
the applicability of the prediction schemes. Examples to demonstrate use of
the concepts and techniques developed in the study are applied to the design
of a new inlet channel and to the modification of an existing channel.

2 UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)

PREFACE

This report is published to provide coastal engineers with an
analysis of the hydraulic response of inlet-bay systems on the Great
Lakes. The work was carried out under the coastal research program of
the U.S. Army Coastal Engineering Research Center (CERC).

The report was prepared by William N. Seelig and Dr. Robert M.
Sorensen, Coastal Structures Branch, Research Division, CERC, under the
general supervision of R.P. Savage, Chief, Research Division.

The authors acknowledge the efforts of the U.S. Army Engineer
District, Detroit and the National Oceanic and Atmospheric Administration,
Lake Survey Center, who callected most of the field data, and the report
review and comments by C. Mason and B. Herchenroder.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th
Congress, approved 31 July 1945, as supplemented by Public Law 172,
88th Congress, approved 7 November 1963.

OHN H. COUSINS
Colonel, Corps of Engineers

Commander and Director

CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI).
SYMBOLS AND DEFINITIONS.
INTRODUCTION .

LAKE AND INLET HYDRAULICS. :
1. Great Lakes and Inlet-Bay System ‘Tavdnaniies ‘
2. Prediction of Inlet Velocities. SAIL

THE FIELD DATA COLLECTION PROGRAM.
1. Field Measurements.
2. Equipment . .
3. Data Reduction and Analysis Techniques.

RESULTS. j ;
1. Seiching of the Grose takes :
2. Predicted Inlet-Bay Response to Monochromatic even
Wave Forcing . 0
3. Observed Lake Level Pieecuacionen Bay ReswOnSe, and
Inlet Velocities . ences oO 0

INLET DESIGN . i

1. New Inlet Ghannell - ‘

2. Inlet Channel Nace Bloaeson,
SUMMARY AND CONCLUSIONS.
LITERATURE CITED .

TABLES
Summary of field measurements .

Inlet and bay geometric measurements.

Modes of oscillation of the Great Lakes .

Influence of Manning's n on inlet-bay response at Pentwater .

Predicted periods of maximum wave amplification and maximum
inlet velocities .

Predicted Duluth-Superior maximum inlet water velocities for

a forcing wave of 1 hour (a, = 3 centimeters).

Numerical Model Prediction of Pentwater response to Lake
Michigan modes of oscillation.

Page

CONTENTS
TABLES --Continued

Summary of Pentwater hydraulic characteristics for selected
inlet depths

FIGURES
Inlet-bay system.
Amplification and phase lag for inlet-bay systems

Three predicted longitudinal modes of oscillation of
Lake Michigan.

Pentwater response to sinusoidal wave in Lake Michigan.
Response to long wave excitation at Pentwater .
Numerical model water level predictions at Pentwater.
Inlet study sites

Data collection sites on Lake Michigan.

Data collection sites on Lake Superior.

Data collection site on Lake Erie, Presque Isle,
Pennsylvania .

Data collection sites on Lake Ontario .
Sample spectra of Pentwater bay water levels.

Measured and predicted inlet velocity cumulative frequency
distributions at Pentwater, Michigan .

Pentwater response to long wave forcing .
Toronto response to long wave forcing .

Predicted response of inlet-bay systems on Lake Michigan
to monochromatic forcing .

Predicted response of inlet-bay systems on Lake Superior
to monochromatic forcing .

Predicted response of inlet-bay systems on Lakes Erie
and Ontario to monochromatic forcing .

Page

IL
21
22
24
25
28

30
31

ail

CONTENTS

FIGURE--Continued

Sample Pentwater and Lake Michigan water levels and
SPE CET ae ata leew oh ieciyie: MPa kal keccta nce nC sntc a) cam Onn Masa sonra

Sample Pentwater and Lake Michigan water levels and
spectra.

Sample water level fluctuations
Water level fluctuations in Little Lake Harbor.

Sample bay levels and inlet water velocities at
Duluth-Superior.

Pentwater inlet cumulative frequency velocity
distributions. a6 .0 0

Inlet velocity cumulative frequency distributions
Crystal Lake, Michigan.

Predicted response characteristics of an inle® for
Crystal Lake .

Predicted inlet velocities at Michigan inlets
Response to long wave excitation at Pentwater .
Response to long wave excitation at Pentwater .

Predicted Pentwater inlet velocities for Lake Michigan
water levels recorded on 18 August 1967. .. .

Page

51
53

57
58

60
62
64

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted to metric (SI)

units as follows:

Multiply by To obtain
inches 25.4 millimeters
2.54 centimeters
square inches 6.452 square centimeters
cubic inches 16.39 cubic centimeters
feet 30.48 centimeters
0.3048 meters
square feet 0.0929 square meters
cubic feet 0.0283 cubic meters
yards 0.9144 meters
square yards 0.836 square meters
cubic yards 0.7646 cubic meters
miles 1.6093 kilometers
square miles 259.0 hectares
knots 1.8532 kilometers per hour
acres | 0.4047 hectares
foot-pounds 1.3558 newton meters
millibars 1.0197 x 10°? kilograms per square centimeter
ounces 28.35 grams
pounds 453.6 grams
0.4536 kilograms
ton, long 1.0160 metric tons
ton, short 0.9072 metric tons
degrees (angle) 0.1745 radians
Fahrenheit degrees 3/9 Celsius degrees or Kelvins!

CE ———— ———————————————_—_—_—S—OorcqCceeeeeeeeeeeeeEeeEeEeEeEeEeEeEeEeee
To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F — 32).

To obtain Kelvin (K) readings, use formula: K = (5/9) (F — 32) + 273.15.

bp.

SYMBOLS AND DEFINITIONS

inlet cross-sectional area at the bay end

bay surface area

inlet cross-sectional area

grid cross-sectional area

inlet cross-sectional area at the sea end

inlet width

grid cell width

wave speed

grid cell depth

water depth

acceleration due to gravity

water surface
water surface
water surface

water surface

elevation

in the
in the

in the

number of channels in the grid

number of sections in the grid

grid cell subscript indicating

mode of oscillation

inlet length

bay
bay at the previous time step

sea

the channel

the cross section

added inlet length in the acceleration terms to account for
long wave radiation from the harbor

basin length

grid cell length

Coe) 2

SYMBOLS AND DEFINITIONS--Continued
number of inlets connecting a bay to a sea
Manning's n
Manning's n of a grid cell
discharge
discharge for the mth inlet
total discharge for all inlets
hydraulic radius of a channel
inlet-bay system Helmholtz period
frictionless inlet-bay system Helmholtz period

period of free mode of oscillation where the subscript
indicates the mode of oscillation

time
mean instantaneous inlet water velocity at a cross section

mean inlet water velocity at a cross section over a
sampling interval

the grid cell weighting function which is the fraction of
total inlet discharge passing through the cell at a time
step

distance along the longitudinal axis of the inlet
distance along a cross section

time step

component of the bottom-stress tensor in the direction of
flow

ron |

mana wy fi
ia belie |

HYDRAULICS OF GREAT LAKES INLETS
by
Willtam N. Seeltg and Robert M. Sorensen
I. INTRODUCTION

Numerous bays and harbors are connected to the Great Lakes by
jettied inlet channels. These inlet channels are important because they
allow (a) access to commercial shipping and recreational boating, (b)
migration of fish, and (c) flushing of pollutants from the bays and
harbors.

Great Lakes inlet-bay systems are generally smaller than those on
the Atlantic, Pacific, and gulf coasts of the United States and respond
primarily to the long wave seiching modes of the Great Lakes rather
than to the astronomical tides. These seiches have smaller amplitudes
and shorter periods than the tides on the ocean coasts.

The major effort of this study involved the collection and analysis
of hydraulic data at several inlet-bay systems throughout the Great Lakes
during 1974 and 1975. Measurements at Pentwater, Michigan, the primary
study location, included simultaneous recording of inlet current veloc-
ities and water levels in the bay and in Lake Michigan. At the other
locations, only bay water levels were measured. However, hydrographic
surveys were obtained for all the inlets investigated, and historic
hydraulic data from selected sources were analyzed.

This study defines the hydraulic mechanisms important to Great Lakes
inlet-bay systems, develops analytical techniques for the prediction of
inlet currents and bay water level oscillations, and presents design
data and system response curves for selected inlets.

Field data were analyzed using (a) a formula for estimating the
seiche periods of the Great Lakes which are important in producing
reversing currents at an inlet; (b) a model that uses bay water level
time histories to predict inlet velocities; and (c) a simplified numer-
ical inlet hydraulic model that, when calibrated for friction effects,
can be used to predict inlet velocities and bay water level oscillations
generated by lake oscillations. These analysis techniques are used with
the field data to develop response curves and cumulative inlet current
velocity distribution curves for the inlets studied.

II. LAKE AND INLET HYDRAULICS

1. Great Lakes and Inlet-Bay System Hydraulics.

An inlet is a relatively narrow channel which connects a "'sea"™ (or
one of the Great Lakes in this study) to a lake or harbor (a “bay" in
this study). The bay is large compared to the inlet (i.e., the radius

of the bay is typically larger than the length of the inlet) and the
surface area of the bay is much smaller than the surface area of the
sea.

a. Causes of Reversing Inlet Currents. Mortimer (1965) and Freeman,
Hamblin, and Murty (1974) show that significant reversing inlet current

velocities are caused by water level fluctuations in the Great Lakes which
generate a hydraulic response in the inlet and bay. The most important
Great Lakes water level fluctuations are due to the resonant seiching or
oscillation of the particular lake at its fundamental and harmonic periods.

Seiches are initiated by storm pressure and wind forces on the sea
which redistribute water in the lake to cause a higher elevation than
normal in some areas and lower levels in other areas. When gravity tries
to restore the water level, seiches are generated. These seiches usually
continue for a number of cycles which may extend over a few days after
the storm has passed.

When a seiche is generated in one of the lakes, the water level
fluctuations outside an inlet cause a head difference across the inlet,
which, in turn, generates a current in the inlet. Water discharge
through the inlet results in water added to or removed from the bay so
the bay level rises and falls in a pumping fashion for most seiching
periods of the lake.

Astronomical tides and other nonseiching long waves cause water level
fluctuations of the Great Lakes; however, these fluctuations generally
have insufficient amplitudes or are at periods that usually do not sig-
nificantly influence inlet hydraulics. Storm surge, particularly on
shallow Lake Erie, may occasionally generate strong inlet currents.

b. Mathematical Description of Inlet-Bay Hydraulics. The response
of an inlet-bay system may be described in terms of the one-dimensional

equation of water motion in the inlet and the continuity equation relat-
ing the rate of bay level change to the inlet discharge.

The one-dimensional equation of motion along the inlet channel axis
can be written:

Yo
any ple ah, av
a 1 J (‘2x )a 8 + 85x * 3p (1)
Y1

where

distance along the channel

=r
il}

water surface level

>
iT}

inlet cross-sectional area

V = water velocity in the inlet

y = distance along the cross section

= component of the stress tensor in the direction of flow
g = acceleration due to gravity

t = time

The inlet has a width, B, depth, d, length, L, and cross-sectional area,
Ag; the bay has a surface area, A . The water levels in the sea and
bay are hg and hp, respectively (Fig. 1).

Equation (1) equates the horizontal driving force due to the water
surface slope with three terms on the right which are the channel fric-
tional resistance, the convective acceleration caused by velocity vari-
ation along the channel axis, and the temporal acceleration (or inertia)
resulting from velocity variation at a point with time. In nearly
prismatic channels, such as many inlets on the Great Lakes, the convec-
tive acceleration is often negligible.

The continuity equation, which relates rate of bay water level change,
dhp/ot, to inlet discharge, Q, is:

dhp
Qe WAS S Apay Gram (2)

A simultaneous solution of equations (1) and (2) for a sinusoidal
sea level fluctuation reveals the important response characteristics of
an inlet-bay system (Fig. 2). In this figure, the phase lag between the
sea and bay water level fluctuations and the amplification of the forc-
ing wave in the bay by the inlet-bay system are plotted as functions of
dimensionless period. Dimensionless period is defined as the friction-
less inlet-bay system Helmholtz period, Ty', divided by the forcing wave
period, T. The Helmholtz period is that period of the forcing wave which
through resonance will cause the largest water level fluctuation in the
bay. The bay water level remains approximately horizontal throughout
this fluctuation.

The inlet-harbor system response is analogous to the response of a
slightly damped spring-mass system or its acoustic counterpart, the
Helmholtz resonator. The motion of the mass of water in the inlet
channel corresponds to the motion of the mass of the spring-mass system,
and the action of gravity on the rising and falling harbor water surface
corresponds to the restraining force of the spring.

At values of Ty'/T approaching zero (long wave periods) the water
level fluctuations in the bay are the same as those in the sea with no

Plan View

inlet cross-
sectional area=Ac BAY

sad a oA surface area = Apay

Profile View

Figure 1. Inlet-bay system.

180°

Increasing
Friction

Phase Lag
O
Z

Increasing
Friction

Amplification

Figure 2. Amplification and phase lag for inlet-bay systems.

phase lag (point A, Fig. 2). At values of Wy approaching 3

(short forcing periods), the inlet-bay system strongly dampens incident
waves and water level fluctuations in the sea have little influence on
inlet hydraulics (point B, Fig. 2). At values of Ty'/T in the range
of 0.25<(Ty'/T)<2, the amount of frictional resistance in the channel
has a major influence on the response characteristics. Inlets with high
friction, e.g., tidal inlets on ocean coasts, typically have amplifications
of less than one. This amplification factor decreases as the forcing
wave period becomes shorter (point C, Fig. 2). At most tidal inlets,
the primary tidal period is large compared to the inlet-bay Helmholtz
period. Low-friction inlets, such as those on the Great Lakes, have
amplifications greater than one and phase lags of approximately 90° for
forcing waves with T,'/Tx1 (point D, Fig. 2) which commonly occur.

2. Prediction of Inlet Velocities.

Prediction of inlet velocities requires (a) the time history of sea
or bay water levels, (b) the geometries of the inlet and bay, and (c) a
friction-calibrated model to relate water level fluctuations to inlet-
bay response.

a. Great Lakes Water Level Fluctuations. In general, no methods are
presently available for inexpensively predicting all important amplitudes
and periods of water level fluctuations at any point in one of the Great
Lakes. Therefore, water levels generally must be measured. However,
inexpensive schemes are available for accurately predicting some periods
and relative amplitudes of seiches of the Great Lakes (Fee, 1968; Rao
and Schwab, 1974). Knowledge of the existing wave periods will aid in
the design of water level measuring systems, analysis procedures, and
preliminary inlet design.

The basic method for estimating one-dimensional fundamental and
harmonic seiche periods, Tz, is to determine the time required for a
wave to travel twice the length of the basin:

Ue © ae (3)

where k is the mode of oscillation, 1, the length of the basin in
the direction of the seiche, and c the speed of the wave. Sample
predicted longitudinal seiches for Lake Michigan, using the Fee (1968)
computer program, are shown in Figure 3 for modes k=1, 2, and 3.

b. Inlet and Bay Geometries. Hydrographic surveys, including Corps
of Engineers dredging records, can be used to determine inlet geometry;
i.e., length, width, and depth field. Maps and aerial photos can be
used to determine bay surface area.

First longitudinal mode ere:
( period ,Qhr ) & Second longitudinal mode

| eee ( period,5.3 hr )

antinode

Portage
Portage u

Ludington.
Pentwater

Ludington
Pentwater

Muskegon ——- oi da

Holland

Holland aa ae
Ak
Third longitudinal

mode
( period ,3.5 hr )

antinode

Relative Seiche Height of Water
(normalized by level at
the southern tip of
Lake Michigan )

Figure 3. Three predicted longitudinal modes of oscillation of Lake:
Michigan (modified from Mortimer, 1965).

c. Methods of Analysis of Inlet-Bay Hydraulics. Inlet current
velocities and bay water surface oscillations may be measured to provide

necessary information on the hydraulic characteristics of an inlet-bay
system. Techniques used for the field measurements in this study are
discussed in Section III, 2.

Several analytical methods are available for predicting inlet
hydraulics, depending on the type of information available and required
results. Three methods are discussed below.

(1) Estimation of Seiche Periods Important to Inlet Hydraulics.
To estimate which of the Great Lakes seiche periods, Ty, may be im-
portant, the frictionless inlet-bay Helmholtz period, Ty', may be

determined from:
[CL+L") A
Way! = 27 Sac, Seen (4)
Aa

where L' is an added channel length determined from:

-B 1B
L! = — In |———], (5)
w % | gd |

L' accounts for the water masses in motion beyond the ends of the inlet
(Miles, 1948). Equations (4) and (5) may be iteratively solved to
obtain a value of Ty'. This approach proved to yield reasonably accu-
rate estimates for the inlets considered in this study.

As a first approximation, seiche wave periods which are approximately
equal to the frictionless Helmholtz period (e.g., between 0.5 and 2
times Tyg'; see Fig. 2) will probably cause the highest inlet reversing
currents.

The seiche node-antinode pattern in the Great Lakes will also influ-
ence the importance of the various seiche modes on an inlet-bay system.
Seiches with antinodes adjacent to the inlet will produce the largest
water level fluctuations. Since ends of the lakes are antinodes for all
modes of oscillation along that axis, bays at the end of a lake will
normally be subject to higher water level fluctuations than those at
other locations; e.g., midway along the longitudinal axis of Lake Michigan,
near Pentwater, the first longitudinal mode of oscillation has a node
(Fig. 3). Therefore, only small oscillations can be generated in the
lake at this point by this mode. The second longitudinal mode of oscil-
lation has an antinode adjacent to Pentwater, so large water level fluc-
tuations in Lake Michigan could be generated outside Pentwater by this
mode.

(2) Estimation of Inlet Velocities from Bay Water Level Records.
A method of predicting inlet velocities, if high-quality bay water level

records are available, is to use the continuity equation. Written in
finite-difference form, equation (2) becomes:

hy > hy
7 2 Samy iE a) (6)

where V is the average inlet current velocity at a cross section of
area Ag over a water level sampling interval, and At, hp', and hp

are mean bay water levels at the beginning and end of the sampling
interval. Measurements of water level at any point in the bay will be
representative of the mean bay level for the Helmholtz mode of oscillation
of inlet-bay systems.

This method for predicting inlet velocities is well suited for Great
Lakes inlets because inlet and bay geometries are simple and level re-
corders are easy to install in the protected bays. The sampling interval
should be one-twentieth of Ty' or shorter and the stilling well care-
fully designed for best results (see Sec. III).

(3) A Numerical Model. A relatively simple but extremely useful
method of modeling inlet-bay hydraulics is to simulataneously solve the
equations of motion and continuity. In this model the inlet channel is
divided by a flow net into a grid of subchannels and cross sections.

The subscripts z and j describe the location of the cell for sub-
channels (IC = number of channels) and grid sections (IS = number of
sections). The equation of motion for an inlet (eq. 1) rewritten

in finite-difference form, and integrated along the axis yields (Seelig,
Harris, Herchenroder, in preparation, 1977):

d 1 ffl 1
$+ (bee) + 20
1s-1f )) L, 4/IC ener!
=
IC
Hail | OY AgE
jal
IS-1
> ee ee
2.208 D, (7)
i=1 7 Aig) j=1
j=l

where A, and Ap are the inlet cross-sectional areas at the sea and
bay ends of the inlet, and W;; is a weighting function for distributing
flow throughout the inlet. The discharge through a grid cell is equal
to the weighting function of the cell, Wea times the total discharge
of the inlet, Q. The Manning's friction factor, njj, is determined

during calibration of the model (see Seelig, Harris, and Herchenroder,
in preparation, 1977).

For M inlets connecting the bay to the sea, the total discharge
for all inlets, Qins is:

M 5
Qe nh: (8)

m=1

The continuity equation is written as:

oh Q
ERR eet 4 (9)

Bay levels and inlet current velocities are determined by solving the
simultaneous differential equations (7) and (9) using a Runge-Kutta-Gill
fourth order finite-difference technique in conjunction with initial con-
ditions and the time history of water levels in the sea. Derivation and
sample applications of this model are given in Seelig, Harris, and
Herchenroder (in preparation, 1977).

To obtain response characteristics similar to Figure 2 for a specific
inlet-bay system, the model can be run by assuming sinusoidal seawater
level fluctuations with a typical amplitude; e.g., 3 centimeters (0.1
foot) at periods covering the anticipated range of lake oscillation modes.
Each run of the model will give predicted bay levels and inlet current
velocities for the wave period used. Sample model results for Pentwater
inlet are shown in Figure 4. The sea level, predicted bay level, and
inlet velocity are shown in the lower part of Figure 4; the importance
of each of the terms in the equation of motion, normalized by dividing
by the magnitude of the largest term at each time step, is shown in the
upper part of the figure. For this condition, the bay level fluctuation
is larger than the sea level fluctuation due to inertia in the system
and the bay level lags the sea level by 84° (Fig. 4). Plotting results
from many runs similar to Figure 4, but with many different forcing
periods, will give the response characteristics of the inlet-bay system.
These curves for Pentwater (Fig. 5) show that the Helmholtz period with
friction, Ty, is 1.8 hours, waves with periods of 1 to 3 hours will be
amplified by the system, and waves with a period of 1.4 hours will gen-:
erate the highest inlet current velocities (3-centimeter wave amplitude
assumed). The effect of friction on Ty is demonstrated by the dash-
line in Figure 2. The frictionless Helmholtz period is also coinciden-
tally 1.4 hours (from eqs. 4 and 5).

The calculated bay amplification and channel velocity in Figure 5
are for a Manning's n = 0.036. The numerical model usually had to be
run for three or four cycles for the bay response to build to equilibrium.
In the prototype harbor it is likely that equilibrium (full amplification)
is never fully achieved. Thus, the calibration curve in Figure 5 forms
the upper envelope of measured prototype data.

Friction

Head Difference

Temp. Acceleration

Pentwater Bay

ake Level

Michigan
Level

Velocity (ft/s)
Water Level (ft)

Time (hr)

Figure 4. Pentwater response to sinusoidal wave in Lake Michigan
(period = 1.5 hours; amplitude = 0.1 foot).

2 |

*(200F T°0 = epnzt[due orem) LoyemjUeg 3@ UOTIEITOXE AEM BUOCT 02 asuodsoy

(4Y) Poised
6) | G € v G

UONDIIJdWYy pajDINI/09

joes M1010 WNWIXDI pajDjNI109
KeZ

"¢ o1n3 Ty

UOIJDIJGwWy PUD (S/J}) AjoojaA wWNWIXxOW

If water level fluctuations just outside the inlet are known, the
numerical model can be used to predict the resulting channel velocities
and bay level fluctuations. Lake Michigan levels just outside of Pent-
water were measured by Duane and Saylor (1967) during August 1967 (Fig. 6).
Although the record appears confused, a spectral analysis shows that the
record is composed of a large number of clearly defined seiche modes of
Lake Michigan. Using this record to force the model of Pentwater, a bay
level time history is predicted which adequately agrees with measured
bay levels (Fig. 6). The inlet-bay system responds primarily to waves
with periods of 1 to 3 hours (near the Pentwater Helmholtz period) ;
shorter period waves are damped. This gives the bay level record a
smoother appearance than the Lake Michigan record. The maximum predicted
inlet current velocity during this episode is 60 centimeters (2 feet)
per second.

Other models that neglect temporary acceleration, e.g., Keulegan
(1967), should not be used for inlets on the Great Lakes where temporal
acceleration, head difference, and friction are important during the
response cycle (see Fig. 4).

III. THE FIELD DATA COLLECTION PROGRAM

Measurements were made at a number of inlet-bay systems throughout
the Great Lakes during 1974 and 1975 (Fig. 7). These study sites were
chosen because the inlets are typical on the Great Lakes, are of special
economic importance, or have maintenance problems.

1. Field Measurements.

Pentwater, Michigan, located midway along the longitudinal axis of
Lake Michigan, was selected as the primary study location because the
inlet-bay system at this location has a simple, fairly common geometry.
Good historical field data are also available at Pentwater from Duane
and Saylor (1967) who simultaneously measured water levels in Pentwater
bay and Lake Michigan as well as inlet velocities during July and August
1967. The U.S. Army Engineer District, Detroit, provided inlet geometry
data of the inlet from hydrographic surveys taken twice a year.

Field measurements during this study (Fig. 8) included water level
measurements at two locations (east and west ends of Pentwater bay) in
1974, and at one other location in 1975. Current velocities in the
inlet were measured concurrently with water levels during both 1974 and
1975. Hydrographic surveys were also taken at Pentwater. Other field
data collection sites were: Portage Lake, Ludington, White Lake, Muskegon,
and Holland on Lake Michigan (Fig. 8); Little Lake and Duluth-Superior on
Lake Superior (Fig. 9); Presque Isle on Lake Erie (Fig. 10); and North
Pond and Little Sodus on Lake Ontario (Fig. 11).

Field measurement types, locations, dates, and data sources for all
the study sites are summarized in Table 1; approximate dimensions of these
inlet-bay systems are summarized in Table 2.

*I9}eM}USg 1B SUOTIOTpoerd [OAST 10}ZBM TOpoOW [eITIOUNN

(44) awiy

paoipaig—~” ©

*g omn3Ty

(43) |@Ae7] 400M

*soqts Apnjs eTuy, “ZZ eansTy

ajST enbsalg © wy 191
lw OO!
pud||OH
B40] OU
0}u010] J8jDM}U8d gOe
uoybuipn’|
26Dj40q
N
By] asst]
2 Jolsadns

DJDG |DNN0JSIH ~4sNjNg
S}UBWOINSDA|’ GJ6] O
SjuawainsDaw vZl6| ©

puabe7

JOIIBONS
a

Legend
O e@ level recorder 1974-75

oa © level recorder 1967
SiR & velocity meter 1974-75

Lake
Michigan

Pentwater

Pentwater Lake, Mich. ~~ a

610 m
| a
2000 ft

Portage Lake, Mich. Weg fe. ae)

Lake
Michigan

Portage

Figure 8. Data collection sites on Lake Michigan.

Outer 0.5 mi
Jetties
Eone m

Jettiés /:-: 2 2
Lake
Michigan

Ludington, Mich.

Ludington

White Lake, Mich.

Lake White Lake

Michigan

Figure 8. Data collection sites on Lake Michigan--Continued.

rad

Muskegon, Mich.

Muskegon

Lake 1000 m
Michigan

| mi

| |

Holland, Mich. ee SON

Lake
Michigan

Figure 8. Data collection sites on Lake Michigan--Continued.

Lake Superior

“-... ,Water Level Recorder
rg Location

i |; Little Lake, Mich.
Little Lake ee

153 m
| a=
500 ft

Duluth Ship Channel
(Inlet 2)

Water Level Recorder
\\ Location

W Duluth-Superior, Wis.
Superior Entry

«— (Inlet |)

Figure 9. Data collection sites on Lake Superior.

Lake Erie

Water Level Recorder
Location

Erie Harbor

Figure 10. Data collection site on Lake Erie, Presque Isle, Pennsylvania.

Newer Inlet 1~>

Water level Je 2:0:
Older Inlet 2% recorder location
North Pond, N.Y.

Lake
Ontario

North Pond

9,000 ft
1,924 m

Lake
Ontario

:.. Little Sodus, N.Y.

ZN 9 900 ft
vival ee ZZ —
ee 610 m

Figure 11. Data collection sites on Lake Ontario.

Table 1. Summary of field measurements.
Water Inlet
Water level Data source level Channel geometry
Location and dates quality velocities (date measured)
Lake Michigan
Portage Portage Lake? This study, fall 1974 Good Sept. 19733
(Oct.-Nov. 1974
Ludington |Pere Marq. Lake? This study, 1975 Good Saas June 19743
Pentwater Lake Michigan and Duane and Saylor (1967) Good Analog June 19673
Pentwater Lake record
East Pentwater This study, 1975 Good Analog Oct. 1974"
Lake? record
Oct.-Nov. 1974
West Pentwater This study, fall 1974 Fair Analog Sept. 1975"
Lake? This study, 1975 record
(Oct.-Nov. 1974,
July-Oct. 1975
White Lake |White Lake This study, fall 1974 Good --- May, Aug. 19743
Oct.-Nov. 1974
Muskegon Muskegon Lake* This study, fall 1974 Apr., Aug. 19743
Oct.-Dec. 1974
Holland Lake Macataw? National Oceanic and Mar. 19743
Nov. 1974 Atmospheric Administration
(analog) Tecords
Lake Superior
Little Lake|Little Lake® June 19753
May-Aug. 1975
Duluth- Duluth Harbor? National Oceanic and Analog re- |] Oct. 19743
Superior j|June 1973 Atmospheric Administratio cord gages
(analog) records had not
Tecently
been cali-
brated?

Lake Erie
Harbor levels?

This study, 1975
May-Oct. 1975
ace) MN CEN |

National Oceanic and Poor
Atmospheric Administratio
records
ue re

\National Oceanic and Atmospheric Administration, Lake Survey Center.
2a S-minute sampling interval.

3Corvs of Engineers records.

“This study, fall 1974.

5A 2-minute sampling interval.

Presque
Isle

Bay levels?
May-Nov. 1972

Little

Bay levels?
May-Oct. 1975

North Pond

5) 2

Sept. 19753

1975 several
surveys made by
University of
Buffalo under
contract

Chart No. used
to determine
bay area!

768

767

763

332

0°26

c° Ce
8° tT

0°09T

OW EL
0°728T
9°16
T°8T
T°8t
0°6T

(g0T x 235)
Aeq fo
eore 90eF Ins

SL6T *8ny
SL6L AeW
SL61I *3dag

pL6T °2°0
GL6I ounr
pL6l
pL61 ‘Sny-*ady
pL6L ‘“8ny-Aey
vL6T °390
L961 oun
pL6T oun
EL6T ‘das

‘Lew

(35)
eole WNUTUTU

18 YIPIM JOTUT

(235)
eole

[T BUOTII9S-SSOID
ZOTUT WNUTUTpW

*sjUuoWsInseow ITIZOWO0eS Aeq pue YoTUI ‘Z aTqeIL

(Z 2eTUT)
(1 29 TUT)
puod yI10N

snpos e[331TT
SITS] enbserg

(Z 2°TuT)
(I 20TUT)
Zotzedns-y3ntng

OYT 913977
pueT Toy
uoseysny
oye] 93 TUM
197eM}U9g
197eM4U9g
u0}SUTpPNT

938 410g

woeasks
Aeq-}9T UT

Pre Equipment.

Small-amplitude sea level fluctuations with a period of approximately
the Helmholtz period of the inlet-bay system may generate relatively high
inlet water velocities as shown in numerical model calculations (see
Sec. II). Therefore, a water level recording system must be carefully
designed for each location to measure small amplitude, but potentially
important long waves. At the same time, this recording system should
eliminate any short-period, large-amplitude noise (e.g., wind waves) that
may mask the long waves in the record. For example, at Pentwater, records
should measure the low-amplitude waves with a period of 1 hour or longer,
and should exclude wind waves and other noise with periods of 1 minute
or less.

One method of designing a stilling well to meet these requirements is
to use the linear damping well design (Noye, 1974). This stilling well
consists of a vertical cylinder with a sealed bottom and open to the
lake through a long, thin tube. Friction in the tube and the relative
cross-sectional areas of the tube and stilling well cause the system to
respond directly to long waves outside the well and to drastically dampen
the short-period noise. Design of stilling wells is discussed by Seelig
(97/77)

Fisher-Porter series 1500 digital float-type water level recorders
with a vertical resolution of 3 millimeters (0.01 foot) and sampling
intervals of 2 or 5 minutes were used to measure water levels in the
stilling wells. Data were collected on punched tape. Water levels were
measured for several months at each location (see Table 1).

Inlet velocities at Pentwater were measured during 1974 and 1975
using a Bendix current meter suspended by a cable approximately midway
along the channel, 4.5 meters (15 feet) from the north wall at mid-
depth. Velocity data were recorded on a strip chart and later digitized
for analysis at the same time interval as the water level data.

3. Data Reduction and Analysis Techniques.

Initially, the Helmholtz period of each inlet-bay system and the free
seiching mode periods.were calculated for the lakes and bays surveyed in
this study (see procedures in Sec. II). These calculations, in conjunc-
tion with a survey of published data on Great Lakes resonance character-
istics, gave an indication of the period and magnitude of important long
waves that could be expected at each location. The information was used
in the design of water level measurement equipment (discussed in previous
section).

When the field data collection program was completed, the digital
punched-tape water level records were mechanically converted to punchcards
for computer analysis. The first procedure for studying these data in-
cluded plotting the records for visual inspection. Then, a fast Fourier

transform and cosine bell function were used to obtain a spectral
analysis of each record (Harris, 1974). A record length of 512 points
was used in these analyses to obtain detailed spectral line resolution.
Spectral analysis indicated the period and amplitude of long waves of
interest (Q.5 to 5 hours) in the record. This analysis is necessary
because several long waves are generally simultaneously present and the
Superposition of the waves gives the impression of a confused record.
Examples of spectra for Pentwater bay water levels recorded during 5 to
14 May 1975 are shown in Figure 12.

If water level records were of good quality, levels measured in the
bay were then used to predict inlet water velocities using the finite-
difference continuity equation (6).

The resolution of the continuity equation should be checked to judge
the usefulness of the predicted velocities; e.g., at Pentwater, the level
recorder has a vertical resolution of 3 millimeters , the sampling interval
is 5 minutes (300 seconds), and the ratio Aba /Aa = 10"; so the velocity

prediction resolution, V,, based on equation (6), is:

= 10" (2:01) = 0.33 foot or 10.1 centimeters per second. (10)

U5 300

Thus, the velocity will be expressed as multiples of 10.1 centimeters
(0.33 foot) per second which may be adequate for many purposes. For
example, if Abay/Ke was 10°, then with the given vertical resolution,

velocities could only be expressed as multiples of 100 centimeters
(3 feet) per second, which is inadequate for most purposes.

At Pentwater, the measured velocities in the inlet were digitized at
a sampling rate of 5 minutes so that a direct comparison could be made
of measured and predicted (eq. 6) velocities. Cumulative frequency
distributions of measured and predicted (eq. 6) inlet velocities are
shown in Figure 13. The 2 months of record in 1974 show that velocities
predicted by continuity are slightly higher than measured velocities,
but adequate for many design purposes.

IV. RESULTS
1. Seiching of the Great Lakes.

Free modes of oscillation of the Great Lakes have been identified by
spectral analysis of water level records in this study and others ©
(Mortimer, 1965; Mortimer and Fee, 1974; Hamblin, 1975; Rao and Schwab,
1974), and have been predicted using numerical techniques (Rockwell,
1966; Mortimer, 1965; Birchfield and Murty, 1974; Rao and Schwab, 1974).
Table 3 summarizes the known modes of oscillation of Lakes Michigan,
Superior, Erie, and Ontario.

Normalized Variance

and 6 May 7 and 8 May 9 and 10 May
1.25 1.44

Wave Period

[2 and 13 May

wl 44

Period (hr)

Figure 12. Sample spectra of Pentwater bay water levels (May 1975).

@ Measured
O Predicted (eq. 6)

ine)
oO

(e)

Pct of time velocity
(equaled or exceeded)
(S))

2
(SG)

0.1

0.5 1.0 1.5 2.0
Inlet channel velocity (ft/s)

Figure 13. Measured and predicted inlet velocity cumulative frequency
distributions at Pentwater, Michigan (October-November 1974).

Table 3. Modes of oscillation of the Great Lakes|.

Mode? Michigan Superior Erie Ontario
1L 8 4.4 4.9
2L 8 Soil 25)
3L .6 5.8 2.4
4L ? 4.1 1.6
Bh .0 3.6 ILS <37/
6L 2 3.0 Zo)
7L of 08) 0.9
8L 38) 22 0.85
OL DP 2.0 AGS

10L .14 1.8 ----
TA
2T :

? : 685
? .66 to 0.68
? : 58
? .52 to 0.53
4 .49

1Qbserved, computed by Fee (1968) program and Rockwell
(1966) data, and compiled from many sources.

2 = longitudinal, T = transverse; ? = other observed
periods, mode unknown. Two-dimensional modes are not
considered.

3Value unknown.

2. Predicted Inlet-Bay Response to Monochromatic Long Wave Forcing.

The response of each of the inlet-bay systems (Table 2) to uniform
long wave forcing was evaluated using the numerical model. In this
analysis, a sinusoidal wave was used to force the model for several
cycles (generally four) until the inlet-bay system response became
periodic. The results will give an upper limit of wave amplification
and inlet velocities because the prototype will generally not reach a
periodic condition.

Each inlet was modeled by using a grid system with one to three
channels and two to seven cross sections. The complexity of the inlet
determines the number of grids used to model the friction; e.g., Pent-
water with a constant width and only slight changes in depth along the
length of the inlet, was modeled using one channel and four cross sections.
Little Lake, with a more irregular inlet, was modeled using three channels
and five cross sections.

Pentwater, Michigan, and Toronto, Canada (Freeman, Hamblin, and
Murty, 1974), are the two harbor systems on the Great Lakes with water
levels recorded simultaneously inside and outside the harbor to provide
the necessary information for calibration of the numerical model. These
models were calibrated by varying the value of the Manning's friction
factor, n, so that the model long wave amplification is a best-fit upper
envelope of prototype measurements. Values of n of 0.036 and 0.062
were found for Pentwater and Toronto, respectively (Figs. 14 and 15).

The numerical model as used in these analyses did not explicitly
account for energy losses due to radiation of long waves into the sea or
entrance and exit losses. These losses are incorporated into the model
in the form of bottom friction through model calibration. Including
these losses in the friction term means that Manning's n calibrated for
Great Lakes inlets is higher than that used in open channel flow compu-
tations.

In the numerical model, the magnitude of Manning's n determines the
amount of energy dissipated. Larger values of n will cause higher
amounts of energy loss which results in less wave amplification in the
bay and lower inlet velocities. The influence of n_ on inlet-bay
response to long wave forcing at Pentwater is shown in Table 4.

Table 4. Influence of Manning's n on inlet-bay
response at Pentwater.

Vereen Che) Se tor
Ely S Wodl sae)

Se)

2.0

Numerical

1.5

5 4 3 2 | 0
Wave Period (hr)

Figure 14. Pentwater response to long wave forcing (n = 0.036).

2.0

0.5

Figure 15.

Predicted
(n= 0.062),

Observed
Freeman, Hamblin, and
Murty, 1974)

3
Time (hr)

Toronto response to long wave forcing.

4 |

Values of n were estimated for other Great Lakes inlets, based on
experience at Pentwater and Toronto. Then, the frictionless Helmholtz
period was estimated for each inlet-bay system using equations (4) and

(5), and the numerical model. Numerical model and frictionless
Helmholtz period results are summarized in Table 5.

The amplitude response curves and predicted maximum velocities are
shown for selected inlets on Lake Michigan (Fig. 16), Lake Superior
(Fig. 17), and Lakes Erie and Ontario (Fig. 18). A 3-centimeter mono-
chromatic forcing-wave amplitude was used in these models. For waves
of different amplitudes, the maximum inlet velocity is approximately
proportional to amplitude. Water level changes throughout the forcing
cycle cause nonlinear effects (i.e., ap/a, is slightly different at
high and low water), so that the mean of ebb and flood conditions is
used in the response curves in this study.

This analysis shows that all of the jettied inlet systems studied
have significant inertial effects because long waves at or near the
Helmholtz period of each system have higher amplitudes in the bay than
in the Great Lakes.

The inlet-bay systems modeled have a wide variation in response
characteristics from one system to another because of the complicated
interactions between the four terms in the equation of motion of the
inlet and the response of the bay to the inlet. Pentwater, for example,
has a moderate amount of wave amplification and produces inlet velocities
greater than 30 centimeters per second (1 foot per second) for forcing
waves of 3-centimeter amplitude and periods ranging from 0.9 to 2.5 hours
(Fig. 16). White Lake has less wave amplification, but the interaction
between the inlet and bay produces higher velocities over a wider range
of forcing periods (greater than 30 centimeters per second for periods
of 1 to 5.6 hours) (Fig. 16). Since Little Lake and Presque Isle have
the capacity to generate reversing currents in only a narrow window of
forcing periods, it is unlikely that significant reversing currents will
be frequently generated (Figs. 17 and 18).

Duluth-Superior has the highest capacity for generating reversing
currents for a given wave amplitude with maximum velocities occurring at
a theoretical forcing period of 1.1 hours. The mean velocities in Duluth
(inlet 2), are approximately 1.5 times larger than in Superior (inlet 1)
(Fig. 17). A unique feature of the Duluth-Superior system is that the
model predicts a net flow into the harbor through the Duluth inlet and a
net outflow through the Superior inlet when the forcing period is near 1
hour (Table 6). This asymmetry in flow throughout the forcing cycle will
generate a small counterclockwise net flow throughout the inlet-bay system
at Duluth-Superior.

North Pond, in 1975, had two short natural inlets connecting a
relatively large bay to Lake Ontario. North Pond does not amplify long
waves because the mass of water in the inlets is small compared to bay
size, and friction in the inlets is high due to the shallow-water depths
(Fig. 18). Since friction is high, North Pond behaves like a traditional
tidal inlet with a balance between head and friction in the inlets.

"INd90 JOU SeOp UOTIeOTFITdwe Jey, OS JUeUTWOP ST UOTIITIY,

pereiqrred,

* (SZ90q0N1T] USD s) 2005 T'0 = %e ‘opnzttdwe yzIM oAeM ITIeWOIYIOUOM eB BUTS,

UeSTYITW eAe]

[epow (s) pue (7) *sbo u s,suTUURBW
Teo TLounN SSOTUOTIITIA

(zy) potzed z1ToyWT eH

(Tepow [eotzoumu)

(ay) ATOOTOA
UNUTXeU FO poTI9g
*, SOTPTOIOTOA

T
ZOTUT uUNUTXeW pue uoTeOTFITdwe oAeM UNUTxeW FO spotzed poeydtperg

03U0LO],
puog YiZON
Snpos 213311

eS] enbseirg

Zotzedns
-yantng
oye] 9T33TT

PueT OH

uoseysny]
sxe] 93 TUM
1978M}U9 q
u07ZS3UTPNT
938110g

UOT}BIOT

"g OTqeL

— — Portage
—--— Ludington
—-— Pentwater

----— White Lake
Muskegon
—-— Holland

Op/d,

6 5 4 3 2 | (0)
Wave Peroid (hr)

— — Portage
—--— Ludington
—-— Pentwater
——-—- White Lake
Muskegon
—-— Holland

Vox (ft/s)

6 5 4 3 2 | (0)
Wave Period (hr)

Figure 16. Predicted response of inlet-bay systems on Lake Michigan to
monochromatic forcing (a, = 0.1 foot).

Duluth - Superior
—-— Little Lake

Superior
—---- Duluth
—-— Little Lake

Vmox, (ft/s)

6 5 4 3 2 | (0)
Wave Period (hr)

Figure 17. Predicted response of inlet-bay systems on Lake Superior to

monochromatic forcing (a, = 0.1 foot).

———-— Presque Isle
Little Sodus

—-— North Pond

3
Wave Period (hr)

———— Presque Isle

Little Sodus
—-— North Pond, North Inlet
—--— North Pond, South Inlet

Vmax, (ft/s)

Wove Period (hr)

Figure 18. Predicted response of inlet-bay systems on Lakes Erie and Ontario

to monochromatic forcing (a, = 0.1 foot).

Table 6. Predicted Duluth-Superior maximum inlet
water velocities for a forcing wave of
1 hour (a, = 3 centimeters).

73 (2.4)
64 (2.1)

43 (1.4)
49 (1.6)

A unique feature of the North Pond inlets is that the maximum velocity
in the inlets is predicted to be approximately the same over a wide
range of forcing periods because of the approximately linear relation
between wave amplitude propagation in the bay and wave period (Fig. 18).
The velocities in the northern inlet at North Pond, the most recently
formed inlet, are predicted to be 1.4 times larger than those in the
older inlet.

The numerical model for Pentwater was run using monochromatic forc-
ing waves with the various modal periods of oscillation of Lake Michigan
(Table 3). The predicted amplification and maximum velocity for a forc-
ing wave of a, = 0.1 foot are listed in Table 7. From this analysis,

the sixth through ninth longitudinal modes of oscillation of Lake Michigan
are predicted to cause the largest wave amplification and generate the
highest relative velocities. However, analysis of the node-antinode
pattern of Lake Michigan shows that only even modes of oscillation will
have antinodes, and cause significant water level fluctuations adjacent

to Pentwater (Fig. 3). Since odd modes of oscillation have a node near
Pentwater (Fig. 3), even the presence of one or more of the odd modes of

oscillation in Lake Michigan will cause only small water level fluctu-
ations near Pentwater. This means that the sixth and eighth longitu-
dinal modes of oscillation of Lake Michigan will probably have the
largest influence on the hydraulics of Pentwater.

3. Observed Lake Level Fluctuations, Bay Response, and Inlet Velocities.

The first obvious characteristic of Great Lakes water level fluc-
tuations is that they are not uniform (as assumed in the previous section).
Therefore, the monochromatic analysis can be used to obtain an upper esti-
mate of bay wave amplification and iniet velocities; however, a complete
analysis is necessary for an accurate estimate of response to a particular
Great Lakes water level time history. Sample water level records from
Lake Michigan and Pentwater bay, along with spectral analysis of 42-hour
records to show typical bay response, are plotted in Figures 19, 20 and
ZA

Figure 19 shows a storm event on Lake Michigan when several modes of
oscillation were excited by meteorological effects. The second longitu-
dinal mode of Lake Michigan (5.3 hours) and a 0.65-hour wave are partic-
ularly dominant. All of the 5.3-hour wave propagates into Pentwater

Table 7. Numerical model prediction of Pentwater
response to Lake Michigan modes of
oscillation.

OF 1.0

Bo 1.0 0.2
3. 1.2 0.5
So 1.3 0.7
2. 1.4 0.9
Bo 1.6 iodl
ite 1.7 1.6
1. 1.6 1.9
alte 1.5 2.0
1. 1.3 1.9
0. 0.8 1.5
0. 0.6 1.1

Notes--L = longitudinal modes of oscillation of
Lake Michigan.

transverse modes of oscillation of Lake
Michigan.

4
i}

? = observed oscillation,:mode unknown.
X = wave has‘a node near Pentwater.

Modes 6L to 9L have modes of oscillation
with large ap,/a, and Vmax for Pentwater.

Period (hr)

0.3

0.4
Pentwater Lake

Lake Michigan

0.5

0.7

1.0

1.4
5.3 Forcing Period (hr)

A0){0) 0)

QIUDIIDA

1.78 2.13 2.48 2.84 3.19 3.54

Frequency (c/hr)

0.38 0.73

Lake Michigan

Pentwater Lake

— See — — — = ==
—<——-—-—-y ==
== == == ‘
= LT
Sa o
as E

: 3 a N
oe “6 fe) ro)
(

(43) |8A87 48}0M

Sample Pentwater and Lake Michigan water levels and spectra.

Figure 19.

Period (hr)

0.3

0.4
Pentwater Lake

1.0 0.7 0:5
----- Lake Michigan

1.4

20.0 3.0

QUDIJOA

[Ou eal Se 2:48 8 25845059 so 4

Frequency (c/hr)

1.43

|.08

OL} O78)

Lake Michigan

Pentwater Lake

_-—— _

Time, 4hr

——_

6)
-0. |
(0),

(43) 18487 49)0M

Sample Pentwater and Lake Michigan water levels and spectra.

Figure 20.

Period (hr)
Z2O)(0) BO) ks ol) 0.7 0.5 0.4 OS)

1.44 Wave Period (hr)

S Seltrs Lake Michigan
io)
= Pentwater Lake
>
AMAL ss
\ y , AA 1 ult rx
0.38 O73 1.08 1.43 1.78 2.13 2.48 2.84 3.19 3.54
Frequency (c/hr)
0.2

Lake Michigan

Pentwater Lake

Water Level (ft)

Time, 4 hr
S|

Figure 21. Sample Pentwater and Lake Michigan water levels and spectra.

3 |

with an amplification factor of 1.0, as predicted by the numerical model,
because the wave period is much longer than the Pentwater Helmholtz period
of 1.8 hours. However, the 0.65-hour wave, which reaches heights of 15
centimeters (0.5 foot), has a negligible effect on the harbor because it
is much shorter than the Helmholtz period of 1.8 hours. Waves of 1.44

and 1.25 hours are slightly amplified by the harbor (shown by the spectral
analysis in Fig. 19), but these waves are difficult to distinguish in the
record because of the mixing of individual wave components.

The storm event in Figure 20 shows that a different set of modes of
oscillation of Lake Michigan is present. The 1.44-hour wave is the
highest, is amplified the most, and probably generates the highest per-
centage of significant reversing inlet current velocities. A 1.8-hour
period wave is also present and is amplified. Waves shorter than 1 hour
are damped by the harbor.

An unusual water level fluctuation at Pentwater where only the 1.44-
hour wave is dominant in Lake Michigan, is shown in Figure 21.

As predicted previously, the 1.8- and 1.44-period waves which are
the sixth and. eighth longitudinal modes of oscillation of Lake Michigan,
cause the highest current velocities.

Figure 22 shows the wide variation of water level fluctuations
occurring in three different harbors along the eastern shore of Lake
Michigan at the same time (Pentwater and Ludington are only 2.3 kilometers
(11 miles) apart). The reasons for the differences are that the forcing
waves outside each location are different as a result of the node-anti-
node pattern of seiching in Lake Michigan (see Sec. II) and because each
harbor responds differently to the forcing that is present (see Sec. IV);
e.g., the 1.44- and 1.28-hour waves in Pentwater and Ludington are not
noticeable in Muskegon harbor which has a Helmholtz period of 5 hours.

The forcing of harbors on the other Great Lakes will be completely
different because the system of seiching varies from lake to lake; e.g.,
on Lake Superior, wave periods of 0.59, 0.68, 0.95, 1.14, and 1.7 hours
occur in Little Lake Harbor (Fig. 23). Shorter period waves may also
occur in Lake Superior, but are not observed in the harbor because the
harbor dampens waves shorter than approximately 0.4 hour.

The 1.7-, 1.14-, and 0.95-hour waves on Lake Superior (the 7th, 10th,
and 11th longitudinal modes of oscillation) were observed to cause high re-
versing currents and associated navigation problems at Duluth-Superior;
e.g., on 10 June 1973, a 1.7-hour wave with a height of approximately
30 centimeters (1 foot) in the harbor, in conjunction with small 0.95-
hour period waves, affected Duluth-Superior. Velocities as high as 200
centimeters per second (6.5 feet per second) were generated in Duluth
inlet and 140 centimeters per second (4.5 feet per second) in Superior
(Fig. 24). High velocities are generated in these inlets because of the
large forcing waves in Lake Superior at this location which have periods

*SUOT}ZENJONTF TOAST Leqem ofdmes °77 omMBTy

== = = 76 | WOH AON == ==
l 2 ———— = 18) HONE Ge

ay G= 4, uobaysnw

ro)
wnjog Aspsjyiquy

Jy QQ’) = 4] JayDMyuad

D8)

1.14

Variance

3.0

Variance

3.0

0.59
28 Aug 1975
= Observed Water Levels
= 0.2
em (0);
au
= -0. |
= -0.2
1.0 06 04

Wave Period (hr)

a) Typical Fluctuations
Computed Spectra fa} Typ

0.68

19 July 1975
Observed Water Levels

Water Level (ft)

oo _ 99S
hm —- O —Pp

10 06 O04
Wave Period (hr)
Computed Spectra

(b) Extreme Event

Figure 23. Water level fluctuations in Little Lake Harbor.

Bay Level al

Duluth
Inlet Velocity (ft/s)

Superior
Inlet Velocity (ft/s)

Figure 24.

Sample bay levels and inlet water velocities at
Duluth-Superior (inlet velocities are approximate;
gages had not recently been calibrated).

near the harbor Helmholtz period. Forcing waves are large because the
harbor is located on the converging end of the lake, which will always
be an antinode of longitudinal oscillations.

Maximum water velocities observed in other inlets are much lower
than in Duluth-Superior; e.g., at Pentwater, all measurements and pre-
dictions show that, velocities are less than 60 centimeters per second
(2 feet per second) for 99.5 percent of the time (Fig. 25). Predicted
inlet velocities for other locations show that Portage, Ludington, and
Pentwater have similar velocity distributions; Presque Isle, Muskegon,
and Little Lake have still lower velocities (Fig. 26).

V. INLET DESIGN

Great Lakes inlet design problems generally fall into one of two
classes: (a) a pond or lake to be connected to one of the Great Lakes
by a new channel, and (b) an existing inlet channel to be modified. The
concepts and techniques developed in this study can be used to aid the
design of an inlet in either class. An example application for each
class is given below.

1. New Inlet Channel.

The procedures for analysis of a new channel that is to connect a
lake to one of the Great Lakes are: (a) determine the approximate inlet
dimensions (length, width, and depth) based on physical limitations such
as the desired navigable depth and width and the distance between the
lake and Great Lakes; (b) estimate a Manning's n for the proposed channel
(see Sec. IV, 2, for typical values of n); (c) use the numerical model
to obtain monochromatic response characteristics of the harbor for the
range of expected lake seiching periods and a typical amplitude; (d)
compare the results to those of other nearby harbors; and (ce) apply the
numerical model to predict inlet velocities, discharge, and bay levels
for the period of record (if Great Lakes water level fluctuation records
are available in the vicinity of proposed site).

For example, suppose an inlet is to be designed to connect Crystal
Lake to Lake Michigan (Fig. 27). Crystal Lake, located on the eastern
shore of Lake Michigan 35 kilometers (22 miles) north of Portage Lake,
has a bay surface area of 4.12 x 107 square meters (4.44 x 108 square
feet). The inlet at this site would be approximately 1,200 meters
(4,000 feet) long. Assume that the inlet would be 61 meters (200 feet)
wide and 5.5 meters (18 feet) deep. Since the inlet is similar to Pent-
water (see Table 5), the Manning's n for this channel is estimated to be
0.036.

The numerical mode] was run for Crystal Lake using Lake Michigan
seiche periods of 9.3, 5.3, 3.5, 2.2, 1.85, and 1.4 hours with an
amplitude of 3 centimeters. The predicted response characteristics of
this inlet-bay system are shown in Figure 28. The model predicts that

a
a. @ Summer |967
oN Q Fall 1974
» @ Summer 1975
XN
20 “
> oO
= 2
23s
=
Co) (<b)
[Se
= Oo
ro
-&
aq S

ORS |. ES 2,0)
Inlet channel velocity (ft/s)

Figure 25. Pentwater inlet cumulative frequency velocity distributions
(1967, 1974, and 1975).

—--— Ludington (Fall 1974)

—-— Pentwater (Fall 1974)
Muskegon (Fall 1974)

— — Portage (Fall 1974)

22 ——-—-— Presque Isle (Summer 1975)
33 —-— Little Lake (Summer 1975)
2
> x
3) cb}
ES
Eo
a
co
oS =}
ao

OL 1.0 1.5 20)
Inlet Channel Velocity (ft/s)

Figure 26. Inlet velocity cumulative frequency distributions (based on
equation 6 and bay water level records).

Crystal Lake

Portage

Muskegon

000 ft

= eS
eee m

OP Or ome Gein

A poy = 4-44x10 ft”

Coke Mj chigan

Crystal Lake, Michigan.

Figure 27.

519

Vmax (G,= 0.1 ft)

0
10 9 8 7 6 5 4 3 2 | 0
Wave Period (hr)

Figure 28. Predicted response characteristics of an inlet for Crystal
Lake (L = 4,000 feet, A = 3,600 square feet, D = 18 feet,
B = 200 feet).

the wave amplitude will be smaller in the bay than in Lake Michigan, pri-
marily because the bay surface area is much larger than the inlet design
cross-sectional area (a ratio of approximately 10°).

The model also predicts that monochromatic seiches with an amplitude
of 3 centimeters will generate maximum velocities of 43 centimeters per
second (1.4 feet per second) for wave periods of 4 to 9 hours (Fig. 28).
Since maximum velocities decrease for waves shorter than 4 hours, a wave
with a 1-hour period produces insignificant inlet velocities.

The first three modes of oscillation of Lake Michigan (9.3-, 5.3-,
and 3.5-hour waves) will generate the highest velocities for the Crystal
Lake inlet design. Portage is located near Crystal Lake; therefore,
forcing amplitudes of the first three modes of oscillation of Lake
Michigan will be similar at both locations (Fig. 3). The predicted vel-
ocities for a given wave are different at Portage and Crystal Lake
(Fig. 29) due to differences in inlet and bay geometry. However, Portage
water level data can be used to estimate Crystal Lake inlet velocities.

To predict inlet velocities at Crystal Lake using Portage data, the
measured Portage bay level fluctuations must first be adjusted to esti-
mate the nearby Lake Michigan wave amplitudes. These amplitudes are then
used to predict velocities at Crystal Lake; e.g., a measured Portage bay
level fluctuation has a period of 3.5 hours and amplitude of 0.15 foot
(4.6 centimeters). This wave was amplified by a factor of 1.3 by Portage
harbor (Fig. 16), so the Lake Michigan wave amplitude was 0.15/1.3 =
0.12 foot. A 0.1-foot wave amplitude in Lake Michigan produces a maximum
velocity of 1.3 feet per second at Crystal Lake inlet (Fig. 28); there-
fore, the 0.12-foot wave produces 1.3 (0.12) = 1.6 feet per second maxi-
mum velocity. This procedure could be followed for other seiche modes
to estimate the maximum velocities expected at Crystal Lake.

If a complete analysis of inlet velocities is required, water levels
should be measured in Lake Michigan adjacent to Crystal Lake for at least
several months. These levels can be used as the forcing function in the
numerical model to produce a predicted time history of inlet velocities,
discharge, and bay levels for the period of record.

2. Inlet Channel Modification.

Procedures for investigating the effect of a modification to an inlet
are: (a) Determine the geometry of the present system and obtain proto-
type hydraulic data (i.e., concurrent bay levels, Great Lakes levels,
and inlet velocities); (b) calibrate the numerical model; (c) obtain
monochromatic response characteristics of the inlet-bay system, (d)
modify the model geometry to reflect the proposed inlet change and pre-
dict the response characteristics of the new condition; and (e) use the
water level records in the Great Lakes to force the model to produce a
time history of inlet velocities, discharge, and bay levels for the
proposed design.

6 |

——_,
-—
>)
=
=

Bf oom
Crystal Loke be

“wo
~
=
So
i=
>
<9
2.0
5 4 3 2 |
Wave Period (hr)
Figure 29. Predicted inlet velocities at Michigan inlets (a, = 0.1 foot).

For example, assume that a prediction of inlet velocities and the
amplitude of bay level fluctuations at Pentwater is desired if (a) the
inlet was deepened to 7.3 meters (24 feet), and (b) the inlet was allowed
to shoal to a depth of 1.8 meters (6 feet).

The monochromatic response of the inlet-bay system for these two inlet
modifications is predicted by changing the inlet geometry in the cali-
brated model of Pentwater. Each model was run with sinusoidal wave periods
between 0.5 and 5 hours and an amplitude of 3 centimeters to predict am-
plification of the wave in the harbor and maximum velocity in the inlet.
Figures 30 and 31 show the amplification and maximum velocities, respec-
tively, for the 1967 inlet geometry, and for inlet depths of 1.8 and 7.3
meters. The results are summarized in Table 8.

Table 8. Summary of Pentwater hydraulic characteristics for
selected inlet depths.

(85/85) maze

T, (hr)
Vi a (ft/s)!

1
ee (hr)

lFor ago = 0.1 foot.

These predictions show that deepening the Pentwater channel from 1.8
to 7.3 meters causes the peak amplification and inlet velocity to in-
crease, and the Helmholtz period and period of maximum velocity to de-
crease. Comparison of the modes of oscillation of Lake Michigan (Table 3)
with the predicted velocities (Fig. 31) suggests that the 0.85-, 0.97-,
1.1-, 1.25-, and 1.44-hour waves will generate the highest reversing
currents at Pentwater if the inlet was deepened to 7.3 meters.

The Lake Michigan water levels recorded on 18 August 1967 (Fig. 6)
were used to force the lumped parameter model for the selected depths
(Fig. 32). The model predicts that for these Lake Michigan level fluctu-
ations, the maximum velocity for an inlet 1.8 meters deep would be 46
centimeters per second (1.5 feet per second); a 7.3-meter-deep inlet
would have a velocity of 107 centimeters per second (3.5 feet per second).

VI. SUMMARY AND CONCLUSIONS
1. Meteorologically generated seiches cause most of the significant

reversing currents at Great Lakes inlets. Seiche periods and node-anti-
node patterns can be predicted numerically. However, water levels must

Amplification

Figure 30.

1967 Condition
(I | ft <depth<20 ft}

Period (hr)

Response to long wave excitation at Pentwater
(wave amplitude = 0.1 feet).

Maximun Velocity (ft/s)

3.0

(2.0)

2.0
1967 Condition
(ll ft<depth<20 ft
[5
1.0 - Design Depth =6 ft
ONS

Period (hr)

Figure 31. Response to long wave excitation at
(wave amplitude = 0.1 foot).

Pentwater

"199F yz pue 9 Jo syqdep
Jeuueys £(9 “3BTy 99S) LOGI YSNSNY BI UO PpepsOIeT STSAST
I9}EM UCBTYITW IY] LOF SAOTILIOTOA JaTUT 19}eMUSg P9}ITPetd

(44) owl
¢| lI 6 2

Ze eins Ty

(S/43) Ayo01aA

be measured near the point of interest to obtain detailed amplitudes and
periods of the active seiche modes.

2. A simple numerical model developed at the Coastal Engineering
Research Center (CERC) can be used to predict inlet velocities, discharge,
and bay levels for Great Lakes inlets. The model was applied to evaluate
the hydraulic characteristics of several Great Lakes inlets, and examples
are given using the model for typical design computations.

3. Numerical modeling of selected inlets showed that head, temporal
acceleration or inertia, convective acceleration, and friction may all
be important in controlling the hydraulics of Great Lakes inlets. Temporal
acceleration may be especially important as it causes bay fluctuations to
be amplified and out of phase with the forcing wave. As a result, a
large head differential may be generated for waves with periods approx-
imately equal to the Helmholtz period of the inlet-bay system. For a
given amplitude, the highest reversing inlet currents will occur for wave
periods slightly smaller than the Helmholtz period. Since even a small-
amplitude seiche may generate significant reversing inlet velocities if
the wave period is near the inlet-bay Helmholtz period, water levels
should be carefully measured.

4. Reversing inlet currents can also be predicted by the continuity
equation from high-quality bay water level records. Cumulative frequency
distributions of inlet velocity developed in this manner are presented
for several Great Lakes inlets.

5. Reversing velocities at most inlets are generally small.
However, velocities may be high if the inlet is located where lake seiche
amplitudes are relatively large and have a period approximately equal to
the inlet-bay system Helmholtz period; e.g., at Duluth-Superior.

LITERATURE CITED

BIRCHFIELD, G.E., and MURTY, T.S., ''A Numerical Model for Wind Driven
Circulation in Lakes Michigan and Huron," Monthly Weather Review,
Vol l025 Now 2 eben S74) eppe Lol Ose

DUANE, D.B., and SAYLOR, J.H., "Water Level and Channel Velocity Data
for Pentwater, Michigan for the Summer of 1967,'' Lake Survey Center,
National Oceanic and Atmospheric Administration, Ann Arbor, Mich.,
1967.

FEE, E.J., "Digital Computer Programs for the Defant Method of Seiche
Analysis,'' Special Report No. 4, Center for Great Lakes Studies, The
University of Wisconsin-Milwaukee, Milwaukee, Wis., July 1968.

FREEMAN, N.G., HAMBLIN, P.F., and MURTY, T.S., "Helmholtz Resonance in
Harbours of the Great Lakes, 17th Conference on Great Lakes Research,
Aug. 1974, pp. 399-411.

HAMBLIN, P., 'Storm Effects on Lake Levels," 18th Conference on Great
Lakes Research, Canada Centre for Inland Waters, Burlington, Ontario,
unpublished, May 1975.

HARRIS, D.L., ''Finite Spectrum Analyses of Wave Records," Proceedings of
International Sympostum of Ocean Wave Measurement and Analysts, Sept.
1974, pp. 106-124.

KEULEGAN, G.H., "Tidal Flow in Entrances, Water-Level Fluctuations of
Basins in Communication with Seas,'' TB No. 14, U.S. Army Engineer
Waterways Experiment Station, Vicksburg, Miss., July 1967.

MILES, J.W., ''Coupling of a Cylindrical Tube to a Half-infinite Space,"
Journal of Acoustical Soctety of Amertea, Vol. 20, 1948, pp. 652-664.

MORTIMER, C.H., "Spectra of Long Surface Waves and Tides in Lake
Michigan and at Green Bay, Wisconsin," Eighth Conference on Great Lakes
Research, 1965, pp. 304-325.

MORTIMER, C.H., and FEE, E.J., '"'The Free Surface Oscillations and Tides
of Lakes Michigan and Superior," University of Wisconsin, Milwaukee,
Wis., 1974.

NOYE, B.J., "Tide-well Systems II: The Frequency Response of a Linear
Tide-Well System,"' Journal of Marine Research, Vol. 32, No. 2, May 1974,
pp. 155-181.

RAO, D.B., and SCHWAB, D.J., "Two-Dimensional Normal Modes in Arbitrary
Enclosed Basins on a Rotating Earth: Application to Lakes Ontario and
Superior," Special Report No. 19, Center for Great Lakes Studies,
University of Wisconsin, Madison, Wis., 1974.

ROCKWELL, D.D., ''Theoretical Free Oscillations of the Great Lakes,"
Ninth Conference on Great Lakes Research, 1966, pp. 352-368.

SEELIG, W.N., “'Stilling Well Design for Accurate Water Level Measurement,"'
TP 77-2, U.S. Army, Corps of Engineers, Coastal Engineering Research
Center, Fort Belvoir, Va., Jan. 1977.

SEELIG, W.N., HARRIS, D.L., and HERCHENRODER, B.E., "A Spatially Integrated
Numerical Model of Inlet Hydraulics," GITI Report 14, U.S. Army, Corps of
Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., and

U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss.,

(in preparation, 1977).

2 ot? Xo anoteetbdaed cone ‘5 taskvet
WwTOaRaR “weoks Bt one 7"

: rE J 5 > a) " ; i i=?
ve * s a conht t tet Westhgitaals eae
Be ah “ Me
: eTOg Ra baa acct )- t yarA Ge a
ree vet y BY (thovied 1 4x08 ceed iy.
Pee ar AAT ee ef co
| : (SHS 504 alban Spree aaa Tt a vabia
ey “ aod Blige. 29) ate OM Lagreus
; ay 4 ova? Preenr ana teRod' sirpaitkga ad
a . ¥ . lot’ sth Sroetireaqua 2 pwxe Taw T4OF rigna yest, Vee he
eel gradutol rete J abuse
, : = : - ‘Tan '¢ ; eg Fd, tapas qeregiae)
| , i Inks Crudles, The.)
ikon, Win, , 1 my 60).
ie inhobts Redenance As
- 7
7 t, y re
ihe ’ el
ss * i
P ' éy } “s
: , tarranrte i
" i bus
i io ~ [avd Pate
i ‘ I iF we i rt”
i f a uly V7,

: iF V9; e fi ie ie Frag rt » Modt~4 invinit Ve Soi
4 we ’ hae ie 8 i] be ud oy Mi ; as: vshineg Pee vet a

T ef W ; oe au 4 awe e a hae wea data fi y
eblenn’s qf 43 ‘Wieden bat, ge cogeeona. on be
: by PY Pea. ; a ww oe a ee. ai : a
. r DY MP:
es } Pet Pe). ea at fia ond
ace 7 . mind Sip ae Lor,’ tmivend t y id Kheqons? Dy M4 Sy

’ mia awa t >) 1S Beir ‘ sa ended saaye "ed
det Sys " Srisna ! f Mepbnw w Benes a.

e~Ti¢ eM
wa’ 2uGe4 4) »
et ne .

‘RAG, OF oe | vt! sc7wes Ment tow th aaa hl
faclosed baeinw on, 4, ¢ jotating Berth:
Kipe rior,” 4 7 , Li port ck an ie, Cer

Wel vewsaey CF Whwcwnidtty erie Pe

il
s 1) 1
7 bh RY 7
: win . ’ ; ; ily

129 8-LL sou daigcn° €0Z0L
*g-// °ou rzeded Teopuyseay
*lJejUueg yoDlessey BupiesuT3uq TeqseoD *s*m :sefses “IJ ‘*1oyAne
Jufof **W Jzeqoy fuesueto0s “TT “eTIFL *I 4 “seyeT 3eeIg *sjeTUT °]
*Teuueyo 3ut
=}87xX9 ue JO UOTIEOTFFpoW sy 02 pue Touueyo JeTUT Mou eB Jo usTsep ayy
03 poeftdde ere Apnjs ay} ut pedoteaep sanbyuyseq pue sqdejuoa ayq jo
26n 23e1}sSUuOMep 02 seTduexq ‘“seTpnqs ustsep pue 3upuuetTd sanqnjz io0z
BIep eseq doTeAep pue Sesuodsez maqsks Aeq-jaTupT 3ufz.oTpetd 10; senbyu
-Y09} YSTTqe3se ‘sap TOOTAaA JoTUT JueoTysTUustTs z0zF uspueyoow Buz ersues
2Yy} pue uoTIeITOXS eABM BUOT Jo 2injeU 2yQ 2qeSTISeAUT 07 SayYeT 3eeID
2y} uo sioqieYy euTU 3 C/=4/6| UT pazoONpuod vieM sjUSMaAnseoM pTeTy
*g9 °d :kyderzorrqrg
(8=LL *ou § SizaRuUeD
yoreesey Suyresuzsug Teqseog *s*n — teded Teopuyoel) “TTF : °d 69
“LL6, ‘2equUa9
yoieesey Buyiseut3uq TejseoD *s*n : “eA S1fFOATEeG 34104 — “‘UesUeTOS °“}{
Jieqoy pue 3TTeeS *N WeTTTTM Aq / sjeTUT seyeT qeeay Jo soppneapAéy
“N WETTTEM ‘37Tee5

£29 8-LL "ou d3.gcn* €0ZOL
*g-/L °ou aeded Teopfuypey,
‘leqjue) yoTeesey Z3uyieeutsug Teqseop *S*n :seattes “TI ‘*zoyzne
jufof ‘*W 3zeqoy ‘uasuer0s “TT “eTIFL ‘1 = ‘“seyel yeerg ‘sqzeTUT *}
*Teuueyo 3uy
-1S}TxXe ue JO UOTIBOTFTpoW 9y} 0} pue TeuUueYyO JeTUT MoU e Jo UsTSep sy}
03 pettdde sae Apnzs ay uz pedoteaap sanbzuyoe7 pue sqydeou0s ay3 jo
asn 93e1}SUOlep 03 setTduexq ‘satpnys ustsep pue Zufuuetd vanqnz 10;
eBjep aseq doTeAep pue ‘asuodsex maqsks Kkeq-jaTuy 3ut,OFpezd 103 senbyzu
-4Y923 YSTTqeqjse ‘sat POOTSA JeTUT JuedTFT_UsTS 10J uszTueyou Buz er9ues
ay pue uotIeRTOXe aAeA BUOT Jo ainjeu ayy a3eTISeAUT 072 SaYyeT e919
294i uo Ssioqiey sUTU 3e C/-4/6| UF pazONpuod o1aM sjuUsleInsesu pTety
*g9 °d :Ayderz0zTqTg
(@-Z£ “ou $ Szeaquag
yoivesey Buyiseutsug TeIseop *g*q — traded Teopuyoey) “TTF : *d 69
“LL6L *t93UaD
yoivesey suyrseuTsuq TeIseop *s*n : “eA ‘AfOATegG 340g — ‘uasuatOS “Wj
Jieqoy pue 3tTeaeSg "N WeTTTEM 4q / SJeTUF sexe] JeeIN Jo soT[NeapAy
“N WETTTEM ‘3T Tea

£29 8-LL *ou d3iecn° €0Z0L
*@=/{ ‘ou azeded Teozuyoey
*ZequeD yoTeesey dufrssuzJug Teqseop *s*n :seytteg “TI ‘“zoyqne
qufof “*y 4aeqoy ‘uesuetos ‘II ‘“aTATL ‘I ‘“seyeT 4eer9 *sjzeTuUL °[
*Touueyo 3ut
-}8}xXe ue JO uoTIeOTJT pow oyz OF pue Touueyod JeTUT Aou e jo usTsep oy}
03 pettdde eze Apnjs ey3 uz pedoTevep senbzuyoe, pue sydeou0s ey3 jo
asn ajei}suowlep 03 seTduexy °setpnjs ustsep pue Zupuuetd eangny 105
Bqep eseq doTeaep pue Sesuodsex wajsks AeqeqoTuT 3upjo-pead 10x senbyzu
-Y222 YSTTGeqse ‘saTRPOOTSA JeTUF JueoFsTUsTs Tos uspueyoou Buz zesr9ues
94} pue uofjeRFOxe sAeM BUOT FO VAnjeu 9yj VzeSTASeauT 07 soYeT eID
ay uo sioqiey sUTU 4e C/-4/6{ UF peONpuod ote sqUeMeANseaM plot”
*g9 ‘d :AydessopTqrg
(G=Z£ *ou { ‘1e3Ua9
yoieesey sgupyitseuzsuqg Teqyseog °s*n — azeded TeoFuyoeT) “TTF : *d 69
*LL6, *te3uUe9
yoreesey Suftiseupsugq TeqseoD *s*n : “eA ‘AFOATeg 340q — c‘uesuatos “pW
Jieqoy pue 3TTeeg *N WeTTTTM 4q / sje TUF sayeT 3eezy jo soppneapsy
°N WETTTEM ‘37TeeS

£29 g-LL *ou daesn: €0Z9L
"Q-£/ ‘ou zeded Teopfuyoey
*teqUe9 YOteesey duprssuTsug Tejyseop *s*m :settesg “II ‘*zo0yRne
quyof “*y yteqoy ‘uesuet0s “TI “eTIFL “I ‘“seyeT qeezg *sqeTUT *|
*Teuueyo sut
-3S}xe UP FO UOTIVOTJTpow ey} OF pue TouUeYyO JeTUT Mou e Fo usTsep ayy
03 pettdde ere Apnys ey, uz pedoTeaep senbzuyseq pue szdeou0s ey3 Jo
esn ejeijsuollep 07 setTdwexy ‘setpnjs ustsep pue 3ufuueqtd eangnz 105
Bqjep eseq doTeaep pue ‘asuodsez waqsks Kkeq-jeTut BufjoTpead 1aoy senbzu
-Y997 YSTTGeISe ‘set ZPOOTAA JeTUT JWUeOTFJT~UsTSs AzoF wusfueyoou Sufzeirsues
ay. pue uoTIeIToOKe |sAeA ZUOT FO J9ANjeuU ay} azeBTASeauT 03 sayeT 7e2eI5
ya uo sLoqzey sUTU 4e G/=-4/6| UF pezZONpuod a1eM sqUeWeINseeU pTETy
*g9 ‘d :AhydesZo0qzT qT
(8-£L “ou £ SzaqueD
yoieesey Buyreeutsuy Teyseog *s*g — reded [Twpuyder) “TTF : *d 69
“216, “293uU99
yoieasoy Zuyiosuzsug Tezseoy “S"n : ‘eA SAFOATEG 310g — *uasuazOS “pW
qieqoy pue 3TTeeS *N WETTTEM Aq / sqeTUT sexe] qeez9 jo sopp[neaphy
“N WETTTEM “3FTees

——— SS

L279 8-LL ‘ou daigcn° €0ZOL
*@-// °‘ou rzeded Teozuyoey
*1eqUeD Yyieesey BuyiseuT3uq TeyseoD *S*n :settes *[I ‘roy Ane
quyof **W Jteqoy ‘uesuetos “TI ‘eTIFL “I ‘“SeyeT 3eeIN “szeTUT *|
*Teuueyo But
-1STX® UB JO UOTIBOTFFpOW 9y} OJ pue TeuUeYyD JeTUT Meu e Jo usTsap ayy
03 pettdde oie Apnjs 943 uf pedotTeasp sanbzuyoeq pue s3zdadu0D ay jo
asn 93e1}suoUep 03 satTduexq ‘*satpnjs us’tsep pue Zuyuuetd oanqnz 105
e3ep aseq dojTeaep pue ‘asuodsei weqsks Aeq-jaTuF B3uT_OTpead 10x sanbtu
-yo22 YSTTqeqse ‘sat POOTeA JeTUT JuedTzTUsTs toy wsTueydou Bup3ere8ues
ey pue uote RTOXe sAeAM BUOT JO ainjeu ay 22e8TISeAUT 02 SeyeT 4eaI9
24} UO Sioqiey suTU 3e C/-4/6| UF pejONpuod 9ieM sqUsMeinseoU prety
*g9 ‘d :SydeasozTqtTg
(@-/L *ou £ Szaquag
yoieesey Suyiseutsuq Teyseop *g°n — 1eded [Teopuydey) “TTT : °d 69
“LL6L *2193U99
yoivesey SutiseuTz3uqg TeIseoD “Stn : “eA SAFOATeg JA10q — ‘uasuar0g “Wj
qieqoy pue 3TTeeS *N WeTTTTM Aq / sje TUT saye] yeeIy Jo sop TneipAy
“N WETTTIM “3TTeeS

£29 G-LL *ou d3aesn° €0Z9L
°Q=£L °ou zeded TeopTuysey
*TeqUueD YyOieesey duptrseuTsug [ejseog *s*n :safzesg “IT ‘soy ne
quyof “*yW yteqoy Suesuetos “II “eTITL “I ‘*seyey qeez9 “sqzeTUT *{
*Teuueyo gut
-3S}xX9 Ue FO UOTIEOTJTpow sy} OF pue Toeuueyo JoTUF Mou e jo usTsep ay
03 pettdde eze Apnjs 9yq uz pedoTeaep senbtzuyse3 pue sqdaou02 ay jo
asn 9je1}suollep 07 seTdwexy ‘setpnjs ustsap pue Zupuuetd eanqny 10;
ejep eseq doTesep pue ‘asuodsez waqsks Aeq-jeTuy 3uzzo~pead aoy senbyu
-Yy297 YSFTGeIse ‘sat RFOOTSA JeTUT JueoTJTUZTS aos usptueyoou Sup ze1r98ues
ay} pue uoze I FOXe aAeM ZUOT FO 9ANjeu ay aIeTTAsSeauT 07 sayeT e215
24} UO Ssioqiey sUTU 3e C/-4/6| UF pezONpuod ei1aM squeMeaNseeM pTeTy
*g9 ‘d :hydezs0zT QTE
(@-LL *ou § Sraque9
yoieesey SsuyiseuT3uy Teqseog *s*g — azeded TeruyIeL) “TIF : “d 69
“LL6, ‘teqUeD
yoreesey dupyiseuzsugq Teqseop *g*n : “eA SAPFOATEg 310g — *ussuetos “W
Jieqoy pue 3TTeeS “N WeTTTTM 4q / sqeTUT seyeT 4eeayN Jo sof [neaphy
“N WETTTEM *3TTeeS

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Hydraulics of Great Lakes inlets"

See other formats