UNDERSTANDING TIDES

UNDERSTANDING TIDES 

by

Steacy Dopp Hicks

Physical Oceanographer

Center for Operational Oceanographic Products and Services

Michael Szabados, Director

December 2006

U.S. DEPARTMENT OF COMMERCE

 National Oceanic and Atmospheric Administration

National Ocean Service
ii

The author, Steacy D. Hicks viewing the William Ferrel tide-predicting

machine, the first tide predicting machine used in the United States

(1885-1914). Photograph taken in early 1960s.
iii

ABOUT THE AUTHOR

Steacy Dopp Hicks

Mr. Hicks retired in December 1990 after 30 years of distinguished

Federal Service, 28 years of which was providing the Nation with

expertise in tides and currents. A graduate of UCLA in 1950 with a major

in meteorology, Mr. Hicks received his Masters Degree in physical

oceanography from the Scripps Institution of Oceanography of the

University of California in 1952. From 1952 to 1962, Mr. Hicks was an

Instructor and Assistant Professor of Physical Oceanography at the

University of Rhode Island. Since entering the Federal Government (old

Coast and Geodetic Survey) in 1962, he has served in research positions;

as Chief, Physical Oceanography Division; and Chief, Physical

Oceanographic Research Group; until assuming duties as Senior Physical

Oceanographer, all in the National Oceanic and Atmospheric

Administration (NOAA) and its predecessor organizations. He also has

been an Associate Professorial Lecturer in Physical Oceanography at The

George Washington University. Mr. Hicks was selected a Fellow in the

first class (1964-65) of the Department of Commerce Science and

Technology Fellowship Program. In the Program, he studied at the

Geophysical Fluid Dynamics Laboratory of NOAA. Mr. Hicks is a 1970

graduate of the senior resident course (College of Naval Warfare) of the

U.S. Naval War College in Newport, RI. He received the Silver Medal of

the U.S. Department of Commerce for his research on tides, long-period

sea-level variations, and tidal datums; and has authored numerous papers

in refereed scientific journals on these subjects. His nine military awards

from WWII, while serving in the Destroyer USS Conway (DD507),

include the Combat-Action Ribbon and the Asiatic-Pacific Campaign

Medal with six major engagement stars.

Michael Szabados

Director, 

Center for Operational Oceanographic Products and Services
iv

Acknowledgments

I am very grateful to Michael Szabados, Director, Center for Operational

Oceanographic Products and Services, for inviting me to have my

manuscript published by his organization. Brenda Via was in charge of

the entire publishing process. Leonard Hickman, my point of contact at

the beginning, was particularly helpful in coordinating the various

elements involved. My sincere thanks to both of them for their

thoroughness on my behalf. Stephen Gill revised and expanded several

important sections. His expertise in the harmonic analysis and prediction

of tides was invaluable. Critical reviews by Chris Zervas on long period

sea level changes and Bruce Parker on tidal theory were very much

appreciated. 
v

TABLE OF CONTENTS

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Introduction ........................................... 1

2. Gravitational Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. The Sun-Earth and Moon-Earth Systems .................... 7

The Sun-Earth System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Primary development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Alternative development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

The Moon-Earth System .............................. 9

Primary development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Alternative development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4. Spring and Neap, Distance and Declination . . . . . . . . . . . . . . . . . . 11

Spring and Neap Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Declination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5. Tide-Generating Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6. Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7. Tidal Waves in Gulfs and Estuaries . . . . . . . . . . . . . . . . . . . . . . . . . 29

Tides in Estuaries and Bays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Tidal Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8. Tidal Harmonic Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9. Harmonic Analysis and the Prediction of Tides . . . . . . . . . . . . . . . 45

Historical Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Prediction of Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Types of Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10. Tidal Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Tidal Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Hydrographic Surveying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Coastal and Marine Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 54

Early Development of Tidal Datums and Marine Boundaries 

in the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vi

Sea Level Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chronology of Significant Tidal Events in the United States . . . . . . . 61

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
vii

LIST OF FIGURES

1. Envelope of tide generating forces . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Spring tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3. Neap tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. Priming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5. Lagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6. The sun-earth system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7. The moon-earth system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8. Diurnal inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9. Lunar node positions for diurnal inequality . . . . . . . . . . . . . . . . . . 18

10. Particle trajectories of progressive wave in deep water . . . . . . . . 26

11. Progressive wave transformation into shallow water . . . . . . . . . . 26

12. Characteristics of a tidal wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

13. Tidal height and current relationships in a progressive tidal wave 30

14. Formation of standing tidal wave from reflection . . . . . . . . . . . . 31

15. Tidal height and current relationships in a standing tidal wave . . 32

16. Damping of a tidal wave in an estuary . . . . . . . . . . . . . . . . . . . . . 34

17. Relationship of tidal waves to size of estuaries . . . . . . . . . . . . . . 35

18. Propagation of a tidal bore in a river estuary . . . . . . . . . . . . . . . . 36

19. Constituent tide curve example . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

20. Phase relationships for spring and neap tides . . . . . . . . . . . . . . . . 41

21. Phase lag and amplitude for a harmonic constituent . . . . . . . . . . 47

22. Coastal and marine boundaries from tidal datums . . . . . . . . . . . . 56

1

CHAPTER 1

INTRODUCTION

This book presents an elementary explanation of tides and tidal

datums. It is written to explain the natural phenomenon of tides in terms

and concepts readily understandable by students as well as those in all

walks of life merely wishing to be enriched by additional knowledge of

their environment. Also, scientists in fields other than physical

oceanography and astronomy may wish to use the book as a primer to the

fundamentals of tidal theory. Although not at all essential to a complete

understanding of tides, a mathematical development of the tide-generating

forces is provided.

Chapter 10 Tidal Datums should be of particular interest to coastal

zone managers, coastal engineers, geologists specializing in beach

processes, attorneys concerned with boundary litigation, and legislators (at

all government levels) representing marine activities and jurisdictions

adjacent to the sea. Finally, it is hoped that new employees in the tides and

tidal datum areas of the National Oceanic and Atmospheric

Administration will find this book a very helpful introduction to their

careers.

The term tide is often restricted to the vertical rise and fall of the water

usually occurring twice in a little more than a day. This rise and fall are

best observed on a breakwater or on the piles of a pier. When the water is

viewed near the time of its lowest point, called low tide or low water, the

extent of marine growth and discoloration indicates the general range of

the vertical excursion. The tidal range varies, from place to place over

time, from almost nothing to many feet.

With tides on a sloping beach, the water moves up the beach landward

and down the beach seaward. Similarly, but on a grander scale, it moves

inland and then seaward across the sloping mud flats and marshes of

estuaries as the tide alternately rises and falls.

The horizontal component of this phenomenon is called the tidal

current. It is best seen at an inlet connecting the ocean with a barrier

sound. The tidal current floods and then ebbs with slack waters in

between. At any particular location, the high and low tides; together with

the floods, ebbs, and slacks of the tidal current; have a particular sequence

of occurrence with near constant time intervals between each other.

The tide (in its restricted sense) and tidal current are both integral parts

of one major phenomenon that will be called hereafter, the tide. Tides
2

should be thought of as being in the form of waves. These waves are

thousands of miles in length. Their crests are the high tides, their troughs,

the low tides, and the horizontal component of the water particles that

make up the wave, the tidal currents. To complicate the matter, these

waves combine to reinforce or interfere with each other in varying

amounts, partially contributing to the wide differences in tidal

characteristics as actually observed.

The tide is fundamentally caused by gravitational interactions between

the sun, moon, and earth. These interactions of the gravitational forces are

the same as those causing the moon and earth to remain in their respective

orbits.

It is often said of science that the ability to predict a natural event is

indicative of understanding. Since tides are one of the most accurately

predictable natural phenomena, it follows from the axiom that the tide is

truly understood. Nothing could be further from the truth. The sciences of

astronomy and geophysics provide very accurate quantitative

determinations of the tide-generating forces on the earth. The science of

physical oceanography provides a detailed understanding of wave

dynamics and the response of the ocean to the tide-generating forces.

The Center for Operational Oceanographic Products and Services

(CO-OPS) of the National Ocean Service (NOS), National Oceanic and

Atmospheric Administration (NOAA) provides very accurate

measurements of the tide over time at numerous locations throughout the

United States and its territories. But, between the fundamental tide

producing forces and the observed tide at a particular place and time, there

is a vast area of numerous unknowns that will be referred to as terrestrial

factors. Some of the factors that the unknowns are associated with are:

1. the restrictive depths of the oceans not allowing the generated 

tidal wave to be in equilibrium with the rotation of the earth,

2. irregular ocean depths over which the waves must travel,

3. reflections and interactions of the waves from irregularly shaped

continents,

4. bottom friction,

5. turbulence, and

6. viscosity of the water.

In addition to the terrestrial factors mentioned above there are also

meteorological factors which, at some locations, cause large differences

between the observed and predicted tide. Weather events such as
3

hurricanes and tropical storms can produce significant differences between

predicted and observed water levels. Wind, rain (associated with river

runoff), and barometric pressure, even when not associated with a storm

event, can also cause major differences between observed and predicted

water levels.

How is it, then, that the physical oceanographers at NOAA are able to

predict the tide with great accuracy? By employing an empirical method

the tide-generating forces can be related to the observed tide with sets of

numerical constants unique to a particular location. Essentially, these sets

of constants embody the terrestrial factors partially listed above. With

astronomical information readily available, extrapolations can be made for

predictions years in advance using the same sets of constants. Why does it

work? First, because it is correct that the tide is fundamentally caused by

the gravitational interactions between the sun, moon, and earth. Second,

because the tide can be observed with great accuracy at a particular

location over time. And third, because the earth happens to be a very

consistent responder to these tide-generating forces, even though all the

answers associated with the processes are not known. However,

incorporating meteorological factors is not feasible for predictions at the

desired degree of accuracy. Only the astronomic tides are considered in

this book. 

The analysis and prediction of tides are covered in the succeeding

chapters. This is not for the purpose of teaching one how to predict the

tides, but rather for the understanding that will be achieved in breaking

down the total system into its simple components and then recombining

these components into the realistic ocean tide.

5

CHAPTER 2

GRAVITATIONAL ATTRACTION

In the ordinary course of human experience, gravitational attraction

seems to be a fundamental and universal force. It maintains the planets

and their moons in orbit and gives rise to the gravity we feel on earth.

Gravitational attraction is a basic force in everyday Newtonian physics.

Newtonian physics describes motion extremely well, providing the motion

does not approach the speed of light and is not in close proximity to a very

large body, like the sun. For example, the motion of the planet Mercury,

closest to the sun, is not described perfectly by this physics. However, as

with the vast amount of observed and experienced motion, tides are best

described and understood through Newtonian physics.

Gravitational attraction possesses several characteristics of

considerable interest. It is well known that objects of differing weights fall

at the same speed in a vacuum. Also, that gravitation is bidirectional. That

is, the moon attracts the earth as the earth attracts the moon.

More obscure characteristics, described thoroughly by Heyl (1954),

are that observations of comets show that gravitation is not affected by

changes in temperature. Pendulum experiments with different bobs show

no differences, whether made of terrestrial or meteoritic material, solid or

liquid substances, radioactive or natural items, or the same substance

before or after a chemical reaction. Similarly, crystalline items of the same

weight in any orientation and magnetized and non-magnetized substances

show no differences.

Gravitation cannot be screened either. Astronomers have looked for

the cumulative effects of minute perturbations in the orbit of the moon

when it is directly behind the earth in a total lunar eclipse, but to no avail.

Newton had no suggestion as to the fundamental cause of gravitation.

Einstein related it to the warping of the space coordinate system in the

presence of large masses. The bending of starlight as it passes close to the

sun is one of the verifications. An analogy, although not entirely

satisfactory, is a large weight placed in the center of a pond with a frozen

surface. The weight will depress the ice such that a bowling ball, thrown

at just the right speed and to just miss the weight, will revolve around it in

an ellipse simulating the effect of gravitational attraction. The idea here is

that the geometric distortion of the ice is analogous to the warping of the

space coordinate system.
6

Gravitational attraction is directly proportional to the product of the

masses involved and inversely proportional to the square of the distance

between the centers of mass. The square in the denominator means that

distance becomes quite important. As will be seen in subsequent chapters,

the development of the tide-generating forces involves a cube in the

denominator, making distance very important in tides.

The words “tide,” as in “high tide,” and “water,” as in “high water,”

have been used interchangeably in the historical development of tidal

activity here in the United States. In the last few decades, “water” has

been favored. Whenever ocean elevation measurements are made, the

highs and lows include the astronomic tides and all other effects such as

those caused by the winds and ocean currents. Therefore, “water” is the

most appropriate term. In analyzing, studying, and predicting, however,

“tide” is the most appropriate since the astronomic tide is the element that

we are able to predict with the greatest accuracy and reliability. Recently,

the term “water level tidal prediction” has been introduced by CO-OPS in

an attempt to solve this semantic problem.

Here, “tide” will be used throughout since the subject is astronomic

tides. However, “water” will be used in such terms as “mean lower low

water” in Chapter 10 Tidal Datums. The latter use is because tidal datums

are precisely defined terms that have been recognized in law as they relate

to coastal and marine boundaries (ownership, jurisdiction, and legal

responsibility).
7

CHAPTER 3

THE SUN-EARTH AND MOON-EARTH

SYSTEMS

To best understand the tide, it is necessary to study each astronomical

motion, together with its associated tide producing forces, separately.

Furthermore, it is necessary to study each as a simple, idealized motion.

Later, these separate, simple, idealized motions will be combined to form

a realistic description of the astronomic phenomena and the tides for

which they are responsible.

The Sun-Earth System

The sun and the earth revolve around their common center of mass in

one year. Although the orbits are elliptical and the earth is inclined to the

ecliptic (its plane of revolution around the sun), these facts are not

considered at this time. The mean distance between the sun and the earth

is about 92,897,400 miles. The center of mass of the sun-earth system is

located about 280 miles (mean) from the center of the sun. Therefore, it

might be easier to visualize the sun as wobbling around this point.

Primary development

The tide-generating forces may be developed by considering

differences in the gravitational attraction of the sun at various locations on

the earth. At the center of mass of the earth the gravitational attraction

must be a value just necessary to keep the earth in its orbit around the sun.

This is F2 in Figure 1. At a point on the side of the earth facing the sun, the

gravitational force, F1, is greater than at the center because it is closer to

the sun. Since water is more fluid than the earth, the water distends toward

the sun. At a point on the side of the earth away from the sun, the

gravitational force, F3, is less than at the center because it is further away

from the sun. Thus, the water distends away from the sun. Quantitatively,

the tide-generating forces are the differences between the gravitational

attraction at the center of the earth and the gravitational attractions at the

points of interest, as altered by distances from the sun.

D1 = F1 – F2 directed toward the sun

D3 = F2 – F3 directed away from the sun

The same development can be used for all points on the earth. The

ends of the difference vectors, D, will form an envelope of the tidegenerating forces. It is a symmetrical egg-shaped distention that many
8

refer to as the tidal bulge of the ocean waters. It is not. Again, it is the

envelope of the tide-generating forces - the tide potential. It is the

distention that the tide-generating forces are trying to make in the ocean

waters but never realize because of the factors mentioned in Chapter 1,

i.e., restriction of wave speed, irregular depths, presence of continents,

bottom friction, turbulence, and viscosity. It is frequently referred to as the

equilibrium tide.

Looking down on the North Pole in Figure 1, the earth spins counter

clockwise in one solar day. The mean solar day is the period of rotation of

the earth on its own axis (spin) referenced to the mean sun and is divided

into 24 solar hours. Starting with the point on the equator toward the sun,

the point experiences a maximum in the solar tide-generating force; six

hours later, a minimum; then a maximum at twelve hours; a minimum

again at eighteen hours; and finally back to the maximum at the

conclusion of the solar day. Note that in one solar day there are two

complete tide-generating force cycles. The distended symmetrical eggshaped envelope of the tide-generating forces stays lined up with the sun

as the earth and sun go around their common center of mass each year.

The earth spins (slips) once within the envelope each solar day.

Figure 1. Envelope of tide generating forces
9

Alternative development

The earth is maintained in its orbit around the center of mass of the

sun-earth system by centripetal acceleration. Centripetal acceleration is

provided, in this case, by the gravitational attraction between the sun and

earth. Centrifugal force, a fictitious or apparent force, may be introduced

for convenience. The concept of centrifugal force is derived from the

inertial tendency of the earth to go flying off into space tangentially to its

orbit. When considered over 360°, it may be represented by a vector

directed away from the center of mass of the sun-earth system.

Centrifugal forces are the same throughout the earth. This is realized

by remembering to take only one motion at a time and to consider

irrotational motion. In irrotational motion, the earth does not spin on its

axis even once during the year. Thus, only the revolution of the earth

around its common center of mass with the sun takes place. Therefore, any

location on the earth has the same radius as any other and the centrifugal

forces at all points on the earth are equal.

Gravitational forces differ throughout the earth. The closer a point on

the earth is to the sun, the greater the force. This is because gravitational

force varies inversely as the square of the distance between the sun and

earth. Since the earth does not fall into the sun or fly off into space

(significantly during historic times), the gravitational force must be

balanced by the centrifugal force (equal but oppositely directed vectors) at

the center of mass of the earth. On the side of the earth facing the sun, the

gravitational force is greater than the centrifugal causing a tide producing

force directed toward the sun. On the side of the earth away from the sun,

the gravitational force is less than the centrifugal causing a tide producing

force directed away from the sun.

It is unimportant which of the above developments is used for

understanding. The two ways of thinking about it are the same and yield

the same results. However, the primary development will continue to be

used for continuity.

The Moon-Earth System

Primary development

The moon and the earth revolve around their common center of mass

in one month. Although their orbits are elliptical and the earth’s equator

and moon’s orbit are inclined to the ecliptic, these facts are not considered

at this time. The mean distance between the moon and the earth is about

238,860 miles. The center of mass of the moon-earth system is located

within the earth about 2895 miles (mean) from the center of mass of the

earth, always toward the moon.
10

Figure 1 shows the gravitational force vectors, F, and the difference

vectors, D. They are essentially the same as for the sun-earth system

except for magnitude. The same development can be used for all points on

the earth as previously explained for the sun-earth system.

Looking down on the North Pole in Figure 1, the earth spins counter

clockwise in one lunar day. The mean lunar day is the period of rotation of

the earth on its own axis referenced to the moon and is 24.84 solar hours

in length. Starting with the point on the equator toward the moon, the

point experiences a maximum in the lunar tide-generating force; 6.21 solar

hours later, a minimum; then a maximum at 12.42 solar hours; a minimum

again at 18.63 hours; and finally back to the maximum at the conclusion

of the lunar day. Note that in one lunar day there are two complete tidegenerating force cycles, just as there were for the sun. However, the solar

cycle took 24 hours and the lunar, 24.84. The distended symmetrical eggshaped envelope of the lunar tide-generating forces stays lined up with the

moon as the earth and moon go around their common center of mass each

month. The earth spins (slips) once within the envelope each lunar day.

Alternative development

The alternative development of the tide-generating forces for the

moon-earth system is the same as for the sun-earth system. However, the

center of mass of the moon-earth system is deep within the earth. By

taking one motion at a time and considering irrotational motion, the earth

wobbles around this point. Since the earth does not rotate (spin) during the

month with irrotational motion, every point on the earth describes a circle

of the same radius. Visualization of the centrifugal force on the side of the

earth facing the moon is difficult, at first. Irrotational motion and

consideration of only the monthly revolution are the keys. The centrifugal

force vector on the side of the earth facing the moon is directed away from

the moon like all the other points on the earth. 
11

CHAPTER 4

SPRING AND NEAP, DISTANCE AND

DECLINATION

Spring and Neap Tides

The range of tide, defined as the vertical difference in height between

consecutive high and low tides, varies from place to place and also varies

over time. The combination of the solar and lunar envelopes during the

synodic month causes spring tides and neap tides. The synodic month is

referenced to the sun (or phase of moon) and is 29.530588 days in length.

By combining the sun-earth system and the moon-earth system as

outlined in Chapter 3, spring and neap tides are developed. Spring tides

have nothing to do with springtime. They are tides with ranges greater

than the average monthly range. Spring tides occur twice each synodic

month due to the approximate alignment of the sun, moon, and earth.

Neap tides are tides with ranges less than the average monthly range. They

occur twice each synodic month due to the sun, earth, and moon forming

right angles (approximately) to each other.

Colloquially, the tide is said to "follow the moon" since it is 2.16 times

more influential than the sun in causing the tide. It is probably more

descriptive to say that the tide, in most locations, is controlled by the lunar

tide-generating forces as modified by the solar.

When the moon is new (on the side of the earth toward the sun) or full

(on the side of the earth away from the sun), the tide-generating forces of

the sun and moon are aligned. In Figure 2, the envelope of the tidegenerating forces of the sun augments the envelope of the moon. The high

tides of the solar envelope occur at the same time as the high tides of the

lunar. This increases the height of the composite high tides. Likewise, the

low tides of the solar envelope occur at the low tides of the lunar. This

decreases the height of the composite low tides. Therefore, larger than

average tidal ranges occur, called spring tides. 

When the moon is in its first or third quarter, however, the tidegenerating forces of the sun are at right angles to those of the moon. In

Figure 3, the envelope of the tide-generating forces of the sun is shown to

conflict with the force envelope of the moon. The low tides of the solar

envelope occur at the times of the high tides of the lunar. This reduces the

height of the composite high tides. Likewise, the high tides of the solar
12

Figure 2. Spring tides

Figure 3. Neap tides
13

envelope occur at the low tides of the lunar. This increases the height of

the composite low tides. Therefore, smaller than average tidal ranges

occur, called neap tides. The change in range between spring and neap

tides amounts to about 20%.

In addition to changes in tidal range, the spring-neap tide phenomenon

also causes changes in the time of occurrence of the high and low tide

phases of the lunar tide. As the moon goes from new to first quarter or full

to third quarter, the solar envelope is behind the lunar such that the

composite is retarded. Thus, as the earth spins, the high and low tide

phases of the composite occur earlier than with just the lunar envelope.

This is called priming. Priming is shown in Figure 4.

Figure 4. Priming
14

Figure 5. Lagging

Similarly, as the moon goes from first quarter to full or third quarter to

new, the solar envelope is ahead of the lunar such that the composite is

accelerated. Thus, as the earth spins, the high and low tide phases of the

composite envelope occur later than with just the lunar envelope. This is

called lagging and is illustrated in Figure 5.

Distance

The sun and earth revolve around their common center of mass in

elliptical, rather than circular, orbits. When the earth is closest to the sun,

the earth is said to be in perihelion. When the earth is farthest from the

sun, it is said to be in aphelion. Perihelion to perihelion occurs in one

anomalistic (a-nom-a-lis-tic) year of 365.2596 days. Perihelion occurs on

January 3rd [dates are for the year 2007, Coordinated Universal Time
15

(UTC)], with the distance being about 91,400,005 miles. Aphelion occurs

on July 7th, the distance being approximately 94,512,258 miles (Figure 6).

It will be shown that the tide-generating forces are inversely proportional

to the cube of the distances between the two bodies. Therefore, when the

earth is in perihelion, the tide-generating forces are larger than average

and greater than average ranges in the tide are experienced. In aphelion,

smaller than average ranges occur.

The moon and earth also revolve around their common center of

mass in elliptical orbits. When the moon is closest to the earth, the moon

is said to be in perigee. When the moon is farthest away, it is in apogee.

Perigee to perigee occurs in one anomalistic month of 27.554,550 days. At

perigee the two bodies are about 221,000 miles apart while at apogee they

are about 253,000 miles away from each other (Figure 7). When the moon

is in perigee, the tide-generating forces are larger than average and greater

than average ranges in the tide occur. The opposite is true when the moon

is in apogee.

Declination

The ecliptic is the plane generated by the earth as it revolves around

the center of mass of the sun-earth system. The axis of rotation (spin) of

the earth is inclined 23.452° to the ecliptic. This gives rise to the seasons

experienced on earth. The cycle of the seasons is the tropical year of

Figure 6. the sun-earth system
16

365.2422 days. It is referenced to the vernal equinox (direction when the

apparent sun passes from south to north over the earth's equator).

On June 21st (2007, UTC) the northern hemisphere is tipped directly

toward the sun (Figure 6). The sun is said to be at its maximum northern

declination (angular height above the equator of 23.452°). This is the

summer solstice and the beginning of the season of summer in the

northern hemisphere.

Three months later, on September 23rd, the earth (still tipped in the

same direction relative to a star) has moved one-quarter of the way around

the ecliptic such that the sun is now over the equator (zero declination).

This is the autumnal equinox signaling the beginning of autumn for the

northern hemisphere.

Winter for the northern hemisphere begins on December 22nd, the

winter solstice, when the sun is at its maximum declination south. The

vernal equinox, ushering in spring for the northern hemisphere, occurs on

March 21st when the sun passes over the equator on its apparent trip north.

The seasons are reversed, of course, in the southern hemisphere.

The distended symmetrical egg-shaped envelope of the solar tidegenerating forces remains lined up with the sun as described in Chapter 3.

However, with the earth tilted, the forces are asymmetrical with respect to

the equator as the earth progresses through the seasons. The differences

Figure 7. the moon-earth system
17

Figure 8. Diurnal inequality

between the two daily maxima of the tide-generating forces for various

locations at the summer solstice are shown in Figure 8. At A the high tide

is higher than at B, 12 hours later. This is called diurnal inequality in the

semidiurnal tide. At the latitude of CD, there is only one high tide per day.

Tides from this situation are known as diurnal tides. At the equator, EF,

the high tides are equal, giving a typical semidiurnal tide. Although not

shown in this figure, the forces are symmetrical with respect to the equator

at the equinoxes and, of course, no solar diurnal inequality occurs from

this effect at these times.

The plane of the orbit of the moon is inclined 5.145° to the ecliptic –

not our equator (Figure 7). The two places where the moon crosses the

ecliptic are called the moon's nodes. The place where it crosses the ecliptic

from south to north is known as the ascending node. Where it crosses from

north to south - the descending node. The lunar declinational effect also

causes diurnal inequality in the tides. Because it is due to the moon, the

effect is greater than that of the sun and its cycle is the nodical month

(referenced to the moon's ascending node) of 27.212220 days.

Looking down on the northern hemisphere, the earth revolves around

the sun counterclockwise, the moon revolves around the earth

counterclockwise, and the earth rotates counterclockwise. The moon's

nodes, however, move clockwise (or westward) on the ecliptic (Figure 7).
18

Figure 9. Lunar node positions for diurnal inequality
19

This is called the regression of the moon's nodes and it takes 18.61 years

for them to go completely around and back to their original position. It is

best understood by visualizing the plane of the orbit of the moon pivoting

on the plane of the orbit of the earth (the ecliptic). The phenomenon

causes the moon to vary in its maximum monthly declination (distance

north and south of the earth's equator) during the 18.61-year nodal cycle.

The maximum monthly declination of the moon will range between:

23.452° + 5.145° = 28.597°, and

23.452° – 5.145° = 18.307°.

During the 18.61-year nodal cycle, times will occur when a line

connecting the moon's nodes (node line, Figure 7) will be at about right

angles to a line connecting the earth and sun. When this happens near the

winter solstice with the moon below the ecliptic on the sun side,

maximum monthly declinations will build up to 28.597° (A in Figure 9)

and large diurnal inequalities will occur. This is also true near the summer

solstice with the moon above the ecliptic on the sun side, as seen in D.

The opposite occurs with the moon below the ecliptic on the sun side near

the summer solstice, B, and above the ecliptic on the sun side near the

winter solstice, C. In the latter two cases the maximum lunar declination

will amount to only 18.307° and relatively small diurnal inequalities will

occur.

This 18.61-year nodal cycle results in slowly varying ranges and

inequalities of observed tides at tide stations. These changes are

significant enough to be measurable and this nodal cycle time-period is

used in the computation of accepted tidal datums (see Chapter 10).

21

CHAPTER 5

TIDE-GENERATING FORCES

The solid earth is not rigid. It undergoes plastic deformation due to the

same gravitational interactions as the oceans. Earth tides, as they are

called, are relatively small but readily measurable. Although an important

and very interesting geophysical phenomenon, they are outside the scope

of this book.

Gravitational force, F, is equal to the gravitational constant, G, times

the product of the masses of the two bodies divided by the square of the

distance between their centers of mass. For the sun and moon,

respectively, the gravitational force can be defined as; 

 Fs = G(SE) / r2

 and Fm = G(ME) / r2

where:

S is the mass of the sun,

M is the mass of the moon,

E is the mass of the earth, and

r is the distance between the centers of mass of the earth, and the sun or

moon.

Fs1 = GSE / (r – ρ)2 is the gravitational force at the surface of the

earth on the side facing the sun directed toward the sun, where ρ is the

equatorial radius of the earth.

Fs2 = GSE / r2

 is the gravitational force at the center of mass of the

earth directed toward the sun.

Fs3 = GSE / (r + ρ)2 is the gravitational force at the surface of the

earth on the side away from the sun directed toward the sun.

The differential or tide-generating force is equal to the differences

between the gravitational force at the center of mass of the earth and the

gravitational force at other points on the earth.

The difference at the surface of the earth facing the sun is;

Fs1 – Fs2 = GSE / (r – ρ)

2

 – GSE / r2

 = GSE[1 / (r – ρ)

2

 – 1 / r2]

 = GSE{[r2

 – (r – ρ)

2

] / [r2(r – ρ)

2

]}

 = GSE{[r2

 – r2 + 2ρr – ρ2] / [r2(r – ρr/r)2]}
22

 = GSE{[2ρr – ρ2

] / [r2[r(1 – ρ/r)]2]}

 = GSE{[2ρr – ρ2

] / [r2r

2

(1 – ρ/r)2]}

 = GSE{[2ρr – ρ2] / [r4(1 – 0)2]} since ρ/r is small

 = GSE{[2ρr(1 – ρ2/2ρr)] / r4}

 = GSE{[2ρr(1 – ρ/2r)] / r4}

 = GSE{[2ρr(1 – 0)] / r4} since ρ/2r is very small

 = GSE(2ρ / r3

)

 = G(2SEρ / r3

)

The difference at the surface of the earth away from the sun is;

Fs2 – Fs3 = G(– 2SEρ / r3)

or;

Ds = ± G(2SEρ / r3)

Similarly the differential or tide-generating force at the surface of the

earth facing the moon is;

Fm1 – Fm2 = G(2MEρ / r

3

) and, at the surface away from the moon the

difference is;

Fm2 – Fm3 = G(– 2MEρ / r

3

)

or;

Dm = ± G(2MEρ / r3

)

where:

Ds is the differential on the earth due to the sun, and 

Dm is the differential on the earth due to the moon at these two locations. 

These differentials, one due to the sun, the other due to the moon, are

the source of the fundamental tide-generating forces.

With the alternative development of using centrifugal force, the

mathematics is the same, since centrifugal force is everywhere equal (but

of opposite sign) to the gravitational force at the center of mass of the

earth.

It is to be especially noted that although the gravitational force varies

inversely as the square of the distance between the two bodies, the tidegenerating force varies inversely as the cube of the distance. This is very

important since it accounts for the moon being 2.16 times more influential

in causing the tide than the sun. This is shown by;
23

Dm / Ds = ± (2MEρ / r

3

m) x (r3s / 2SEρ)

where:

rm is the distance (in miles) between the center of mass of the earth and

center of mass of the moon, and 

rs is the distance (in miles) between the center of mass of the earth and the

center of mass of the sun, so;

Dm / Ds = (M / S) x (r3

s / r3m) 

If the mass of the moon is taken as one, then;

Dm / Ds = (1 / 27,069,106) x (92,897,4003

 / 238,8603)

≅ (1 / 27 x 106

) x [(93 x 106)

3

 / (24 x 104)

3

]

≅ (1 / 27 x 106

) x (804,357 x 1018 / 13,824 x 1012)

≅ 2.16

Thus, although the sun is 27 x 106 times greater in mass than the moon

and the moon is only 389 times nearer to the earth than the sun, the moon

is still 2.16 times more influential in causing the tide than the sun (or, the

sun’s influence is .46 that of the moon). Again, this is because the tidegenerating force varies inversely as the cube of the distance. In the tidegenerating force equation, the cube of 93 x 106

 is a very large number in

the denominator. It dominates the mass in the numerator to make a

relatively small value of the ratio.

For simplicity, the tide-generating forces at the points on the earth

directly beneath and away from the sun or moon have been developed.

Forces act in varying amounts throughout the surface of the earth forming

the envelopes previously described and may be similarly computed.

However, the angles formed by the force vectors directed toward the

centers of mass of the sun and moon must be considered.

While the solar and lunar envelopes are thought of as representing the

actual ocean waters, another very important factor must be recognized.

The components of the tide-generating forces acting tangentially along the

water surface turn out to be the most important. Just as it is easier to slide

a bucket of water across a floor rather than to lift it, the horizontal tractive

components move the waters toward the points directly beneath and away

from the sun or moon far more effectively than the vertical components

can lift them. These tractive forces are most responsible for trying to form

the ocean into the symmetrical egg-shaped distensions (the tide potential,

the equilibrium tide). They reach their maximums in rings 45° from the

points directly beneath and away from the sun or moon. 

These tide producing forces set the waters of the ocean basins into

periodic motion. The open ocean tides then progress onto the continental
24

shelves and up into the various estuaries and bays where they undergo

even more complex transformation. The actual response of the ocean

basins to these tide producing forces can be quite complex and is

discussed in the following chapters.
25

CHAPTER 6

WAVE MOTION

Since tides are in wave form, it is essential to examine the mechanism

of wave propagation in the ocean. This is particularly true for an

understanding of tidal currents and tidal bores.

The crest is the top or high point of a wave. The trough is the valley or

low point of a wave. Waves are described by their:

1. Height, H, the vertical distance between the level of a crest and the

level of a trough (amplitude, A, is one half the height),

2. Length, L, the horizontal distance from one crest to the next,

3. Period, T, the time interval between the occurrence of two 

successive crests at a fixed point, and 

4. Speed (celerity), C, the length divided by the period, C = L / T,

expressed in distance per unit time.

The internal workings of waves and their transformations in shallow

water are best understood by following swell from the open ocean to

shore. Swell are wind waves that have moved out of the wind generating

area or have continued after the wind has died down or changed direction

in the generating area. Swell are progressive waves. Although each wave

form moves in distance on the ocean, the water particles that make up the

wave do not. Under the crest the water particles making up the wave are

going in the same direction as the wave form. Under the trough the

particles are going in the opposite direction. Half way between the crest

and the preceding trough, the water is moving up to form the next crest

one-quarter of a period later. Half way between the crest and the following

trough, the water is going down to form the trough one-quarter of a period

later. Thus, over one wave period, the particles making up the wave

describe circular orbits. Swell in the open ocean are deep water waves.

That is, the bottom does not affect them. The orbits decrease

logarithmically in diameter with depth (Figure 10).

Actually, there is a slight overlap in these orbits that accounts for a net

mass transport of water in the direction of the waves. This, in turn, gives

rise to the longshore and rip current system that is necessary to remove the

excess water that accumulates in the surf zone. This net mass transport can
26

Figure 11. Progressive wave transformation into shallow water

 Figure 10. Particle trajectories of progressive wave in deep water
27

become very significant in high wave conditions accompanying hurricanes

and other severe storms.

When swell move into water of depths less than about half of their

wave lengths, the circular orbits are restricted by the presence of the

bottom. The orbits become elliptical. As the water gets progressively more

shoal, the orbits become more and more elliptical until the particulate

motion is just back and forth (Figure 11).

Waves in shallow water propagate at a speed equal to the square root

of the product of the acceleration of gravity and the depth, C = %gh. As a

wave moves toward shore, h gets smaller so C becomes less and less. T is

constant since the same number of waves that enter the surf zone must

equal the number that break on the shore (or combine with other waves).

Since C is decreasing, T is constant, and C = L / T, L is decreasing. Since

L is decreasing, H is increasing (the accordion effect, Figure 11). Thus,

waves get higher in the surf zone. A time finally arrives when C becomes

less than the speed of the particulate motion under the crest. The wave

breaks starting at the crest where the particulate speed is greatest. The

wave curls over and plunges onto the foreshore. This process is

particularly important in understanding tidal bores (Chapter 7).

Tidal waves are always shallow water waves. That is, the length of a

tidal wave is very much longer than the depth. Therefore, the particulate

motion that makes up the wave is just back and forth. This back and forth

particulate motion in a tidal wave is called the tidal current.

Tidal or tide waves are often confused with tsunamis. The word

tsunami (meaning great harbor wave in Japanese) is reserved for ocean

waves caused by earthquakes, undersea bottom slumping, large meteorite

impacts, or undersea volcanic eruptions. The term tidal wave is used for a

wave caused by the astronomic tide-generating forces.

29

CHAPTER 7

TIDAL WAVES IN GULFS AND ESTUARIES

Tides can be described in terms of waves. For a tidal wave, the crest

is a high tide and the trough is a low tide. The height of a tidal wave is the

range (or tidal range) and the mean of the high and the low tides is called

mean tide level. Figure 12 illustrates some of these fundamental

characteristics of tidal waves. For a semidiurnal tide (two highs and two

lows per tidal day), the time period (T) between successive high waters

(crests) is about 12.42 hours. For a diurnal tide (one high and one low tide

per tidal day), the time period between successive high waters is about

24.84 hours. In most gulfs and estuaries of the world, the tidal waves are

one of two types or a combination of the two. The first is called a

progressive wave, the second a standing wave.

In a progressive tidal wave in the open ocean, the horizontal

component of the particulate motion under the high tide area is going in

the same direction as the tidal wave form and is known as the flood tidal

current (or flood tide). Its maximum, directly under a high tide, is called

strength of flood. Similarly, in a progressive tidal wave, the horizontal

component of the particulate motion under the low tide area is going in the

opposite direction as the tidal wave form and is known as the ebb tidal

current (or ebb tide). Its maximum, directly under a low tide, is called

strength of ebb.

Half way (in time) between high tide and low tide (and low tide and

high tide) at the mean tide level, there is no horizontal flow. That is, no

tidal current. These points are known as slack, ebb begins (or slack before

ebb) and slack, flood begins (or slack before flood) respectively. Observed

from a fixed point, the time relationships of the tide and tidal current for a

progressive tidal wave are shown in Figure 13.
30

Figure 13. Tidal height and current relationships in a progressive tidal wave 

 Figure 12. Characteristics of a tidal wave
31

Tides in Estuaries and Bays

When a progressive tidal wave encounters a barrier such as a coast,

embayment, head of an estuary, etc., it reflects upon itself to form a

standing wave. Figure 14 illustrates the process. Column one shows a

progressive tidal wave entering an estuary from the open ocean. Column

two shows the reflection from the previous wave moving out of the

estuary. Column three shows the resulting combination of the two waves.

The presentation assumes complete reflection at the head of the estuary

and no attenuation due to friction.

In the standing tidal wave, the wave form does not progress. Half way

(in distance) between the crest and trough (and trough and crest) there is

no tidal range. These locations are known as nodes. The wave form

merely goes up and down in a standing tidal wave, except at the nodes.

There are no horizontal tidal currents under the crests or troughs, called

the antinodes. Under the nodes the tidal currents reach a maximum for the

entire wave. Choosing a fixed location somewhere between the crest and

node near the head, for example, the time relationships of the tide and

tidal current are shown in Figure 15.

 Figure 14. Formation of standing tidal wave from reflection
32

Bottom and internal friction complicate the situation. As a tidal wave

moves into an estuary, it is attenuated. Its reflection continues to dampen

as it moves seaward. The combination of an incoming damped wave and

the damped reflection from the previous wave produces a tidal wave

whose characteristics are somewhat complicated. It is not a perfect

standing wave and tends to have a progressive component. An example is

shown in Figure 16.

How much of a tidal wave an estuary can hold is dependent on the

estuary’s length and depth. From Chapter 6 we know that the formula for

the speed of a tidal wave, since it is a shallow water wave, is C = %gh.

Since C = L / T and T is a constant, L is determined and compared with

the length of the estuary. The driving force at the entrance to the estuary is

the ocean tide. If a node of the estuary wave happens to be near the

entrance, the range of the estuarine tide will be greatly amplified. If an

antinode is near, there will be no significant increase from this effect

(Figure 17). The tides in the Bay of Fundy are an example of this process.

The open Atlantic Ocean tides drive and amplify the tides in the Gulf of

Maine. These, in turn, drive and further amplify the tides in the Bay of

Fundy. This results in the highest tides of the world, amounting to a spring

tide range up to 50.5 feet in Minas Basin. 

The Coriolis force affects all motion relative to a rotating earth. In the

northern hemisphere, it is directed 90° to the right (looking in the direction

of motion). In the southern hemisphere, it is directed 90° to the left. It

Figure 15. Tidal height and current relationships in a standing tidal wave 
33

does not exist for motion along the equator. As a progressive tidal wave

enters an estuary, the flood tidal current under the crest is deflected to the

right (looking toward the head of the estuary) causing the water to slope

down to the left. The ebb tidal current under the trough is deflected to the

left (looking toward the head) causing the water to slope down to the right.

This accounts for the range of tide being greater on the right side of an

estuary than on the left in the northern hemisphere and is known as the

Kelvin wave. It also accounts for the tendency of the system to form an

amphidromic point – an across estuary nodal line reduced to a point. The

motion tends to cause a rocking oscillation of the total estuary and a notide point, the amphidromic point, near the center. Actually, in a real

embayment, it is probably a minimum tidal-range area due to all the

complicating factors such as an irregular embayment and bottom

topography, internal and bottom friction and river discharge.

Amphidromic lines, points, or areas also exist in the open ocean. The

interactions of various tidal waves in defined oceanic basins cause them to

occur. The area near Tahiti, for example, is a well-known one.

The damped incoming tidal wave and its combination with the damped

reflection of the previous wave, together with the Coriolis force and

amplification, usually result in a standing type tidal wave with a

progressive component and a slightly greater tidal range on the right. The

amount of wave contained in the estuary (location of node) and the

driving force from the open ocean tide determine, in general, the tidal

ranges, strengths of tidal currents, and time relationships throughout the

estuary.
34

Figure 16. Damping of a tidal wave in an estuary
35

Figure 17. Relationship of tidal waves to size of estuaries
36

Tidal Bores

The tidal bore or eager (mascaret, French; pororoca, Brazilian) is a

tidal wave that propagates up a relatively shallow and sloping estuary or

river in a solitary wave form. The leading edge presents an abrupt rise in

level, frequently with continuous breaking and often immediately

followed by several undulations. An uncommon phenomenon, the tidal

bore is usually associated with very large ranges in tide as well as wedgeshaped and rapidly shoaling entrances.

Bores occur in river estuaries with little or no reflections at the head.

One of the models for a tidal bore is analogous to the process involved in

an ordinary plunging breaker in the surf zone as described in Chapter 6.

When a progressive tidal wave slows down to the point where the

particulate motion (flood tidal current, u) in the crest (high tide) area

exceeds the speed of the wave form itself, the tidal wave breaks. This is at

a specific depth since C = %gh. There is a tremendous amount of water

making up the high tide area that keeps coming in on the flood tidal

current in contrast with the plunging breaker. As the tidal wave continues

to propagate, the tidal height increases and, with a sloping riverbed, the

point of breaking is just a little bit further up the estuary. This is a

continuous process of infinitely small increments and it may continue for

many miles up a river as a constantly breaking tidal bore (Figure 18).

Figure 18. Propagation of a tidal bore in a river estuary
37

A very large tidal range is usually required. Bores tend to occur in

river estuaries where funneling in the entrance topography helps to

increase the range and the river flow tends to retard, and therefore build

up, the wave. Also, tidal bores occur more often during spring tides and,

especially, spring tides when the moon is in perigee.

Tidal bores occur in several river estuaries throughout the world.

Before engineering modifications such as dredging, diversions,

entrainment, and the construction of barriers and dams, many more

existed. A few of the well known bores, together with the location in

which they occur are:

 Qiantang River, Haining Province, China

 Petitcodiac, New Brunswick, Canada

Salmon River, Nova Scotia, Canada

Shubenacadie River, Nova Scotia, Canada

Solway Firth, between England and Scotland

Severn Estuary, England

River Trent, England

Orne River, France

Gironde River, France

Hooghly River, West Bengal, India

Amazon River, Brazil

Turnagain Arm, Cook Inlet, Alaska

39

CHAPTER 8

TIDAL HARMONIC CONSTITUENTS

Each of the tide-generating motions described in the above chapters

can be represented by a simple cosine curve as illustrated in Figure 19.

The horizontal axis represents time and the vertical represents the

magnitude of the tide-generating force. The crests give the times of the

maximums in the tide-generating force and the troughs, the minimums.

For example, in the sun-earth system outlined in Chapter 3, noon, with the

sun overhead, is at the first crest. Six hours later a minimum occurs at the

trough. The second maximum is at midnight with the second crest.

Another trough is for dawn and then back to the original noon crest. The

vertical excursion of the curve is the range that the tide-generating forces

are trying to cause in the waters from this component of the total tide.

Figure 19. Constituent tide curve example
40

Each one of the tide-generating motions, represented by a simple

harmonic cosine curve, is known as a tidal component, tidal constituent, or

harmonic constituent. A letter or letters and usually a subscript is used to

designate each constituent. The tidal constituent described above, for

example, called the Principal Solar semidiurnal constituent, is designated

S2. The Principal Lunar semidiurnal constituent is designated M2. S is for

sun and M is for moon, of course, and the subscript 2 means that there are

two complete tidal cycles for each astronomic cycle. Thus, these are said

to be semidiurnal constituents. Constituents are described by their tidal

period (the time from maximum to maximum), T. The period for the S2 is

12.00 solar hours (hr.) and the period for the M2 is 12.42 solar hours.

In tidal work, each constituent (cosine curve) is more often described by

its speed (or frequency in degrees per hour). The cosine curve is divided

into 360° (from crest to crest). The speed, n, of the constituent is 360°/T.

Thus, for S2, n = 360°/12.00 = 30°/hr. For M2, n = 360°/12.4206 =

28.984°/hr.

We could go on and on to describe almost all of the perturbations in

the relative motions of the sun, moon, and earth (including the distance

and declinational aspects described in Chapter 4). However, after about

37, the effects of these motions in representing the actual tides are

extremely small in most locations. For tidally complex areas inside

estuaries, such as Anchorage, AK, and Philadelphia, PA, it takes over one

hundred constituents to adequately describe the tide curve. These

additional constituents are artifacts that combine the fundamental diurnal

and semidiurnal constituents to produce high frequency (from 3 to 13

cycles per day) constituents that attempt to describe the complex nonlinear effects of bottom friction and shallow water. 

Some of the prominent tidal harmonic constituents are now examined

in considerable detail. Their representations of the various astronomical

events and the development of their periods and speeds are essential in

understanding the harmonic analysis techniques discussed the Chapter 9.

The development of frictional, shallow water and compound tidal

constituents are outside of the scope of this book. For more information

about these and other constituents and a detailed discussion about

harmonic analysis consult “Tidal Hydrodynamics” by Parker (1991). 

As mentioned above, the Principal Solar semidiurnal constituent, S2,

represents the earth spinning relative to the sun. The earth rotates once in 24

mean solar hours or, since around the world is 360°, it is going at the rate of

360° / 24 = 15° / hr. However, there is a maximum in the solar tide

producing force under the sun and again on the opposite side (midnight).
41

So, the period (maximum to maximum) of the constituent is 12 mean solar

hours and the speed is:

S2 360° / 12 = 30° / hr.

The Principal Lunar semidiurnal constituent, M2, represents the earth

spinning relative to the moon. Since the moon is moving eastward, it takes

24.8412 mean solar hours to bring the moon back overhead. Again, there

are two maximums in this lunar day, so the period is only 12.4206 mean

solar hours and its speed is:

M2 360° / 12.4206 = 28.984° / hr.

S2 and M2 get into phase (maxima lined up) and out of phase

(maximum of one lined up with the minimum of the other) to produce

spring and neap tides, respectively (Figure 20). Spring tides occur at the

times of full moon and new moon while neap tides occur at the times of

the first and third quarter moons. The revolution of the moon around the

earth relative to the sun takes 29.5306 days (called the synodic month or

one lunation). Since there are two maximums, spring tides occur every

29.5306 / 2 = 14.765 days and neap tides 7.383 days later than the springs.

Figure 20. Phase relationships for spring and neap tides
42

The Larger Lunar Elliptic semidiurnal constituent, N2, and the Smaller

Lunar Elliptic semidiurnal constituent, L2, are two constituents designed to

simulate the cycle of perigee to perigee. These are completely artificial

constituents in contrast with S2 and M2 that have realistic relationships to

the solar and lunar envelopes of the tide-generating forces. Perigee to

perigee occurs every 27.5546 days (the anomalistic month) or 661.31

mean solar hours. The speed of perigee to perigee is thus 360° / 661.31 =

.544° / hr. This is a lunar event and the speed of M2 is 28.984° / hr. The

constituent speeds are, therefore:

N2 28.984 – .544 = 28.440° / hr.

L2 28.984 + .544 = 29.528° / hr.

Thus, when N2 and L2 are in phase every 27.5546 days (the

anomalistic month) they add to M2 to simulate the near approach to the

moon (perigee). Also, 13.7773 days later they are out of phase simulating

apogee (the moon farthest away).

The Luni-solar Declinational diurnal constituent, K1, and the Principal

Lunar Declinational diurnal constituent, O1, are also artificial constituents

designed to simulate the cycle of maximum declination to maximum

declination of the moon. Maximum north to maximum north occurs every

27.3216 days (the tropical month) or 655.72 mean solar hours. However,

both north and south declinations produce the same results. The north to

south (and south to north) cycle is 655.72 / 2 = 327.86 hrs. The speed is

360° / 327.86 = 1.098° / hr. The speeds of the constituents, as they modify

M2, will be the speed of M2 plus and minus the speed of the north to south

cycle. Since the maximum is only felt once per day as the earth spins, the

constituent speeds are half the sum and difference:

K1 (28.984 + 1.098) / 2 = 15.041° / hr.

O1 (28.984 – 1.098) / 2 = 13.943° / hr.

Thus, when K1 and O1 are in phase, every 13.6608 days (one half of

the tropical month; i.e., the month with respect to the vernal equinox),

they add to M2 to simulate the maximum declination of the moon north or

south. They account for the diurnal inequality due to the moon (the two

high tides and/or the two low tides being unequal in height each tidal day)

and, at extremes, diurnal tides (one high tide and one low tide each tidal

day).

The Luni-solar Declinational diurnal constituent, K1, and the Principal

Solar Declinational diurnal constituent, P1, are designed to simulate the

cycle of maximum declination to maximum declination of the sun.

Maximum north to maximum north occurs every 365.2422 days (the

tropical year) or 8765.81 mean solar hours. However, both north and south
43

declinations produce the same results. The north to south (and south to

north) cycle is 8765.81 / 2 = 4382.91 hrs. The speed is 360° / 4382.91 =

.082° / hr. The speeds of the constituents, as they modify S2, will be the

speed of S2 plus and minus the speed of the north to south cycle. Since the

maximum is only felt once per day as the earth spins, the constituent

speeds are half of the sum and difference:

K1 (30.000 + .082) / 2 = 15.041° / hr.

P1 (30.000 – .082) / 2 = 14.959° / hr.

Thus, when K1 and P1 are in phase every 182.62 days (one half of the

tropical year, i.e., the year with respect to the vernal equinox), they add to

S2 to simulate the maximum declination of the sun north or south. These

constituents also contribute to diurnal inequality.

The speeds of the tidal harmonic constituents also may be derived by

combining the speeds of certain fundamental astronomic elements. The

classic description of the tide-producing forces uses a reference frame for

which the earth is the center and projections of the movements of the sun

and moon are made upon the celestial sphere. The fundamental

astronomic elements are:

1. Mean rotation of earth relative to sun, T = 15° / mean solar hr.

2. Rate of change of moon, s = 0.549° / mean solar hr.

3. Rate of change of sun, h = 0.041° / mean solar hr.

4. Rate of change of lunar perigee, p = 0.005° / mean solar hr.

The speeds, n, of the same constituents described above are computed

by this method:

Principal Solar semidiurnal constituent, S2, n = 2T = 30° / hr.

Principal Lunar semidiurnal constituent, M2, n = 2T – 2s + 2h =

28.984° / hr.

Larger Lunar Elliptic semidiurnal constituent, N2, n = 2T – 3s + 2h + p =

28.440° / hr.

Smaller Lunar Elliptic semidiurnal constituent, L2, n = 2T – s + 2h – p =

29.528° / hr.

Luni-solar Declinational diurnal constituent, K1, n = T + h = 15.041° / hr.

Principal Lunar Declinational diurnal constituent, O1, n = T – 2s + h =

13.943° / hr.

Principal Solar Declinational diurnal constituent, P1, n = T – h =

14.959° / hr.
44

These are only two of several ways to develop tidal harmonic

constituents and to compute their speeds. By going through each

development, visualization and understanding is more easily obtained.

The theoretical relative magnitudes of the various constituents are also

of interest. It must be remembered, however, that they are computed from

the tide-generating forces and are not necessarily the values in the

observed tide. They are based on the value, one, for M2, since M2 is

usually the dominant constituent. The relative magnitude values, together

with the periods of the constituents (360° / speed), are:

M2 1.00 12.42 hrs.

S2 0.46 12.00 hrs.

O1 0.41 25.82 hrs.

K1 0.40 23.93 hrs.

N2 0.20 12.66 hrs.

P1 0.19 24.07 hrs.

L2 0.03 12.19 hrs.

The tidal harmonic constituents described in this chapter are only

a few of many constituents. In practice, 37 constituents are used by

CO-OPS for most locations to simulate the principal motions and

perturbations in the sun-moon-earth system.
45

CHAPTER 9

HARMONIC ANALYSIS AND THE

PREDICTION OF TIDES

Historical Organization

CO-OPS, and its predecessors, has always been the government

agency responsible for water level and current observations, tide and tidal

current analyses, tide and tidal current predictions, and sea level

monitoring for the coastal waters of the United States and its possessions.

The organization was known as Survey of the Coast from its founding in

1807 to 1836, Coast Survey to 1878, Coast and Geodetic Survey to 1970,

and National Ocean Survey to 1982. In 1982 it was renamed National

Ocean Service (NOS). From 1965 to 1970 the Coast and Geodetic Survey

was a component of the Environmental Science Services Administration

(ESSA). NOAA became the successor to ESSA in 1970.

Tide predictions by this organization began in 1844 using the lunitidal

interval method. This method involved the time interval from the transit of

the moon over the local meridian to the first high tide. Time adjustments

were made for the phases of the moon in order to take into account some

of the solar effects. In 1867 the harmonic method was introduced. The

tedious additions of the cosine curves encouraged the invention of the

mechanical Maxima and Minima Tide Predictor of William Ferrel in

1885. This machine summed 19 constituents. It is now in the National

Museum of American History of the Smithsonian Institution. A second

machine, developed by Rollin A. Harris and E. G. Fischer, summed 37

constituents and was used from 1912 through 1965. This one is now

located in the NOS facilities in Silver Spring, MD, along with the first

Tide Tables (1867) and other items of interest in the history of tides and

tidal currents.

Harmonic Analysis

A water level (tide) station is a geographic location and facility at

which water level heights are observed over time. Generally, the water

levels are measured by a mechanical, acoustic, and/or electronic water

level (tide) gauge of some sort. Peripheral equipment automatically

records, stores, and/or transmits the data. The measurements are

permanently referenced (vertically) to a series of bench marks on the

adjacent land by terrestrial leveling.
46

Detailed information on these stations, their locations, the actual data,

analyses of these data, tide and tidal current predictions, tidal datums, and

general information on the tides are readily available at the CO-OPS web

site, http://www.tidesandcurrents.noaa.gov/. Also included at this site is

the Physical Oceanographic Real-Time System (PORTS®). PORTS®

provides real-time water levels, currents, and certain meteorological and

water (temperature and salinity) data over telephone and the Internet for

harbor complexes that have PORTS® installed.

When water level data are plotted on a graph of height against time, a

curve, somewhat resembling a cosine curve, is revealed. This curve

contains all kinds of things. It includes the effects of currents, water

density changes, and numerous meteorological factors as well as the

hydrological effects from the proximity of rivers and estuaries.

Also included within this actual water level curve is the astronomic

tide. To study it, the physical oceanographers of CO-OPS must remove

the astronomic tide, usually the dominant signal, from the actual water

level curve. They do it by taking out one constituent tide at a time. The

process is a mathematical one called harmonic analysis. By knowing the

periods of the constituents, it is possible to remove them, providing there

is a series that is long enough. Generally, one year is desirable but one

month can provide adequate results with dominant semidiurnal tides.

Standard analyses are made for 37 constituents by CO-OPS, although

several of them may be quite small at many of the stations.

From a harmonic analysis of the observed water level series, two

values are obtained for each tidal constituent. Amplitude, the vertical

distance between mean tide level and the level of the crest (when plotted

as a cosine curve) is one of the values. The other value is the phase lag

(Epoch). The phase lag is the amount of time elapsed from the maximum

astronomic event to the first maximum of its corresponding constituent

tide. It is usually expressed in degrees of one complete cosine curve

(360°) of that constituent. These two values are known as harmonic

constants and are illustrated in Figure 21. It must be remembered that they

are unique to the particular station location from which they were derived.

Also, the harmonic constants are treated as a constant even though in the

strictest sense they are not because the computed values are affected by

noise in the signal, the length of the series analyzed, etc. The accepted

constants that are used are considered the best estimates of the actual

(unknown) values. When any natural event or engineering project occurs,

such as erosion, deposition, dredging, and breakwater construction that

has the potential to cause major alterations in the adjacent topography,

new measurements and a new harmonic analysis must be made. The
47

accepted harmonic constants that CO-OPS has for a number of station

locations are listed in their web site under Observations – Other -

Harmonic Constants.

Prediction of Tides

To predict the tide, say for a calendar year, it is necessary to know the

harmonic constants (amplitudes and phase lags) for the constituents at

each location for which predictions are desired. These are obtained from a

harmonic analysis of the observed tide at each station as described above.

Adjustments are made for the astronomic configurations for the beginning

of the year. Knowing the phase lag of each constituent from the harmonic

analysis, the first maximum of each cosine curve occurs after the event by

the amount of its phase lag. The amplitude of each cosine curve is that

found from the harmonic analysis. Finally, at each hour of the year, the

heights of all the cosine curves are added. When plotted, the resulting

curve is normally very similar (in shape and size) to the original observed

curve.

The times and heights of the high tides and low tides are placed as

predictions for the forthcoming year on the CO-OPS web site previously

given. Predictions of the tide, called Water Level Tidal Predictions by

CO-OPS, are given for over 3,000 locations. Tidal current predictions are

given for over 2,700 locations. The vast number of predictions is possible

Figure 21. Phase lag and amplitude for a harmonic constituent
48

by applying corrections to those stations for which harmonic constants

have been determined - the Primary Control Stations (Reference Stations).

Subordinate Stations (those without harmonic constants) are referred to

their nearby Reference Stations by empirical constants. Thus, predictions

are also obtained for these Subordinate Stations.

Mathematically, the process of prediction may be described as

follows. To predict the tides at a particular location, it is simply a matter

of adding the cosine curves of the tidal constituents. These constituents

were obtained from the harmonic analysis of the tide, previously observed

at that location. The harmonic analysis provides the tidal constants

(amplitude of cosine curve and phase lag from the astronomic event) for

each tidal constituent. This straightforward addition may be developed by:

h = A cos (at + φ) for one simple harmonic cosine curve where:

h is the height at any time t,

A is the amplitude of the curve, 

a is the frequency (speed), and

φ is the phase at time 0.

For tides,

h = H0 + A1 cos (at + E – κ)1 + A2 cos (at + E – κ)2 + AAAA An cos (at

+ E – κ)n

where:

H0 is the height of mean water level above a selected datum,

A1,2,n is the amplitude of the constituent,

E is the equilibrium argument of a constituent at t = 0 (from the

astronomic event to t = 0),

k is the phase lag or Epoch (from the astronomic event to the

maximum amplitude of the constituent), and

n is the number of constituents used in the prediction - usually up to 37.

E = V0 + u = Greenwich (V0 + u) + aS / 15 – pL

where:

V0 is the uniformly varying portion,

u is the slow variation due to changes in the longitude of the moon’s 

node,

S is west longitude of the time meridian,

L is west longitude of the station, and

p = 0,1,2,...n for long period, diurnal, semidiurnal, and other tides, 

respectively.
49

Types of Tide

Qualitatively, there are three basic types of tides. They are

semidiurnal, mixed, and diurnal. When the two high tides and two low

tides of each tidal day are approximately equal in height, the tide is

semidiurnal. When there is a relatively large difference in the two high

and/or low tides of each tidal day (i.e., large diurnal inequality), the tide is

mixed. When there is only one high tide and one low tide each tidal day,

the tide is diurnal.

The type of tide at a given location is largely a function of the

declinations of the sun and moon. The declinations are constantly varying

such that the type of tide changes throughout the month and year at many

of the locations (see also Declination, Chapter 4).

A more rigorous classification system is available using the

amplitudes of the major constituents at each location. Quantitatively,

where the ratio of (K1 + O1) to (M2 + S2) is less than 0.25, the tide is

classified as semidiurnal; where the ratio is from 0.25 to 1.5, the tide is

mixed mainly semidiurnal; where the ratio is from 1.6 to 3.0, the tide is

mixed mainly diurnal; and where the ratio is greater than 3.0, it is diurnal.

51

CHAPTER 10

TIDAL DATUMS

Tidal Datums

A datum is a reference from which linear measurements are made. It

can be a physical point, line, or plane. It can also be an invisible point,

line, or plane positioned by a statistical treatment of the numerical values

of a particular natural phenomenon. Datums are usually specified as

references for either horizontal or vertical measurements.

For marine applications, the sea surface is the vertical datum. It is used

as the base elevation from which measurements of height and depth are

made. However, since the sea surface moves up and down in time

intervals of less than a tenth of a second to geological ages, and in height

from less than an inch to many feet, this plane must be statistically

stopped in its vertical excursions. When the sea surface is stopped by a

statistical application to the numerical values of a particular phase (such as

lower low water) of the tide, it is called a tidal datum.

As explained in previous chapters, there are usually two high waters

(tides) and two low waters in each tidal day. The two high waters and/or

two low waters are seldom equal in height, with the amount of the

difference varying primarily with the moon's declination north and south

of the equator over the month, and secondarily with the declination of the

sun through the year. The difference in height between adjacent high

and/or low waters is called diurnal inequality. The names of these phases

are higher high water, lower high water, higher low water, and lower low

water.

In some locations, the diurnal inequality becomes so large that the

lower high and higher low waters actually disappear leaving only one tide

per day, known as a diurnal tide. Diurnal tides are the least common. They

usually occur in certain topographically susceptible regions and only

during times of maximum declinational effects. Tides that are always

diurnal are extremely rare.

The definition of a tidal datum, a method definition, generally

specifies the mean of a particular phase or phases over a specific series

length. For example, the tidal datum of mean lower low water is defined

as the average of the lower low water heights (or only low water height) of

each tidal day observed over the National Tidal Datum Epoch as defined

by CO-OPS.
52

In order to make a datum stable, it is necessary to obtain an observed

time series of sufficient length to average out the meteorological,

oceanographic, and hydrologic variability as well as all significant tidal

periods. A series length of 19 years was chosen because it contains all the

tidal periods that need to be considered in tidal computations; i.e., through

the 18.61-year node cycle (the period for the regression of the moon's

nodes westward through 360° of longitude relative to the vernal equinox).

Actually, the real problem was finding a rationale for selecting a series

length that would effectively handle the meteorological, oceanographic,

and hydrologic variability. A series length of 18.61 years seemed

sufficient to handle the variability, and it did have an astronomic

significance for the rationality. The series was then extended to 19 years

because the amplitude of the annual cycle is huge in comparison with the

amplitude of the node cycle. Also, the annual cycle is very short (very

steep curve) in comparison with the 18.61-year node cycle (very flat

curve). In other words, the change in height of the node cycle from 18.61

to 19 years is insignificant.

A specific 19-year period is necessary for standardization because of

relative apparent secular trends in the sea level series. Relative means

changes in the sea surface with respect to the adjacent land. Secular means

non periodic and apparent secular means that it is not known whether the

trend is truly secular or merely a segment of a much longer oscillation yet

to be revealed. Also, a specific 19-year period is needed for

standardization because relative apparent secular sea level trends are

seldom linear; thus, if not used, meaningful comparisons between stations

would be difficult. The specific 19-year period adopted by CO-OPS as the

official time segment for tidal datum determinations is called the National

Tidal Datum Epoch. It is currently 1983 through 2001. After 15 to 20

years or so, relative apparent secular trends in sea level cause tidal datum

elevations to become unrealistic. Therefore, it is the policy of CO-OPS to

update the tidal epoch every 25 years, or when needed to take into account

the apparent secular (non-periodic) changes in sea level. Changes in sea

level are monitored on a continuing basis and yearly changes are usually

too small to have any practical significance.

Tidal datum is the accepted term for any one of a family of water

surface elevation references. There are several tidal datums in common

use today. The tidal datum of mean high water, for example, is the base

elevation for structure heights, bridge clearances, etc. The tidal datum of

mean lower low water has been officially designated chart datum for the

United States, its territories, Commonwealth of Puerto Rico, and Trust

Territory of the Pacific Islands. Chart datum is the datum for soundings
53

and isobaths on all nautical charts and bathymetric maps, and for tide

predictions.

Hydrographic Surveying

Hydrographic surveys are conducted to determine bottom topography.

For each survey, numerous soundings are obtained, usually by ship or

launch in a fairly small area along the coast and in a relatively short period

of time. Horizontally, the soundings are controlled by precise positioning

systems. Vertically, the soundings are controlled by tide measurements.

The tide measurements are used in two parts. The first part is to obtain

chart datum (mean lower low water). The second is to correct each

sounding for the elevation of the water floating the ship above chart datum

at the time each particular sounding is made. Thus, the corrected

soundings will all be at depths of the water below chart datum.

It is not feasible to obtain a 19-year (let alone, specific 19-year) tide

measurement series to determine chart datum in each segment of the coast

for which hydrographic surveys are to be made. However, there are many

tide stations distributed along the coasts that have been in operation for

over 19 years. These stations are continuous and are considered permanent

installations of CO-OPS. Each station is called a primary control tide

station and collectively they comprise the present 117 marine stations of

the National Water Level Observation Network (NWLON).

Essentially, each station consists of a tide gauge and a suite of tidal

bench marks on the adjacent land. A large amount of ancillary equipment

can also be present. The tidal bench marks are leveled between each other

and to the gauge. Thus, the gauge measurements are all related to the

relatively stable land nearby.

Secondary control tide stations (operated for at least one year) and

tertiary tide stations (operated for at least one month) fill in the geographic

gaps between primary control tide stations. Whether the gauge at a

secondary or tertiary station is in operation or has been removed, its bench

marks are retained in place. They are protected from destruction caused by

vandalism, construction projects, and deterioration through respect for

NOS (hopefully) and by the infrequent maintenance visits by CO-OPS

tide personnel.

Even the secondary control and tertiary tide stations are probably not

spaced at close enough intervals to be adjacent to many of the future

hydrographic survey areas. Therefore, a method of comparison of

simultaneous observations has been developed to extend the tidal datum

system to new locations. Note, this method is also used to compute

accepted datums at secondary and tertiary stations.
54

Whenever a primary control, secondary control, or tertiary tide station

is not adjacent to an area in which a hydrographic survey is to be

conducted, a temporary hydrographic tide station is established at a

representative location for the area. The station is operated during the

hydrographic survey. At the nearest control tide station, the vertical

difference between the elevations of the tidal datum of mean tide level

(the mean of all the high and low tides) over the 19-year National Tidal

Datum Epoch (NTDE) and mean tide level over the same time period that

the temporary hydrographic tide station was in operation, is computed.

This difference is then applied to mean tide level at the hydrographic tide

station to give the equivalent mean tide level over the NTDE at the

hydrographic tide station location.

At the nearest control tide station, the mean diurnal range (mean

difference between mean higher high water and mean lower low water)

over the NTDE divided by the mean diurnal range over the same time

period that the temporary hydrographic tide station was in operation,

multiplied by the mean diurnal range at the hydrographic tide station,

gives the equivalent mean diurnal range over the NTDE at the

hydrographic tide station location. One-half of the equivalent mean

diurnal range below the equivalent mean tide level is the elevation of

mean lower low water – chart datum – at the hydrographic tide station.

The other half of the navigation equation is tide predictions. The

nautical chart and bathymetric map provide the mariner with information

on bottom topography over large areas through the use of printed discrete

depth values and isobaths. These are below chart datum. Tide predictions

provide the mariner with water level heights above chart datum as a

function of date, time, and location. They are readily available on the COOPS website. With this depth information and tide predictions, the

mariner can easily determine the distance from the water surface to the

bottom for any time; both at his present location and at any other location

he expects to go.

Coastal and Marine Boundaries

Coastal boundaries are formed by the intersection of the ocean surface

with the land at the elevation of a particular tidal datum. To designate the

boundary, the word “line” is used after the tidal datum name. For

example, the boundary between private and state land, in most states, is

the mean high water line. However, it should be remembered that the

beach itself may erode or pro-grade due to such factors as wind waves,

alongshore currents, storm surges, etc. Although this does not affect the

tidal datum itself, it does affect the tidal datum line. In other words, in our

example, the tidal datum of mean high water may remain constant, but the
55

mean high water line (used as a coastal boundary) may move significant

horizontal distances with erosion and depositional processes.

Marine (offshore) boundaries are also controlled by tidal datums.

Basically, they are all measured from a tidal datum line. For example, the

mean lower low water line forms the baseline from which most offshore

boundaries are measured. Points on the mean lower low water line, as

depicted on large scale nautical charts of NOS, are then used to strike

distance arcs offshore to delineate the boundaries. These construction

procedures follow predetermined national policy or international

convention agreements. Figure 22 illustrates the many coastal and marine

boundaries presently used by the United States.
56

Figure 22. Coastal and marine boundaries from tidal datums.
57

Early Development of Tidal Datums and Marine Boundaries in the

United States

President Thomas Jefferson appointed Ferdinand R. Hassler, a

distinguished Swiss geodesist, the first Superintendent of the Survey of

the Coast in 1807 on the recommendation of the American Philosophical

Society. Essentially, the Survey was to delineate the shoreline and to

conduct topographic and hydrographic surveys of the coast. The work did

not begin until 1816 and consisted of baseline measurements and

triangulation of greater New York Harbor. It lasted only to 1818. In 1832,

the work began once more and by 1834 hydrographic surveys

commenced. In all of his activities, Hassler used the most thorough,

precise, and accurate procedures known in the world in his day. Similarly,

he obtained the best instruments in existence anywhere and developed

new ones when they did not meet his exacting standards.

The extreme precision required of Hassler for his associates and

himself, was intrinsic to the determination of geographic positions (by

astronomic observations), direct baseline measurements, triangulation,

leveling, and tide measurements. With the geographic positions and key

baseline measurements, triangulation could be conducted. The basic

triangulation network would then be used for horizontal position control

of the coastal topographic and hydrographic surveys.

A zero elevation was also needed for the topographic and

hydrographic surveys. The surface of the ocean was the obvious answer

but, with a tidal range of about 4 to 5 feet in the greater New York region,

Hassler would have to mathematically stop the ocean surface in its vertical

excursion in order to achieve his desired precision. This artificially

stopped elevation of the ocean surface became what is now known as a

tidal datum.

Hassler had another problem. The shoreline boundary that he was

charged to delineate is formed by the intersection of the surface of the

ocean with the land. Since the ocean surface went up and down with the

tide, the shoreline moved up-the-beach-landward and down-the-beachseaward. Furthermore, with a gently sloping beach, the horizontal

excursion of the shoreline would be greatly magnified. The obvious

solution was to stop the vertical motion with a tidal datum and thus, the

horizontal motion would also be stopped.

Actually, our predecessors in England and along the east coast of

North America were well aware of the tide and its importance to

boundaries long before Hassler’s day. They saw the rise and fall of the

ocean amounting to several feet and the associated movement of the

water's edge both inland and seaward amounting to many yards. Terms
58

such as ordinary high tide line for the boundary between private and state

lands (derived from English common law) were in general use. But it fell

to Hassler and his successors down through the years to develop

technically sound definitions for these vague legal terms. They would

have to be method definitions and remain acceptable to the courts. These

efforts slowly developed into our present tidal datum system. In fact, the

system is still evolving. Increased economic pressure (population and

offshore oil resources) on our coastal zone has stimulated ongoing

research efforts to understand certain aspects of water level fluctuations

and to continue incorporating the results of these investigations into the

legal-technical tidal datum system.

Sea Level Variations

Sea level has a profound effect on coastal and marine boundaries. If

changing sea level becomes great enough to cause CO-OPS to update the

National Tidal Datum Epoch the datum elevations at some locations will

change significantly. On a gently sloping beach, the horizontal

displacement of the datum line will be greatly magnified. However, the

boundaries do not change automatically with a new Epoch.

An increase in carbon dioxide accumulation in the upper atmosphere,

primarily from the burning of fossil fuels, is a recent concern of the

scientific community, Congress, and the general public. The concern is

that the greenhouse effect will be intensified by this accumulation, thus

slightly increasing the mean temperature of the earth. The result of this

warming could include such things as shifts in climate belts with the

concomitant adjustment of agricultural and fishery areas. Also predicted

by model is an increase in the height of the surface of the ocean and the

associated inundation of low lying coastal areas. The rise would be due to

thermal expansion of the ocean waters and to the melting of glacial ice.

The sensationalism of this predicted phenomenon tends to mask and to

confuse a very important question; what has sea level been doing against

the shores of the United States?

For very practical reasons, this question is extremely important. Since

the rates of sea level rise have been fairly uniform, in most cases, short

term projections are probably quite accurate. The size and/or amount of

shore protection structures and beach nourishment required to mitigate

that portion of erosion due to sea level rise is dependent on the local trend

and variability of sea level. Private property lines, as well as state and

federal coastal and marine boundaries, are dependent on changes in sea

level elevation. Future requirements for drainage pumping and salt water

intrusion prevention are a function of sea level trends. Hydrographic

surveying is also controlled by the sea level stand. In conservation, our
59

extremely biologically productive marshlands are threatened by the

constant narrowing of the zone between the ocean and developed land due

to rising sea level on the gently sloping shore. Finally, changes in trend

will enable physical oceanographers to detect the beginning of the

predicted greenhouse induced sea level rise.

Fortunately, CO-OPS keeps track of what sea level has been doing

against the shores of the United States (Zervas, 2001). It operates 45 tide

stations (from which sea level statistics are derived) that have been in

continuous operation since at least 1940, 60 since at least 1950, and 122

since 1975. The longest sea level series is from San Francisco. It dates

from 1854. The New York series began in 1856 but has only been

continuous since 1893. The series at Seattle began in 1898. However, for

station comparisons, the common 1950 to 1999 series is used.

The following abbreviated list gives the trend [mean rate of rise (or

fall)] of sea level for various representative stations using the common

1950-1999 series (Zervas, 2001):

mm/yr

Sewells Point (Norfolk), VA 4.48

Charleston, SC 3.05

Pensacola, FL 2.00

San Francisco, CA 2.23

Seattle, WA 2.26

Juneau, AK –12.25 

Honolulu, HI 1.44

The trends are what the ocean surface is actually doing on our beaches

as observed by a person standing on a specific beach (relative sea level

trends). They are the trends that affect our coasts and individual beaches

and what coastal communities must react to.

Two of the largest components of relative sea level trends are: 1) local

vertical land movements, and 2) changes in height of the sea surface

relative to the geographic center of the earth. Models of the greenhouse

effect predict an acceleration in the rise for the latter component.

Therefore, one of the most important activities of CO-OPS is to study the

changes in these sea level series each year. By careful analysis, it is hoped

that the beginnings of the predicted acceleration (if it should occur) can be

detected. If local vertical land movements do not have uniform

accelerations themselves, then the ocean signal should be able to be

separated out. However, with data through 1999, the predicted

acceleration has not yet been detected.

61

CHRONOLOGY OF SIGNIFICANT TIDAL EVENTS

IN THE UNITED STATES

When applicable, dates refer to the Tide or Tidal Current Table

volume containing predictions for the stated year.

1807 The Survey of the Coast established.

1830 Tide predictions begin in the United States. One high tide time

prediction per day for Boston, New York, and Charleston are

published in The American Almanac. Time differences for 96

stations and spring ranges predictions for 84 stations are also

given.

1836 The Survey of the Coast became Coast Survey.

1844 Tide Notes, including lunitidal intervals, appear on nautical charts

of United States coasts and harbors.

1853 Tables for obtaining tide predictions by the lunitidal interval

method are published in the Appendix of the Annual Report of the

Superintendent of the U. S. Coast Survey for the first time.

1854 Tidal Division formed.

1864 Last year of publishing lunitidal interval tables. One thousand

copies provided to Union Navy.

1867 First Tide Tables published by U. S. Coast Survey. 

 

1868 Low tide predictions begin for west coast of Florida and Pacific

coast.

1878 Coast Survey became Coast and Geodetic Survey.

1885 William Ferrel’s Maxima and Minima Tide Predictor introduced. 
62

1887 Low tide predictions included for all stations in the Tide Tables.

 

1890 Tidal current predictions begin for New York harbor and vicinity.

 

1896 Tide Tables extended to include numerous ports throughout the

world.

1910 Tidal Research Section formed.

1912 Harris-Fischer Tide Predicting Machine introduced. 

 

1914 Last year Ferrel’s Maxima and Minima Tide Predictor used. 

 

1922 Tide Table format changed from sequential listing to separate

columns for high and low tides.

1923 Tidal Current Tables first published separately from Tide Tables.

Published in two volumes – Atlantic Coast, North America, and

Pacific Coast, North America.

1928 Single port miniature tables introduced. 

 

1932 Last year of single port miniature tables (revived from 1940

through 1944 for New York harbor and vicinity only).

1940 Special restricted tables for war effort begin. 

 

1951 Last year of special wartime occupation tables. 

 

1955 Special tide tables for selected places in Greenland, Canada, and

Alaska begin.

1958 Format changed from separate columns for high and low tides to

sequential listing.

1959 Tide predictions added to Small Craft Chart series.
63

1961 Motor drive and automatic readout installed on Harris-Fischer

machine. Last year of publication of special tide tables for

selected places in Greenland, Canada, and Alaska.

1963 Established Deep Sea Tide Program.

1965 Last year Harris-Fischer Tide Predicting Machine used. Analogto-digital recorder (ADR) tide gauges and computer processing

introduced.

1966 Electronic digital computer introduced for predictions. 

 

1967 International Symposium on Mean Sea Level of the IAPO and

UNESCO held. Electronic digital computer introduced for

harmonic analysis of tides.

1970 Coast and Geodetic Survey became National Ocean Survey.

1973 Sea Level Service begins. Established National Tidal Datum

Epoch.

1977 Gulf Coast Low Water Datum adopted.

1978 Water Level Telemetry System introduced for marine coasts.

1980 National Tidal Datum Convention of 1980 adopted.

.

1982 National Ocean Survey became National Ocean Service.

Personal computer software introduced for local user access to

water level telemetry stations.

1988 Personal computer software introduced for local user access to

current telemetry stations. Operational Next Generation Water

Level Measurement System (NGWLMS) field units introduced

using air acoustic ranging sensor and satellite telemetry.
64

1995 Last year of Tide Table and Tidal Current Table publication in

hard copy by CO-OPS, NOS, NOAA. Partial tide prediction

provided on telephone bulletin board.

1996 CO-OPS tide and tidal current predictions begun exclusively on

Internet and CDs. Private publishers provided CO-OPS

predictions for various tide and tidal current publications. 

CO-OPS tide and tidal current predictions provided to National

Imagery and Mapping Agency for publishing in hard copy for

U.S. Navy, U.S. Coast Guard, etc.
65

BIBLIOGRAPHY

Cohen, I. Bernard, Isaac Newton. Scientific American, v. 193, n. 6, p. 73-

80, Dec. 1955

Defant, Albert, Ebb and Flow. Ann Arbor, Univ. of Michigan Press, 121

pp., 1958

Doodson, A. T. and H. D. Warburg, Admiralty Manual of Tides. London,

H. M. Stationary Office, 270 pp., 1941

Forrester, Warren D. Canadian Tidal Manual. Ottawa, Dept. of Fisheries

and Oceans, 138 pp., 1983

Gammow, George. Gravity. Scientific American, v. 204, n. 3, p. 94-106,

March 1961

Groves, Gordon W. Tides. Encyclopedia Britannica. Chicago, William

Benton

Hatfield, H. R., Tides and Tidal Streams. Taunton, Somerset, The

Hydrographer of the Navy, 119 pp., 1969

Heyl, Paul R., Gravitation–Still a Mystery. The Scientific Monthly, v.

LXXVIII, n. 5, May 1954

Hicks, Steacy D., Tide and Current Glossary. Rockville, MD, National

Oceanic and Atmospheric Administration, U. S. Dept. of

Commerce, 30 pp., October 1989

Hicks, Steacy D., Richard L. Sillcox, C. Reid Nichols, Brenda Via, and

Evette C. McCray, Tide and Current Glossary. Silver Spring, MD,

NOAA, NOS, CO-OPS, 28 pp., January 2000, http://www.coops.nos.noaa.gov/

Knauss, John A., Introduction to Physical Oceanography, 2nd Ed., Pearson

Education, 309 pp., 1996

Lisitzin, Eugenie, Sea Level Changes. New York, Elsevier, 286 pp., 1974

Marmer, Harry A., The Tide. New York, D. Appleton, 282 pp., 1926

Parker, Bruce B., Tidal Hydrodynamics. New York, J. Wiley, 883 pp., 1991

Russell, R. C. H. and D. H. Macmillan, Waves and Tides. New York,

Philosophical Library, p. 165-335, 1953

Schureman, Paul, A Manual of the Harmonic Analysis and Prediction of

Tides. 

Washington, U. S. Coast and Geodetic Survey, Sp. Pub. 98, 416 pp., 1924

(now Silver Spring, MD, NOAA, U. S. Dept. of Commerce)

Sverdrup, Harold U., Martin W. Johnson, and Richard H. Fleming, The

Oceans. Englewood Cliffs, NJ, Prentice-Hall, 1087 pp., 1942
66

Von Arx, William S., Introduction to Physical Oceanography. Reading,

MA, Addison-Wesley, 422 pp., 1962

Zervas, Chris, Sea Level Variations of the United States 1854-1999. Silver

Spring, MD, NOAA Technical Report NOS CO-OPS 36, 186 pp.,

2001