NAVY SCIENTIFIC PAPERS, No. 2. 
TIDES AND TIDAL PHENOMENA: 
FOR THE USE MORE PARTICULARLY OF U. S. NAVAL OFFICERS. 
BY 
HENRY MITCHELL, 
Assistant U. 8. Coast Survey. 
BUREAU OF NAVIGATION, 
NAVY DEPARTMENT. 
WASHINGTON: 
GOVERNMENT PRINTING OFFICE. 
1868. TIDES AND TIDAL PHENOMENA. 
BY HENRY MITCHELLL, 
Assistant U. S. Coast Survey. 
Tidal phenomena present themselves to the observer under two 
aspects : as alternate elevations and depressions of the sea, and as 
recurrent inflows and outflows of streams. 
The word tide, in common usage, is applied without distinction to 
both the vertical and the horizontal motion of the sea, and much 
confusion has arisen among navigators from this double application 
of the term. Careful writers of modern times, however, use the 
word tide in strict reference to the changes of elevation, while they 
distinguish the recurrent streams as tidal currents or tidal drifts. 
With regard to the words ebb and flood, no distinction of application 
has been observed. We advise, however, that in the field-book 
these terms should be used in reference to the current, and that the 
words rise and fall should replace them when the tide is referred to. 
The words stand and slack are often confounded; the former, cor- 
rectly used, designates that period of time during which the height 
of the tide remains essentially the same; while the latter indicates 
the interval of time between the cessation of flood or ebb current 
and the commencement of the opposite stream. 
It is often the case that the elements of the tide, obtained from a 
series of observations at a single port, may represent, in a general 
way, the conditions for a considerable extent of the adjacent coast, 
and even be applicable to other ports in the neighborhood; whereas 
the elements of the tidal current are so strictly local that the same 
port may require several distinct series of observations in its differ- 
ent approaches and avenues. From these simple considerations the 
importance of a strict use of terms and phrases will be obvious to 
the voyager who may design to collect data for the use or information 
of others. 
Although it is often desirable that observations of the tides and 
their dependent currents should be made simultaneously, we must, 
for the sake of perspicuity, refer to them separately in this article. 4 
TIDES. 
Tidal observations consist simply of a record of the heights of the 
water above a fixed zero at stated times. These heights are meas- 
ured by gauges of different forms according as the station is more or 
less exposed to waves, ice, &c. In the Appendix to this article we 
have enumerated the different kinds of gauges which have been used, 
and described them briefly. The field-book should contain descrip- 
tions of the instruments used, locality of the station, means of obtain- 
ing the true time, together with notes on the weather, &c. 
§ 1. To form a clear notion of the tide, it must be conceived to 
be a long wave or undulation, propagated to our shores from mid- 
ocean without necessarily causing visible agitation of the water par- 
ticles.* Its range is the vertical distance between the low and high 
water levels; its length the distance measured on the surface of the 
sea from one low water to the next. The tide wave, considered as 
an undulation, is always moving in a single direction, but is subject 
to changes of figure from point to point in its journey, by which its 
range and its length are altered. The rate with which the tide wave 
moves depends upon the depth of water ; it is measured by comparing 
the times of high water observed at different determined points along 
its path. 
§ 2. When we speak specifically of the form or figure of the tide 
at any given place, we refer to the curve which the observations de- 
scribe when plotted upon paper, using the heights and the hours of 
the day as co-ordinates. The tidal curves for ports in the same 
neighborhood differ but slightly, but those of different seas in some 
cases present striking contrasts of figure. 
§ 3. The observed tide is not a simple wave; it is a compound of sev- 
eral elementary undulations, rising and falling from the same common 
plane, two of which can be distinguished and separated by a simple 
grouping of the data, to which we shall hereafter refer. These two 
waves are known as the semi-diurnal and the diurnal tides, because 
the first, if alone, would give two high and two low waters in a day, 
while the second would give but one high and one low water in the 
same or an equivalent period of time. In nearly all ports these two 
tides coexist, but the proportion between them varies remarkably 
for different seas. 
Along our Atlantic coast the semi-diurnal wave predominates, and 
* For descriptive articles on tidal phenomena we may refer the reader to Laplace's 
" Exposition du Systems du Monde" and to the American Encyclopaedia, Article "Tides." 5 
the diurnal is so small that its presence is made scarcely sensible in 
the small differences which it causes between the morning and the 
afternoon tides-diurnal inequalities. In some parts of the Gulf of 
Mexico it is the diurnal wave which usually prevails, and so much 
does it exceed the semi-diurnal in height that, upon the average, but 
one high and one low water occur in twenty-four hours. Upon our 
Pacific coast, the two elementary waves have nearly the same range, 
so that although there are usually two tides per day, they are very 
unequal. 
In undertaking to investigate the tides of a port, it is important to 
ascertain as early as possible the form of the tide; that is, whether 
it resembles the semi-diurnal, the diurnal, or the mixed type; be- 
cause not only may this information be itself of scientific value, but 
the knowledge thus gained at the outset will enable the observer to 
fix upon the best method of keeping his record. 
§ 4. The type forms to which we have referred are illustrated in 
the annexed diagram, where the figures of the waves are plotted in 
curves, using the times as abscisses and the heights as ordinates: 
In this diagram the curve traced in dotted line is a tide wave of 
the semi-diurnal type; that traced by the broken line one of the 
diurnal, while the full line is one of the mixed type-in this case the 
compound of the two others. It will easily be seen that upon the 
magnitudes and relative positions of the two elementary waves de- 6 
pend not only the relative, heights, but also the relative times of the ob- 
served high and low waters. 
The semi-diurnal tide increases in magnitude as the moon ap- 
proaches her conjunction or opposition with the sun, and diminishes 
as she nears the quadratures. 
To sketch the history of the diurnal wave in the least critical way, 
it may be said to increase with the angular distance of the moon from 
the equator-that is to say, with her declination (north or south)-to 
reach a maximum when this declination is greatest, to decline as it 
declines, and to vanish when it vanishes. Unlike the semi-diurnal 
wave, it is evanescent.* 
In order to determine the type to which the tide of any port be- 
longs, it is usually only necessary to make hourly observations for a 
day or two at the date of the moon's maximum declination, and to 
repeat the series about a week later when the moon crosses the 
equator. 
Information relative to the type forms of the tides among the Pa- 
cific islands has long been desired. From the few observations made 
at the Sandwich Islands and at some other points, it would appear 
that these tides are generally of the diurnal type and very peculiar 
in many respects. 
§ 5. The reported irregularities of the rise and fall at any place 
should not deter persons from careful investigation. We could men- 
tion many instances where the phenomena are apparently so compli- 
cated that the local pilots could not formerly detect any sort of order; 
yet, when subjected to careful tabulation, the skein has been so com- 
pletely unraveled that useful predictions can be made. It is natural 
that changes following the phases of the moon should be easily re- 
cognized; but those following the moon's declination must everywhere 
appear accidental, or at least inconsistent, to the ignorant. Upon 
our Atlantic coast two popular rules for the time of the tide are 
found among the fishermen and coasters: to the northward of Nan- 
tucket " South moon makes high water;" to the southward, "South- 
east moon makes high zoater.'1' These two rules are nearly as correct 
as they are convenient; for, between Nantucket and the Bay of Fundy, 
high water occurs in the various ports shortly before the moon reaches 
the meridian; while along the southern coast, between Nantucket and 
* Both of these elementary waves are, in reality, compounds of lunar and solar tides-the 
latter usually small-and here, as in other natural phenomena, there are considerable inter- 
vals of time between the causes and their observed effects. 7 
Florida, high water visits the ports four or five hours before the moon's 
transit, or while she is seen in the southeast. Another general rule 
which applies to our entire Atlantic coast is, that " the greatest tides 
(springs) follow soon after the full and new moon, the least (neaps) soon 
after the quarters.11 
In the Gulf of Mexico no such rules as those we have mentioned 
would apply, because the tides belong to the diurnal and mixed 
types. If, for instance, the new or full moon occurs at her zero de- 
clination, the tides, instead of increasing as the time of syzygy draws 
near, undergo a decline and perhaps vanish. The phenomena of 
spring and neap tides are not recognized in some parts of this sea, 
and yet the tides are generally obedient to the influences of the sun 
and moon. 
To collect data for the construction of prediction tables has been 
the leading object of tidal observations, and a multitude of such tables 
have been constructed for different ports. Those for the North At- 
lantic, and those for the Pacific coast of America, are practically 
complete, but for other parts of the world often inaccurate. 
We shall sketch out the most recent and approved plans for tabu- 
lating the field-notes, taking up each type separately. 
Tides of the semi-diurnal type. 
§ 6. Let us consider the case of a station where the tides are semi- 
diurnal in type, considerable in magnitude, and generally well defined. 
The data required for practical use are the times and heights of 
high and low water, together with such meteorological notes as may 
bear upon the matter. 
To obtain the true times and heights, the observations for each tide 
should be made every two minutes, commencing at least twenty min- 
utes before and continuing till twenty minutes after the stand. The 
middle of the stand is the true time of the high or low water. 
With tides of small range the culminations are often very slow, so 
that the observer is doubtful of the exact time when the vertical 
changes cease; a change of wind or a squall may interrupt the normal 
conditionsand disturb his confidence in his record. Under such cir- 
cumstances, it is a good plan to plot upon profile paper the observa- 
tions obtained, using the times and heights as co-ordinates, (very 
much as we have done in § 4,) so that irregularities may appear to 
the eye and may be struck out by drawing a smooth curve through 
them. This method of ' ' graphical correction11 is a guide to the 8 
judgment in estimating the probable instant of culmination. It was 
introduced by Dr. Whewell in his tidal essays, and has been long 
familiar in many other studies of variable numerical data. 
§ 7. Having carefully reviewed the field-book and marked the 
true times of the high and low waters, the computer is prepared to 
take the first step in the compilation of a tide table. This first step 
is the drawing up of a tabular statement in the form given below. 
This form is called the " First Reduction," and, in the specimen we 
have given, we have placed a few observations to illustrate its use. 9 
1 
Date. 
2 
Moon's 
Transit. 
3 
Time of- 
4 
lunitidal Interval. 
5 
Height. 
6 
Duration. 
7 
Stand. 
8 
Meteorological Data. 
High water 
Low water. 
High water 
Low water. 
High water 
Low water. 
Rise. 
Fall. 
High 
water. 
Low 
water. 
Wind. 
Barometer. 
Temperature. 
Hrs. 
Mins. 
Hrs. 
Mins. 
Hrs. 
Mins. 
Hrs. 
Mins. 
Hrs. 
Mins. 
Feet. 
Dec. 
Feet. 
Dec. 
Hrs. 
Mins. 
Hrs. 
Mins. 
Mins. 
Mins. 
Direc'n. 
Force 
In. 
Dec. 
Temp, 
o 
Air. 
W ater. 
1860. 
23 
12 
49 
Aug. 31 
09 
9 
07 
2 
37 
9 
18 
4 
81 
2 
33 
6 
30 
01 
05 
NW. 
2 
30 
15 
68 
74 
65 
8 
21 
25 
14 
46 
9 
16 
14 
57 
5 
14 
2 
35 
6 
39 
5 
39 
03 
15 
S.SE. 
1 
30 
04 
67 
65 
65 
Sept. 1 
0 
29 
9 
39 
3 
00 
9 
10 
14 
51 
5 
24 
2 
30 
6 
39 
5 
35 
01 
04 
W. 
4 
29 
92 
69 
78 
65 
12 
48 
21 
50 
15 
36 
9 
02 
15 
07 
5 
16 
2 
61 
6 
14 
5 
57 
07 
06 
sw. 
3 
30 
07 
63 
81 
66 10 
The first entries in this table are made directly from the field-book, 
and occupy, in order, spaces numbered 1, 3, 5, and 7. 
Next, the space numbered 2 is filled from the tables of the moon 
given in the Almanac, after due correction. In the "American 
Ephemeris and Nautical Almanac" the upper and lower transits for 
each astronomical day are given in mean time as calculated for the 
meridian of Washington. For our use these transits must be reck- 
oned back to the civil day, by reversing the rule laid down in the Al- 
manac, ("Use of the Tables,") and corrected for the longitude of the 
tidal station by adding about two minutes for each hour of west, and 
subtracting the same for each hour of east longitude from Washington. 
The "Lunitidal Interval" is the time elapsed between the transitof 
the moon and the occurrence of high or low water. It is customary to 
refer both the high and the succeeding low water to the same transit, 
viz: that which immediately precedes the high water; for this reason 
the lunitidal intervals of low water appear in our table about six 
hours longer than those of high water. On the 31st of August it 
will be seen by our table that the first high water occurred at 97i. 07m., 
and the succeeding low water at 14A. 46m.; both of these are referred 
to the transit which occurred at 237z. 49m. on the preceding day, and 
the intervals are, respectively, 9A. 18m. and 147z. 57m. By these sim- 
ple calculations the space 4 is filled. The sign (opposition) is in- 
troduced to fill the blank, and, as we use it, simply indicates that the 
upper transit occurs after midnight so as to fall in the next civil day. 
If it were the other transit that passed over, we should use the sign 
6 (conjunction) with the same specific meaning. The proper motion 
of the moon to the eastward causes her transits over the same meri- 
dian to be later each day about forty-eight minutes in the average, so 
that in the course of every fifteen days one of her transits (either the 
upper or loioer) fails to fall within the same civil day as the other. 
There are not, then, two transits in every calendar day; on the con- 
trary, in twenty-nine and one-half days (the average time elapsed 
between one new moon and the next) there are but fifty-seven. In 
the same way the tides, which follow the moon, do not always give 
us two high and two loio waters within the same day; but in the 
course of a month two days occur in each of which only one high 
water obtains, and two also in each of which but one low water takes 
place. 
The space 6 is filled by reckoning the time between each high 
water and the succeeding low water for duration of fall; and between 
each low water and the succeeding high water for the duration of 
rise. 11 
To this table are usually attached columns of meteorological data 
transferred from the field-book. They serve occasionally to explain 
disorderly results, and in nicer computations appear as elements of 
correction. 
§ 8. If the observations of several weeks are placed in this table, 
and reduced as we have described, a general inspection will discover 
traces of certain laws which govern the phenomena: 
1st. With regard to the heights, the greatest elevations and de- 
pressions (spring tides) will be found to occur shortly after those 
dates when the moon crosses the meridian at noon or midnight, 
(when she is near her conjunction or opposition with respect to the 
sun;) and the least (neap tides) shortly after those dates when the 
moon's transits occur at Gh. or 18A. (when she is near her quadra- 
tures.) 
2d. The lunitidal intervals will be found similarly, and even more 
closely, dependent upon the hours of transit; the least intervals occur 
at or near the 3/j. or 15A. transits, and the greatest intervals at or 
near the 9/z. or 21A. transits. 
§ 9. In order to exhibit more clearly the dependence of the tidal 
variations in height and interval upon the moon's hours of transit, 
another table, called the " Second Reduction," is necessary, in which 
the results are grouped according to those hours. An example of 
such a table, arranged for high water, is given below, the data for 
which is borrowed from actual observations of the Coast Survey in 
Hampton Roads. 
Moon's tran- 
sit. 
Lunitidal in- 
terval. . 
High water. 
Height of 
high water. 
observations. 
*5 
Hrs. 
Mins. 
Hrs. 
Mins. 
F. 
Dec. 
K 
0 
30 
8 
58 
5 
43 
48 
1 
30 
8 
46 
5 
42 
48 
2 
30 
8 
38 
5 
35 
48 
3 
30 
8 
30 
5 
20 
48 
4 
30 
8 
26 
5 
10 
48 
5 
30 
8 
32 
5 
04 
48 
6 
30 
8 
44 
4 
97 
48 
7 
30 
9 
08 
5 
12 
48 
8 
30 
9 
18 
5 
14 
48 
9 
30 
9 
21 
5 
06 
48 
10 
30 
9 
19 
5 
26 
48 
11 
30 
9 
09 
5 
32 
48 
Means 
8 
54 
5 
20 12 
§ 10. In this second reduction each interval and height is the mean 
of all those observed, referable to the same hour of transit; the tran- 
sits between Oh. and 1A., 12A. and 137?., being classed as Oh. 30m.', 
those between 17?. and 27?., 13A. and 147?., being classed as 17?. 30m., 
and so on. At the foot of each column is given the arithmetical mean 
of the quantities above. The mean interval is called upon our charts 
the " corrected establishment," and is the leading expression of the 
ordinary tide table, such as we give below, which requires now but 
little further explanation. 
Corrected establishment 
. VIHh. XVIIm 
Rise of highest tide observed above the plane of reference 
4.8 feet. 
Fall of lowest tide observed below the plane of reference 
1.7 feet. 
Fall of mean low water of spring tides below the plane of reference 
0.2 feet. 
Height of mean low water of neap tides above the plane of reference 
0.2 feet. 
Mean rise and fall of tides 
2.5 feet. 
Mean rise and fall of spring tides 
3. 0 feet. 
Mean rise and fall of neap tides 
2. 0 feet. 
Mean duration of rise, ? Reckoning from the middle of one stand to the ? 
6h. 01m. 
Mean duration of fall, $ middle of the next. J 
6h. 25m. 
Mean duration of stand 
Oh. 14m. 
The greatest high water in the table of second reductions is as- 
sumed to be that of spring tide; the least, that of neap tide. The 
reverse holds for low water. The durations of rise, fall, and stand, 
as well as heights of highest and lowest observed tides, are taken 
from the first reductions. 
The plane of reference mentioned in this table is that from which 
the soundings of the chart are measured; the depths, as now repre- 
sented upon the charts of our coast survey, are referred to mean low 
water. We shall have further remarks to make in this connection 
when we come to speak of the mean level of the sea. 
With regard to the "stand" there may be some uncertainty. Its 
very existence, as a measurable tidal phenomenon in the strictest 
sense, is very doubtful; but as there must always be a period when 
the vertical changes are slow, it is well to express it in the table, in 
order that the pilot may estimate about what length of time he may 
expect to carry a certain maximum or minimum draught across a bar 
or flat. 
§ 11. It will be seen by an inspection of our second reduction that 
the variations of intervals and heights, to which we referred in § 8, 
are distinctly indicated; and if we plot these quantities, using the 
hours of moon's transits as abscissae, as in the following sketch: 13 
we discover that curves possessing some degree of regularity appear. 
These are called the curves of " half-monthly inequality," because the 
moon's hours of transit advance from OA. to 12h. in a half lunar 
month.* 
In our sketch we have given only the curves for high water, be- 
cause our limited space does not admit of extending the diagram. In 
actual practice it will be found convenient to present, on a larger 
scale, the intervals and heights both for high and low water upon the 
same diagram, so as to furnish at a glance a compend of the tidal 
laws of the port. In drawing up a report of tidal observations, or 
in furnishing the statistics of a seaport, a diagram like that we have 
described will add scientific value to the tide table, and may also en- 
gage the attention of the reader where an array of numerical data 
would fail. 
§ 12. One of the most striking features of the curves of half- 
monthly inequalities is the retard of the tide which they make appa- 
rent. The highest tides and the mean interval do not occur at full 
and change as might be expected, but a day or so after these events- 
a considerable period of time has elapsed between the creation of the 
tide and its propagation to our shores. This at once suggests to us 
that, in referring the tide of to-day to its next preceding transit, we 
* The synodical lunar month, the time from one conjunction or new moon to the next, is in 
the average 29 days, 12 hours, 44 minutes, and 3 seconds. 14 
are not connecting cause and effect in their natural order; for the 
tide which reaches us to-day is but an offset or derivative from the 
true tide wave of mid-ocean, which was perhaps excited or raised by 
the sun and moon of several days ago. 
Long series of observations are necessary to determine accurately 
the age of the tide, but curious and valuable results have already 
sprung from the inquiry.* At exposed stations on either shore of 
the North Atlantic the tide is at least a day and a half old, while on 
the Pacific coast of the United States it is scarcely a half day old. By 
determining the ages of the tide at different stations we may obtain 
a clue to its path, and ascertain whether the tides of different seas 
are of distinct origin, whether they interfere, Ac. 
§ 13. With regard to the number of tidal observations requisite 
for the construction of a tide table ve cannot speak definitely. Al- 
though the principal variations are half-monthly, they are not co- 
incident, so that it can very rarely happen that the deviations in 
the heights and intervals from the true mean will cancel in a single 
half month. By reason of the changes in the distance of the moon 
from the earth and the alterations of her angular distance from the 
equator, the heights and intervals of consecutive springs differ ma- 
terially at times upon some coasts ; the same may be said of consecu- 
tive neaps. As a general rule, at least fifty-seven consecutive high iva- 
ters and the corresponding low waters should be observed before a suitable 
tide table can be made for a chart. 
§ 14. When several tidal stations are occupied in a river, a bay, or 
an arm of the sea, the march of the phenomena is best represented 
by referring all the tides to the same transit, viz : that which pre- 
cedes the high water at the entrance. The stations, in this case, 
should be furnished with time-pieces set to one standard, viz : the 
mean time of the entrance. This matter will require our attention 
again when we reach the subject of the disturbances of the tide 
wave. 
§ 15. The tide table offers the means of making a rude prediction 
of the principal tidal phenomena, and the diagram of half-monthly 
inequalities gives the variations for different hours of transit. If the 
series of observations extends through a sufficient period, we may add 
three other important corrections, viz : those for the changes in the 
moon's declination, those for changes in the moon's distance from the 
* The true age of the tide is that derived from the heights. Airy, Phil. Trans., 1845. 15 
earth, and those for the changes in the sun's declination.* These 
three astronomical elements are independent of each other and of the 
moon's hours of transit, so that, taking observations for long periods, 
we may tabulate our tidal data according to the increase or decrease 
of any one of them while the others remain nearly constant at mean 
values; and thus, by successive groupings of the observations, we 
may evolve the corrections for the different positions of the heavenly 
bodies without recourse to special theory or the use of formulae. The 
manner of procedure is as follows: 
1st. To the table of first reductions add three columns, in which 
state the moon's declination, sun's declination, and the moon's hori- 
zontal parallax for each, from the Nautical Almanac. The second 
reductions made out for the whole series will give, as we have shown, 
the values for different hours of transit; and the differences between 
these values and the general means will be the corrections for those 
half-monthly inequalities which depend upon the moon's phases. 
2d. From the first reductions take out the intervals and heights 
and arrange them in groups for every three degrees of declination, 
(irrespective of sign,) 0° to 2° inclusive, 3° to 5° inclusive, and so 
on to maximum. We shall then have, on taking the means in each 
group, eight values for intervals and heights, the differences between 
which and the grand means from the whole series will be the correc- 
tions for another class of inequalities of a half-monthly character, 
dependent upon declination. 
3d. By another arrangement of the tabulations just mentioned, 
class the elements following superior transits of north declinations with 
those following inferior transits of south declinations; then class the 
elements following inferior transits of north declinations with those fol- 
lowing superior transits of south declinations; making two classes of 
eight groups each, the differences between the means of the corre- 
sponding groups of which give the corrections for diurnal inequali- 
ties-the differences in intervals and heights of consecutive tides, t 
4th. From the first reductions take out the elements for each min- 
ute of horizontal parallax and group separately; then take the means 
* The corrections for variations in the sun's distance from the earth are too small to re- 
quire consideration in ordinary computations. 
t In making the first reductions, with these subsidiary tables in view, it is well to distin- 
guish superior transits and their corresponding heights and intervals from inferior transits 
and their elements by using differently colored inks. 
Although in tides of the semi-diurnal type these inequalities are always presumed to be 
small, the cases are very rare in which they are wholly absent. 16 
of each group and compare with the grand means. By this compu- 
tation the corrections for the different distances of the moon from the 
earth, represented by her parallax, will be exhibited. These cor- 
rections are half-monthly in long averages. 
5th. From the first reductions take out the average intervals and 
heights for the six pairs of opposite months, grouping January with 
July, February with August, and so on; then compare these with the 
grand means to obtain the corrections for half-yearly inequalities 
dependent upon the sun's declinations. 
The five corrections which we have described are different for dif- 
ferent ports, and especially so for different seas, so that no tables for 
general application have yet been arranged. A great deal has been 
done by Professor Airy and others towards the construction of for- 
mula} which should make it possible to predict the tides of a port 
from a limited series of observations; but a great deal remains to be 
learned of the general laws of tides. For the full application of the 
numerical method we have described, we must refer the reader to 
the articles of Sir John Lubbock and Dr. Whewell, in the Philosophi- 
cal Transactions of the Royal Society of 1834 and 1835. For the 
application of this method, as far as consistent, to a very short se- 
ries, (four months,) and comparisons with theory by application of 
Professor Airy's formulas, see an article by Charles A. Schott, Esq., 
entitled "Tidal Observations within the Arctic Circle," in the 
Smithsonian Contributions to Knowledge, 1865. This article is the 
most exhaustive of the data which it employs, and is perhaps the best 
example in the treatment of short series of observations.* 
Tides of the Mixed Type. 
Thus far we have spoken only of the tables for the tides of the 
semi-diurnal type, in which the differences in magnitudes and inter- 
vals for the two tides of the same day are insignificant; we now pro- 
pose to consider tides of the mixed types, whose diurnal inequalities 
are so large that the tide table described in § 10 would be quite in- 
adequate for practical use. For these tides we are obliged to resort 
to a different arrangement of the data, and more extended tables; 
but as the reader has become familiar with our phraseology in what 
has gone before, he will be prepared to comprehend each step with- 
out special explanation. 
* We are indebted to the Smithsonian Institution for an advance copy of this article. 17 
As the tables for low water are constructed upon precisely the same 
plan as those for high water, we shall confine ourselves particularly 
to the description of the latter. 
§ 16. Having made the first reduction, as in § 7, separate the 
heights and intervals following the upper from those following the 
lower transit. Next separate each of these classes into two others, 
distinguishing those belonging to north from those belonging to south 
declinations. It will be seen after this that the tidal elements fol- 
lowing upper transits with south declinations are similar to those fol- 
lowing lower transits of north declinations, and those following loiver 
transits of south declinations similar to those following upper transits 
of north declinations. The same statement applies to the smaller tide 
and to the greater and smaller intervals. 
From the data, classified as we have directed, two final tables are 
made out. These two tables differ only in the general heading. We 
give the form of one of them below. 18 
Intervals and heights of high water for upper transit of south declination or loiver transit of north declination. 
Hours of transit. 
Moon's 
Days before maximum and after, 
declination, 
7 
6 
5 
4 
3 
2 
1 
0 
1 
2 
3 
4 
5 
6 
7 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 
Int. 
Ht. 19 
Into the above table are put the heights and intervals which fol- 
low the upper transit of south declination and the loiver transit of north 
declination, arranged according to hours of transit and days before or 
after the maximum declination of the moon. In a second table the 
heights and intervals following lower transit of south declination and 
upper transit of north declination are placed with similar arrangement. 
One of these tables will be found to contain the greater tide and 
the lesser interval, while the other will contain the lesser tide and the 
greater interval. 
If to these, two other tables of similar headings are added for low 
water, all the elements are complete. The navigator has then only 
to look in the almanac for the hour of transit, its kind, (whether upper 
or lower f) with the number of days from maximum declination, and 
then entering the table, he can at once reckon the time when the tide 
will occur, and its height.* 
The heights in these tables are measured from a fixed plane, which 
should be stated. Mean low water, as computed from the data in 
the first reduction, is the best plane of reference for practical use, 
since it is the datum for the soundings upon modern charts. 
§ 17. The highest high water and the lowest low water occur, usu- 
ally, when the moon's maximum declination happens at full or change. 
If the zero declination happens with the moon in quadratures, the 
nearly equal tides are at their least average. 
§ 18. The diurnal inequalities do not, at any station yet visited, 
vanish exactly at zero declination, but at some places they become 
insensible several days before and at others several days after the 
moon crosses the equator; in other words, the epochs of these in- 
equalities do not coincide with those of declination. Again, the 
epochs of the intervals do not correspond with those of the heights for 
either high or low water separately; but the intervals of high water 
seem to vary with the heights of loiv water, and the heights of the 
former with the intervals of the latter. These peculiar relations of 
epoch and magnitude can only be studied critically by decomposing 
the observed tide into its diurnal and semi-diurnal components-a 
matter to which we shall presently refer. 
* In the " Tide Table for the use of Navigators, prepared by Prof. A. D. Bache, Superin- 
tendent U. S. Coast Survey," a different order of tables is given to avoid the reference to the 
lower transit. As we have all along used both transits, we are able to indicate a less com- 
plex system without fear of misconception and without the long explanation which a de- 
scription of the method of computing the Coast Survey tables would require. Seamen use 
the word "southing" for the upper transit and "northing" for the lower transit in northern 
latitudes, but reverse those terms in southern latitudes. 20 
§ 19. When the inequalities are at or near their maximum, we find 
at the stations on the northwest coast of the United States that the 
greater fall is that which succeeds the greater rise, and the lesser rise is 
followed by the lesser fall. A rule of this kind, when stated distinctly 
and concisely, has great practical value for the navigator; it assists 
him in the use of his tide table, and enables him to foretell at what 
time he may expect the greatest or the least depth of water over 
bars, and whether for any length of time he is likely to have a favor- 
ing or adverse current. This rule for our northwest coast seems to 
hold, less constantly, for the few tidal observations which we have 
seen from the Pacific coast of Asia. 
In connection with the study of tides, we know of no field of in- 
quiry about which so little is known as this of diurnal inequalities. 
In the Pacific Ocean, even where the tides are of medium height and 
well defined, the variations in the relative positions and magnitudes 
of the diurnal and semi-diurnal waves from point to point are not com- 
prehended. From a few stations we have ample data, so that accu- 
rate local tide tables have been drawn up; but until we have obser- 
vations from many other points we cannot hope to possess a connected 
view of the phenomena. 
§ 20. In the present state of our knowledge information is wanted 
from the tropical islands of the Pacific and from the extreme northern 
portions of the same ocean; also from the .coasts of Mexico, Central 
America, and the southern borders of the Caribbean Sea. 
In the attempt to trace the march of the phenomena connected 
with the diurnal inequalities, it has been found a great advantage to 
separate the two elementary waves which compose the observed tide, 
and study their changes independently. 
§ 21. Let us suppose that we have a connected series of very fre- 
quent observations for equal intervals of time extending through three 
lunar days, and that we designate these days as A, B, and C. We 
fiist take the mean of all the heights of each lunar day to find the 
mean level) then take out the ordinates of the tidal curves for each 
inteival as referred to this axis for each day. If now we move the 
ordinates of A a half lunar day forwards, and those of C the same 
amount of time backwards, and add the corresponding ordinates to 
those of B, we shall have a column of sums which, divided by two, 
will give the ordinates of the semi-diurnal waves for the day B. By 
inspecting the diagram given in § 4, it will be readily seen that by 
this process the diurnal wave is eliminated. 
M e use lunar days and lunar hours because they represent equal 21 
intervals of tidal time very nearly. A lunar day is the time elapsed 
between two successive transits over the same meridian, and a lunar 
hour is one twenty-fourth of this period. 
In order to avoid the half-monthly inequality in this method of de- 
composition, we move each set of half-day ordinates both forwards 
and backwards, obtaining in this way two sets of semi-diurnal waves, 
of which we take the mean. 
As the mean level alters a little from day to day, it is well to re- 
gard our determination of it as the position for the middle of each 
day, and interpolate for the hours intermediate between one midday 
and the next, so that our axis of ordinates becomes an inclined or 
curved line. 
Having obtained the ordinates of the semi-diurnal waves for any 
day, we have simply to subtract these from those observed on that 
day to obtain the ordinates of the diurnal wave. 
By plotting the observed tides just as they are taken from the field 
book, and shifting the curves backwards and forwards with tracing 
paper, this method may be made a graphical one. In this case we 
need not compute the mean level, and we may regard half the time 
between one low water and its correspondent of the next day as a 
tidal half day.* 
Tides of the Diurnal Type. 
§ 22. Tides of the diurnal type are usually small, and it is not, 
therefore, necessary to draw up elaborate tables for the use of navi- 
gators. The first reductions should be made in the usual manner both 
for high and low water, with columns of moon's declinations appended. 
In these reductions the transit used is shifted on passing the date on 
which the tide vanishes. If the superior transit has been used with 
north declinations, the inferior transit is to be used for south decli- 
nations, and vice versa. * 
The diurnal tide follows the moon at a decreasing interval. If we 
were to reckon by lunar days and lunar hours, we might say that the 
diurnal tide comes on earlier each day than on the day previous. 
The lunitidal interval is nearly constant for a few days at or near the 
time when the tide has its maximum range, (usually when the moon 
* This direct and simple method of decomposing tides is essentially the same as that de- 
vised by L. F. Pourtales, Assistant U. S. Coast Survey. In his own work this gentleman 
prefers the graphical mode to the numerical. See description in Mr. Schott's article on 
''Tidal Observations in the Arctic Seas," Smithsonian Contributions, 1860. 22 
is at or near her greatest declination.) The greatest change of in- 
terval occurs on the day when the tide vanishes or passes its minimum 
range. At this time the variation is so great that the reappearing 
or newly growing wave seems to follow a preceding transit. By 
shifting back one transit for each reappearance, we are able to obtain 
similar intervals for the maximum tide of each half month. 
§ 23. It is not a settled question whether the diurnal tide is contin- 
uous or actually evanescent. In the curves plotted from observations 
upon the northwest coast of America, the diurnal wave does not be- 
come insensible. Its range gets very small near the time of zero de- 
clination, and the point of culmination is ill defined, but the diurnal 
vertical change is always to be traced.* 
§ 24. The information which the navigator requires relative to 
tides of the diurnal type is the following: 
1st. The number of days before or after maximum declination at 
which the greatest tide of the half month occurs. 
2d. The average range of this "greatest" tide. 
3d. The lunitidal interval of this " greatest tide," and the daily 
variation of the interval for two days before and two days after. 
4th. The number of days before or after zero declination at which 
the tide becomes insensible or passes its minimum. 
The value of the first three data is obvious. The fourth is also im- 
portant, since it furnishes to the navigator the dates of the tideless 
sea when the water remains nearly at mean level, and neither favoring 
nor adverse currents are to be expected. 
§ 25. By a coincidence, the high water of the greatest diurnal tide 
falls nearly in the same sidereal hour, whenever it occurs, at the same 
place; so that a rule for the time of the tide may often be given 
which shall refer it to the southing of a familiar star or constellation- 
a rule like the following: 
"At this pert the greatest tide of each half month occurs (three) 
days after date of moon's maximum declination with its high water, 
at the time when (the belt of Orion) crosses the meridian."! 
§ 26. For the proper study of tides of the mixed and diurnal types, 
frequent and continuous observations are necessary. It will not suffice 
to observe only the high and low waters, even if these are distinct 
* Since there is a solar as well as a lunar diurnal tide, an absolute evanescence can very 
rarely occur. 
t Pourtales' Rule. 23 
phases, but the whole tidal curve for each day should be developed 
by recording the heights at least every fifteen minutes. Series of 
observations of less than a month are of little value for practical use 
in drawing up tables and rules for navigators, although they may 
serve, as we have before said, to indicate to what type the tides of a 
place belong. In classifying tides according to type, no very exact 
rule can be followed. We should call those tides semi-diurnal in 
which the maximum difference between successive high waters does 
not exceed the difference between springs and neaps; and those 
diurnal in which the existence of two tides in the same day cannot 
ordinarily be traced. From the data thus far collected, it would 
appear that the tides of our globe may, for the most part, be classed 
with the mixed type; notwithstanding that those of the Caribbean 
Sea, Gulf of Mexico, and China Sea are diurnal, and those of the 
Atlantic semi-diurnal. In the central islands of the South Pacific one 
station at least (Tahiti) is visited by semi diurnal tides with the anomaly 
of an excess of the solar wave, so that, for the larger part of the year, 
high or low water occurs about the same hour of day; springs and 
neaps, however, are well marked.* 
Mean Level. 
§ 27. We have already used the term mean level in § 21, and ex- 
plained the manner of computing it for one lunar or tidal day when 
continuous observations at short intervals of time have been made. 
It is the plane from which the tide waves rise and fall, and corre- 
sponds nearly with half tide, as this phrase is ordinarily used, if the 
diurnal inequality is insignificant. If the observations comprise only 
the heights of high and low water, and considerable inequality ap- 
pears, the following method, employed in the discussion of the Arctic 
observations, (already referred to in § 15,) may be used: Having 
plotted the observations upon profile paper, take the mean of two 
successive high waters and plot with the proper ordinate directly 
over the intermediate low water. Repeat this till a point appears 
over each low water of the series; then through these points draw a 
line-this will be the line of average high water. Obtain the line of 
average low water in a similar manner-the points representing the 
* We refer here to the observations of Captain John Rodgers, U. S. N.-a remarkable 
series covering six months. If the opposite six months could be added to this series, the his- 
tory of the solar tide would be complete. 24 
mean of successive low waters being plotted directly under the inter- 
mediate high water. A line drawn midway between the two already 
described will be mean level. With the proper care in the grouping 
of the figures representing the averages, this method of obtaining 
mean level may be used numerically, and the diagram dispensed 
with. 
§ 28. The plane of mean level is subject to changes of elevation, 
which appear to be independent of the tidal range, except in shallow 
places where the low water is considerably more obstructed as a 
travelling phase than high water. Not only does this plane remain 
at the same elevation for springs and neaps, other things equal, but 
it is known to be nearly the same for different derivations of the tide 
wave in several remarkable instances. The British Association as- 
certained from levelings between stations on the Bristol and English 
channels that the mean levels for these arms of the sea differ in height 
but 9 inches, although the mean tidal ranges are feet in one case, 
and 10 feet in the other. The writer of this article, in a recent phys- 
ical survey of the approaches to the site of the proposed Cape Cod 
canal, determined that mean level during the dry season is essentially 
a common plane for Cape Cod Bay and Barnstable Bay, notwithstand- 
ing that the tidal range of the former is double that of the latter, and 
the establishments of high and low water, respectively, three and 
four hours later for the former than for the latter. These two tide 
waves were also observed upon at several points on the inside and 
outside shores of the islands of Nantucket and Martha's Vinyard, 
and the stations connected by lines of levellings which gave the same 
plane for mean level.* 
Results quite at variance with those we have given were obtained 
by Mr. Airy from a discussion of the tides observed upon the coast 
of Ireland by the Ordnance Survey. If the levellings used by Mr. 
Airy are reliable, a variation in the height of mean level exceeding 
two feet may occur in the propagation of the same derivation of the 
tide wave in a distance of fifty miles.t 
We have mentioned these results from different inquiries to show 
the want of further knowledge In this matter. In the location of 
railroads and canals, opportunities are offered for connecting different 
harbors and bays by lines of levels; and if, in these cases, the benches 
of tidal stations are referred to, and more than usual care exercised 
* Appendix No. 37, Coast Survey Report, 1856. 
t Mr Airy on the Laws of the Tides on the Coast of Ireland. Phil. Trans., 1845. 25 
in the levellings, very useful and important information may be 
gained. 
§ 29. Theoretical investigations upon the laws of the tides show 
that periodical variations of mean level may be expected, dependent 
upon the declinations of the sun and moon; and those dependent upon 
the moon certainly appear at some of the stations at which careful 
series of observations have been made. At Plymouth, England, and 
at Boston, U. S., averages of many observations give about three 
inches rise in the height of mean level as the moon's declination 
increases from 0 to 25°, (north or south.)* Prof. Bache, in a discus- 
sion of the tides of Key West, (Coast Survey Report, 1853,) discov- 
ered half-yearly and half-monthly variations following, respectively, 
the declinations of the sun and moon, but in the reverse order, the 
greatest altitudes being reached at zero declinations. There are 
many stations at which no variations of this class can be distinctly 
traced; and unless the computer is furnished with at least six months' 
observations, he can scarcely tri st the results he may obtain except 
as the most general indications of the existence and character of such 
variations. The uncertainty which attends the investigation of these 
small periodical oscillations arises chiefly from the difficulty of elimi- 
nating such irregular causes of disturbance as winds and barometric 
changes. 
§ 30. In deep and open seas it is supposed that the winds produce 
only short undulations, but upon the boldest coasts long-continued 
gales from seaward raise the general level of the water sensibly. In 
shallow districts, in sounds and bays where the depths are inadequate 
for the restoration of equilibrium by the retrograde motion of par- 
ticles lying below the influence of the winds, the water is actually 
driven forward and accumulated to leeward; and although at most 
tidal stations the irregularities of mean level caused by winds are 
often so large as to be detected in a general manner by the most cur- 
sory inspection of the data, they offer to the computer great perplexi- 
ties. These effects are not usually eliminated by such a grouping of 
the observations as will cause the recorded velocities and directions 
of the wind to cancel, because winds from certain directions act at 
much greater advantage than those from others. Abay, for instance, 
may be sheltered by hills from off-shore winds, but particularly ex- 
* The results from Plymouth observations are to be found in the Phil. Trans., 1839; those 
from Boston in unpublished computations of the Coast Survey Office. 26 
posed to winds from seaward, so that the observer's record cannot 
be relied upon to represent accurately the forces which act upon the 
bay as a whole. 
Observations of long duration are requisite for the proper measure- 
ment of the effects of winds; because the computer should be able 
to group, first, the data from a selection of quiet days, and, after this, 
the observations from dates on which strong winds prevail from each 
quarter separately; and yet, in all these groupings, there should be 
data enough to cause the other inequalities to cancel. In working 
up short series of observations for tide tables, &c., the tidal records 
made during storms should be rejected. 
§31. The mean level is observed to undergo changes which resem- 
ble in character, and inversely correspond with, those of the mercury 
in the barometer. These changes are generally attributed to the 
unequal and varying pressures of the atmosphere upon the ocean. 
With this theory of the case, it may well be doubted whether the 
local barometer and the simultaneous reading of it are proper criteria 
for the pressures under which the changes in the elevations of the sea 
occur at a tidal station; but the comparisons have thus far been made 
in this way, and with pretty good agreements for different series at 
the same place, but with great discrepancies when results from differ- 
ent points have been compared. At Brest 1 inch fall in the mercurial 
barometer is attended by 16 inches rise of mean level, and the rise 
of the barometer by a fall of mean level in the same ratio. At Liver- 
pool the ratio is 1 to 10; at London 1 to 17, &c.* Here we observe 
that stations not far apart give different results, notwithstanding that 
tidal series of many years' duration have been used in each case. 
With the theory of unequal pressures of the atmosphere we have to 
conceive of waves of unequal length, propagated, therefore, with un- 
equal velocities, and differently modified by resistances, so that the 
ratios might be expected to vary from time to time and from place to 
place. 
§ 32. Aside from the fact that these barometric variations of mean 
* Results given by Captain J. C. Ross in a paper " On Effects of Barometric Pressures on 
the Mean Level of the Sea." Phil. Trans. R. S. At Port Foulke, Greenland, the ratio is 1 
to 4 nearly. 
The Peninsula of Aliaska, and the Aleutian Islands, which separate the Sea of Behring from 
the Pacific, offer positions where (witA the barometers on either shore in agreement-because not 
far apart) the effects of atmospheric pressures upon different oceans can be compared, and 
the propriety of referring these effects to the local barometer ascertained. 27 
level are often too large to be neglected in tidal computations, the 
peculiar interest which attaches to them, in connection with meteor- 
ology, would seem to recommend them to special investigation. There 
are some familiar meteorological facts and theories that have peculiar 
significance in this connection. The oscillations of the barometer are 
much greater near the sea than elsewhere, and at many points on the 
coast a great rise or fall of the barometer is a forerunner of a long 
blow from a certain direction. Meteorologists conceive that the 
equilibrium of the atmosphere is restored by compensating currents 
which lie side by side in the temperate zones, but one above another 
in the torrid zone. All of the long series from which these varia- 
tions of mean level have been thus far computed lie in the North 
Atlantic. 
Barometric changes of mean level should not only be studied from 
observations at single stations, but from comparisons between stations 
simultaneously occupied, in order that the local features may be 
evolved and the general features connected. Comparisons between 
stations but a few miles apart on the coast would be useful for tracing 
individual cases, and the mean results from stations widely separated 
in latitude are much wanted for general comparison. Again, the 
variations for rising and falling stages of the barometer should be 
separately compiled and then compared. 
The mean level rises from station to station as the point of obser- 
vation moves up a tidal river, giving, when determined, the true 
slope of the stream precisely as it would have been if no tides trav- 
versed it. 
§ 33. In all projects for the improvement of tidal harbors the mean 
level is an important datum plane, so that a careful determination of 
it is indispensable to the engineer. It is for him the proper plane 
of reference for the soundings and the tides, especially in places 
where no fresh-water feeders exist, since it is not only more general 
and permanent, but more easily obtained and recovered than any other 
stage. In preparing charts for navigation, however, the plane of 
low water has been found most acceptable to the pilot who chiefly 
desires to know the minimum depth under ordinary circumstances. 
For the charts issued by our government, the plane of reference is 
the mean of all the low waters observed if the tides belong to the 
semi-diurnal type, and the mean of all the lower low waters if the 
tides belong to the mixed type. It would be a good custom to give 
upon all charts the height of mean level. In the case where the tides 
belong to the mixed type, the few tidal observations that can be made 28 
during the progress of the survey are usually insufficient for the ac- 
curate determination of the low-water reference, but might suffice 
for obtaining the mean level very closely. In places where the tides 
are diurnal, there are several days in each fortnight when the surface 
of the sea remains at or near the mean level. In this case a state- 
ment of the position of this plane would enable the navigator to cor- 
rect his chart accurately on certain dates by applying the constant to 
each sounding. 
Bench Maries. 
§ 34. At each tidal station a bench marie should be made, to which 
the zero of the tide gauge should be referred. If, in the neighbor- 
hood, there is a solitary sharp-topped rock, known by name, or easily 
described, which is only covered at a certain stage of the tide, it 
may serve as a bench if the observer records its position, name, and 
the height of the water upon his gauge at the instant when its high- 
est point disappears. If the waves make this record a little uncer- 
tain, the observation should be repeated on several quiet days. A 
bench of this kind should not be too distant from the shore, because 
such marks are often used in the location of railroad tracks, canals, 
and other structures where levellings are required. 
Where a bench is made on shore it should be marked by a circle 
of two or three inches diameter with a cross in the centre, indicating 
the reference point. The levellings between this point and the gauge 
should be run over twice, and the details recorded. A bench made 
upon a wharf or other perishable structure is of little value, but in 
the absence of permanent objects it is better than nothing. Portions 
of piles, never exposed above water, remain sound for many years. 
In all cases the marks should be cut in as deep as convenient, if on 
stone, and if on wood copper nails should be used. 
§ 35. A bench mark should always be made in connection with a 
hydrographic survey. It affords the means of recovering at any 
future time the exact plane of reference of the soundings, and makes 
the chart valuable for comparisons when the discovery of new rocks 
or the formation of new shoals makes a partial resurvey necessary. 
Many of the old charts, bearing evidences of careful work, are now 
useless for comparison because the reference planes for the soundings 
cannot be recovered. 
A bench mark has value from a geological point of view. In 
many parts of the world the relative elevations of the land and sea 
are changing by the gradual rising or subsidence of the former; and 29 
notwithstanding that these changes are usually very slow, they have 
already been accurately measured in some parts of Europe. 
Propagation of the Tide Wave and the Changes of its Figure. 
§ 36. Wherever upon the same coast the observations have been 
carefully made at several stations equally exposed to the sea, there 
has been discovered a connection or mutual relation between the 
tides of different points, so that they could be recognized as pro- 
gressions or propagations from a single-wave system. At such 
stations the visits of the tide preserve nearly the same relative order 
of times, indicating that the approach of the wave is always from the 
same direction. We may conceive of high water as the crest and low 
water as the hollow of this great undulation. If upon any straight 
shore the high waters occur at the same time from point to point, we 
conclude that the direction of the tide is at right angles, or, what 
amounts to the same, that the wave advances with its crest parallel 
to the beach. If, on the other hand, the times of high water are 
later and later from point to point on this shore, we conclude that 
the approach is oblique to the coast line. It is by comparisons of 
this sort that the positions of the crest lines of the tide wave for 
stated hours have been plotted upon charts, under the name of 
cotidal lines, by Young, Lubbock, Whewell, and Bache.* 
These charts of cotidal lines may be regarded as essentially com- 
plete for the shores of Europe and of the United States, but for other 
parts of the world comparatively little has been done. Indeed, the 
cotidal lines already determined furnish little or no clue to the course 
and rate of travel of the ocean tide. Even for the North Atlantic 
no connection has yet been discovered between the tides of the 
opposite coasts, and the absence of islands to furnish the intermediate 
links makes the problem a very difficult one. The Pacific Ocean 
offers a better field for efforts of this kind, because, in its broadest 
portion, the numerous islands may furnish stepping stones for the 
inquiry. It is probable that the tides in the middle of this ocean 
are very small and much disturbed by prevailing or periodic winds, 
but these things should not discourage the efforts of the inquirers. 
As the tide approaches a continent it is retarded; and its length, 
although still essentially the same as measured in time, is much re- 
duced in space upon the earth's surface. 
*They were first suggested by Lord Bacon in an essay called "The History of the Wind." 30 
The velocity of the tide wave is dependent, almost exclusively, 
upon the depth of water in which it is propagated; but the changes 
of range appear to depend less upon variations of depth than upon 
those of width. 
§ 37. The true velocity of the tide wave is that of its crest. Com- 
parisons of the times of high water along the tidal path show fewer 
inconsistencies with each other, and with theory, than those of low 
water. The cotidal lines drawn from the -mean lunitidal intervals of 
high water (using the same transit over the same meridian) seem to 
confirm Mr. Airy's rule that "the velocity is the same as that which a 
free body ivould acquire by falling from rest, under the action of gravity, 
through a space equal to half the depth of the tvater.' ' * 
As the tide wave approaches a shallow coast its range is, as a rule, 
slightly augmented, and this augmentation is perceived to bear some 
relation to the diminution of distance between the cotidal lines; but 
where the wave enters a bay or arm of the sea whose shores con- 
verge, the increase of range is often very great, and in rivers an 
increase of range is often found at points where the area of the cross 
section lessens. The contrasts of range which the tides of different 
ports present, however, are not fully accounted for. A very interest- 
ing and instructive method of pursuing this inquiry is to prepare 
charts with lines of equal range. This has been done for our Atlantic 
seaboard, and the dependence of the heights upon peculiarities in the 
configuration of the coast well illustrated.f 
§ 38. If, by simultaneous observations at many different stations, 
an entire tide wave could be followed through an arm of the sea of 
which a thorough survey has been made, it might lead to a better 
understanding of the circumstances under which the changes of range 
occur. We should be able to compare the volumes from point to 
point with the widths, depths, sections, &c., at each stage of the 
tide. We have not yet obtained the history of an entire tide wave, 
except in certain rivers-and bays where the conditions are compli- 
cated. 
§ 39. It has been ascertained from the little data we possess, that, 
in localities where the diurnal inequalities are large, on passing 
through straits or into arms of the sea, the two elementary waves, the 
* "Tides and Waves." Enc. Metropolitan a, page 292. 
t See paper "On the heights of the tides of the Atlantic Coast of the U. S., from observa- 
tions in the Coast Survey, by A. D. Bache, Supd't." Proceedings of the Am. Association of 
Science, 1857. 31 
diurnal and semi-diurnal, alter their relative positions and ranges in 
a manner that thus far seems to baffle inquiry. Several suppositions 
may be made respecting the causes of these changes, and we shall 
mention the most probable of them in order to explain some of the 
suggestions we shall have to offer for the manner of pursuing the 
investigation. 
1st. Since the waves have different masses, they may be supposed 
to be differently affected by resistances-the diurnal may outlive the 
semi-diurnal and appear without its companion; or the same causes 
may augment one more than the other. 
2d. The two elementary waves may have different rates of travel 
as the wind waves are observed to have, which would alter their 
relative position from point to point, and also give contrast to the 
effects of resistance. It would appear from the observations made 
in Puget Sound, that in this place the diurnal waves travel with 
double the velocity of the semi-diurnal. Within certain limits the 
length of a wave enters into the function of velocity. The diurnal 
wave is the longer, and in a shallow sea might be expected to pre- 
serve a greater velocity than the semi-diurnal. 
3d. Two branches of the tide may interfere in such a manner as to 
reduce one of the elementary waves more than the other. Let us 
instance an extreme case: suppose two equal tides, whose intervals 
differ six hours, should meet; the resulting tide would be diurnal in 
character, because the semi-diurnal waves would cancel each other, 
the crest of one falling directly upon the hollow of the other, and a 
compound of the two diurnal waves would remain. 
If the change of diurnal inequalities occurs high up in a bay or 
in a lagoon, make several stations, one of which should be at the 
mouth, another at the head, and others intermediate. These should 
be simultaneously occupied, and the results arranged for decomposi- 
tion. When these decompositions are made, observe whether one of 
the elementary waves steadily decreases, or is retarded, or is dis- 
torted. Ascertain the depths along the channel-way, and with these 
depths and the distances between the stations endeavor, by repeated 
trials, to discern relations between them and the tidal changes ob- 
served. If the bay or lagoon is very broad, but with only a narrow 
inlet, both of the elementary waves will undergo degradation ; that is 
to say, they will be flattened out laterally so as to have less range. 
Observe whether the shorter wave suffers more degradation than the 
longer. By watching the ordinary wind wraves as they enter lagoons, 
one may notice that the long swells of the ocean penetrate much 32 
farther before becoming insensible than the short ones. Even on 
the supposition that the diurnal and semi-diurnal waves are not dis- 
tinct undulations, it is obvious that in a lagoon the elevations and 
depressions will represent most conspicuously those features of the 
tidal figure which are of the longest duration, other things being 
equal. An elevation of the sea above its mean level for six hours 
may not raise the surface of a lagoon perceptibly, because the inlet 
may be too small to admit a sufficient supply of water in this short 
period of time, but a rise of double this duration might, perhaps, 
sensibly elevate the water in the basin. We may conceive an ex- 
treme case where the surface of an extensive lagoon is unaffected by 
the tides of a single day, but which, nevertheless, rises or falls 
gradually with the mean level of the ocean as the moon's declination 
increases or decreases. 
If the peculiar change of diurnal inequality is observed in a strait 
having two entrances from the sea, make simultaneous observations 
at three stations-one at each entrance and the third midway. Then 
ascertain by trials whether the intermediate tide is a compound of 
the extremes. The first step in this undertaking will be the reduc- 
tion of the three tides to the same zero of times and heights. The 
mean level of the sea may be assumed to be a plane common to all, 
and the ordinates referred to this axis taken out for the same equal 
periods of time for the several stations. The simple addition of the 
corresponding ordinates of the extreme stations, with due regard to 
signs, will give a compound curve, which should be compared with 
the middle tide observed. It will probably differ considerably in 
form, because it is hardly likely that the two waves approaching 
from different directions can, in their journeys from the entrances to 
their meeting point, have suffered equal delays and changes of range. 
This comparison will suggest what constant of time and coefficient of 
height may be required by one or both of the components to make a 
combination more nearly resembling the middle tide observed. This 
amounts simply to shifting the relative positions of the extreme tidal 
curves upon their axis of abscissae and multiplying the ordinates of 
one or both by a whole or fractional quantity, and computing success- 
ively different compound curves for each change. By repeatedly 
subtracting the newly computed compounds of the extreme tides 
from the observed intermediate tide we may obtain at last a combi- 
nation giving the least possible residual; and if this residual, upon 
plotting, indicates no recurrent forms, and is very small, we conclude 
that we have decomposed the interference tide and fully accounted 33 
for its peculiarities. In this manner of resolving interferences, we 
are not obliged to decompose the extreme tides into diurnal and semi-' 
diurnal waves, so that we avoid many perplexities. 
In the cases of interferences which have been examined on our 
eastern coast, where the tides are semi-diurnal in character, the 
observed interference wave has been found to bear evident traces on 
its profile of its twofold character. The meeting waves are usually 
of dissimilar heights, and their difference of outline, hardly percep- 
tible when single, is made evident and exaggerated by combination. 
There are four high waters each day at the meeting point of the tides 
near the junction of the Nantucket and Vineyard Sounds, and these 
are distinctly exhibited in the profile of the observed tide. A com- 
bination has been made of the observations at the extreme limits of 
these sounds which compares well with the observed tide at their 
meeting place. In this instance a fractional coefficient of heights had 
to be applied to each of the simple waves to make their compounds 
agree with observations; that is to say, by the local action of gravity, 
the contrasts of heights, offered by the two tides, were found to be, 
in part, reconciled. 
§ 40. The tidal curve undergoes changes of figure after leaving the 
open sea and passing into harbors or rivers, and these changes have 
been studied at places where improvements to navigation or the 
reclaiming of lands have given practical value to the information. 
In some instances the distortions of the tide are very marked, and 
afford a curious insight to the physical relations existing between it 
and the avenue traversed; so that information of this kind has often 
more than local value.* 
Before we proceed to speak of the manner of exhibiting these 
changes, it is necessary to define the terms which we shall have 
occasion to use. Let us suppose that we have plotted upon profile 
paper a day's tides of the semi-diurnal type. If we draw the line 
of mean level across our curves, we cut them into two portions-positive 
and negative waves. The positive wave lies above the mean level and 
culminates in high water; while the negative wave lies below the 
mean level, and its apex is low water. That portion of the line of 
mean level which lies between the.rising and the falling curve is 
*For method of combining the curves of different dates, so as to ascertain the average 
form, and represent this by a mathematical expresion, see Airy, Tides and Waves, § 479; 
and in this connection see also "Development of Bessel's Functions for the Effect of Periodic 
Forces." Appendix No. 22 of Coast Survey Report of 1862. 34 
called the diameter. The distance between two successive low waters 
as we have already stated, is called the length; it is measured in time 
along the base of the tidal figure as we plot it. Lines parallel to 
the diameter are called chords, and a line drawn down from high 
water so as to bisect the chords, the diameter, and the base, is called 
the vertical axis. 
§ 41. The figure of the normal tide, that observed near the open 
coast, is nearly a curve of sines from a circle of the same diameter 
whose radius is half the entire range of the tide. This statement 
applies to the tide at the time when no diurnal inequality exists, or 
to the decomposed waves at other times. 
Let us suppose the diameter of a certain tide to be divided into six 
equal parts which we may call tidal hours, (they will be a trifle over 
one clock hour each,) then the ordinates of the theoretical curve will 
be simply four repetitions of the multiples of the sines of 0°, 30°, 
G0°, and 90°, the first and last set having the minus sign. If the half 
range of the tide is unity, then the theoretical tide will be- 1.00, 
(L. W.,) - 0.87, - 0.50, 0.0, + 0.50, + 0.87, + 1.00, (H. W.,) 
4- 0.87, + 0.50, 0.0, - 0.50, - 0.87, - 1.00, (L. W.) 
A mere inspection of the plotted curve of observations will usually 
show whether it approximates to the normal form. If the profile of 
the tides corresponds with the curve of sines exactly we shall have the 
following results: The rise and the fall will be reciprocals, the dura- 
tions and the rates of rise and fall for corresponding stages will be 
equal, and the chords in each case will be bisected by the high-water 
ordinates. The tide upon the open coast answers to these tests un- 
der favorable circumstances; but an advance of only a few miles into 
a bay or harbor usually changes them all in a remarkable manner. 
The fall is no longer the reciprocal of the rise; the maximum rates of 
both rise and fall are obtained later-that is to say, at greater interval 
after high or low water-and the chords are no longer bisected by the 
high-water ordinate. Moreover, if we obtain data from several sta- 
tions at different distances from the open sea, we may follow the pro 
gress of these changes somewhat as follows: the length of the wave, 
measured in time, remains pretty much the same, but the lengths of 
the chords grow less; in other words, the wave sharpens and the axis 
becomes inclined to the horizontal. This last change may be stated 
in other words, thus: the positive portion of the wave preserves more 
of the original impulse and hastens forward at a greater speed than 
the negative. By plotting the tide and drawing the chord, say for 
every vertical foot of the entire height, then drawing the axis, this 
tendency of the tide wave may be exibibited to the eye. It is essen- 35 
tially the same change as that which we observe in ordinary waves 
upon the sea shore. Their crests appear to hasten forward as they 
come upon shallow ground, the front of the wave gets steeper, and 
ultimately the crest falls over, forming a breaker. The phenomenon 
called the bore is supposed to be a similar action of the tide, but we 
have no careful observations to show this. 
The sharpening of the tide wave is often observed at points nearer 
the sea than those which give a reduction of range; showing that the 
order of the influx or reflux is altered, but not the total effect. Some 
quality of the channel is accountable for this change. 
§ 42. The changes in the figure of the tide to which we have 
referred are those which seem to be caused by the resistances of 
shallow and confined paths, but the precise manner in which these 
causes act is not known. In some bays which have few or no fresh- 
water feeders the rise of the tide occupies a longer period than the 
fall. Whether this is a general rule for bays of this character we are 
unable to say-it would seem opposed to the idea we have formed of 
a shoreward impulse of the tide wave. It may be, however, that the 
inclined bed of a bay resists the influx but favors the reflux of the 
tide. In rivers the opposite relations for durations of rise and fall 
obtain. 
§ 43. It will sometimes be found, on analyzing the tide, that the 
vertical axis is curved out of place near the middle of the figure. 
When this is the case, inquire if any obstacle lies in the path of the 
tide, and whether this obstacle affects the rise more than the fall. Tides 
above bridges have their vertical axes curved as we have described, 
and the amount of disturbance caused by a bridge has been measured 
in this way.* 
We speak of any tidal phase as an approaching phenomenon, and 
we speak correctly, for the times of both high and low water are 
later as wTe go up a bay or harbor; but we are not to conceive that 
the water particles are always pursuing a single direction ; on the 
contrary, they approach the land on the flood and recede from it on< 
the ebb. It is no doubt true that the tide wave, in mid-ocean, is. 
propagated without requiring any essential alteration in the geograph- 
ical position of water particles, it being simply necessary that each 
particle should move in an ellipse whose axes measure but a few 
feet. 
The best idea that we can present of the tide of a harbor is that 
* Tenth Report of U. S. Commissioners on Boston Harbor. 36 
of a bodily movement of water under pressure. When, on a rising 
tide, the sea without is higher than that of a harbor, a grand move- 
ment of the water takes place, a pushing in to restore equilibrium. 
In this operation a single water particle may have to make a consid- 
erable advance, but the motion is communicated to the particles 
beyond by impact, so that it is not necessarily a pouring in of an 
entirely new supply, although, in small creeks, it degenerates into this. 
So long as there remains the smallest horizontal film of water in a 
channel the tidal propagation does not wholly degenerate into a cur- 
rent-the phases will still travel faster than the water particles. High 
water and low water are phases that are always moving up a river, 
but the current may never actually cease to run down. This current 
must, however, receive at least a check before any rise can take place, 
and this check causes the "backing upwhich is the usual phrase 
applied to river tides. 
§ 44. In the tides of harbors or confined channels, the greatest 
activity among the water particles is excited at or soon after half 
tide, because then the ocean rising or falling most rapidly creates the 
greatest disparity of heights between itself and the basin, and induces 
the most powerful efforts of gravity to restore equilibrium. If the 
axis of the tide is thrown out of place near the plane of mean level, 
the observer will at once perceive that something obstructs the free 
passage of the water and breaks up the continuity of its transmitted 
motion. 
§ 45. The tides of rivers are but imperfectly understood. Their 
peculiar characteristics maybe summed up as follows: 
The duration of the fall is much greater than that of the rise; or, 
in other words, the phases of the rise travel up the stream much more 
rapidly than those of the fall, if the inland or fresh-water supply is 
considerable. 
The ranges of the tide vary with the width, depth, and irregular 
course of the channel, but in a manner not yet reduced to rules. 
The mean level rises as we go up a tidal river, but to what limit is not 
known. Instances are known of rivers up which the tides advance 
till the low water of the upper reaches attains the same absolute 
height as the high-water level at the mouth. 
In some rivers double tides occur, giving two high waters within 
a few hours of each other, with an intermediate fall of small extent. 
In some rivers the tides at certain intermediate stations are ob- 
served to have greater ranges than at points above or below, where 
no causes connected with form or dimensions of the channel are 
reported. 37 
River tides should be studied from simultaneous observations at 
different points, and careful gaugings of inflows and outflows. Many 
of the phenomena connected with the varying ranges may be traced 
to alterations in the length of the base, and the chords of the tidal 
figure measured in space as well as in time. If the stations are suffi- 
ciently numerous and well disposed, profiles of sections along the 
channel-way can be plotted for every hour, so as to reveal the experi- 
ences of every phase of the rise and the fall. If changes in the rate 
of travel for different phases between the same stations are observed, 
they may be traced to the alterations of depth and width of channel 
due to the tide itself, or they may result from the setting back or 
return of the current higher up the river. A relation may, perhaps, 
be found between the slope of the river bed and the rate of travel- 
a comparison of the rising with the falling phases may be suggested 
in this connection. 
The tides of high stages of the river should be compared with those 
of low stages. If the sea water underruns the river in dry seasons, as 
it does in the Hudson for many miles, some curious facts may be 
evolved from this comparison, especially as regards the surface slope 
and that of mean level. 
Observations of river tides should be, when convenient, accompa- 
nied by current observations at different depths, and by hourly trials 
of specific gravity at the surface and near the bottom. A complete 
history of a tidal river has not yet been compiled, and till this is done 
by patient labor, there is little hope that the various phenomena will 
be connected and interpreted. 
§ 4G. There are some facts easily collected with regard to rivers 
which are always of practical value, viz: The farthest point reached 
by the tide at springs and at neaps, in dry and in wet seasons; 
whether at any former period-before the building of a bridge or 
the formation of a sand-bar or other encroachments-the tide was 
felt higher up the stream. The distance to which the flood tidal 
current and slack waters extend during the dry and during the wet 
seasons. The point where the ranges begin to lessen, and the average 
rate of this decline above. The difference of mean level between 
the lowest and the highest fresh-water stages at several points, and 
the approximate rates of outflow. The limits of salt or brackish 
water in wet and in dry seasons. The limits of ice, duration of its stay, 
usual time of breaking up, danger to vessels at anchor, &c. 
§ 47. At some stations a singular indenture occasionally appears in 
the tidal figure about the time of high water-instead of a stand, there 38 
is a falling away of the water for a brief interval, and then a return to 
about the same height. Interferences sometimes produce such effects, 
but these are repeated from day to day. Where the phenomenon is 
only occasional, it may be the effect of long swells from sea which be- 
come conspicuous near the stand, but would not be noticed during the 
rise or fall. On our Atlantic coast this indenture of the profile is only 
observed at stations very much exposed. Under ordinary circum- 
stances the ocean is not traversed by any waves intermediate in mag- 
nitude between the tide and the "rollers" which break along our 
shores, and the comparative lengths of these two wave systems may 
be judged from their times; the tide passes any given point once in 
twelve hours, while the time of the ordinary swells at the same point 
will not exceed as many seconds. 
At rare periods the sea is troubled by earthquakes, which create 
undulations which, once in motion, continue for several days, and 
extend perhaps from continent to continent. These waves are too long 
to produce any sensible roughening of the sea at a distance from 
their origin; but they discover themselves by the unusual elevations 
and depressions of the "water in quiet ports-not perhaps unusual in 
the matter of height, but more especially in that of duration.* 
Now that permanent tidal stations are kept up on both the 
Atlantic and Pacific shores of our country, the visits of these strange 
waves cannot escape critical observation; and if they are accounted 
for by distant earthquakes of which the time and place are reported 
by voyagers, they may lead to much knowledge of the character of 
such undulations, their rate of travel, longevity, depth of the ocean, 
&c. 
Tidal Currents. 
Tidal currents are rarely sensible far from land, since they are 
called into activity by the delays and the distortions of the tide wave 
consequent upon the resistances of shoals, or confined and shallow 
* The earthquake waves of December, 1854, which were observed on the gauges of the Coast 
Survey on our Pacific coast, were perhaps the longest swells ever measured, yet the longest 
of these did not exceed 35 minutes of time, or less than one-twentieth the length of the tide. 
These waves crossed the Pacific from the neighborhood of Japan, at a rate of travel exceeding 
350 miles per hour. (See notice of earthquake waves on western coast of the U. S., 23d and 
25th Dec. 1854, by A. D. Bache, Supt. U. S. C. S., Am. Ass. Science.) 39 
channels. As a general thing the observer's station will be a vessel 
or boat at anchor within sight of the shore. 
§ 48. The elements for a table of tidal currents are obtained by 
very simple means, and the following set of rules will be found a 
sufficient guide to the observer: 
I. The observer designing to measure the currents of any particular 
channel should select straight portions of the water-way, and anchor 
or moor his vessel in the axis or thread of the stream. 
II. For position he may take angles from his vessel on three or 
more determined objects on shore, suitably situated; or if these do 
not exist, he may measure a base line on shore, and by triangulation 
fix the position of his vessel and develop the neighboring shore line 
on either side of the channel-way. The article on surveying in 
Bowditch's Navigator may be referred to in this connection. 
III. The immediate objects of the current observations are three- 
fold. 
1st. The determination of the velocity, which is obtained by the 
use of the log-line and a time-piece. 2d. The epochs of slack water. 
3d. The direction of the flow, which is determined by the usfe of a 
sextant or a compass. 
IV. The log-line must be examined each day, and the error in its 
divisions due to its stretching or shrinking must be ascertained and 
noted. 
V. The log-line is divided into a certain number of parts, and these 
are so arranged that, passing off the reel during a period of thirty 
seconds, they measure the motions of the log in nautical miles and 
decimals per hour. The larger divisions are marked by leather straps, 
in which holes are perforated indicating the number of miles. The 
straps are secured to the line at intervals of fifty feet, eight and one- 
half inches (more nearly 50.72 feet.) Each mile is divided into tenths 
by nine knotted cords placed at equal distances between the leather 
straps. The number of tenths is indicated by the number of knots 
upon each cord. Each of these smaller divisions is five feet and 
seven-eighths of an inch in length, (more nearly 5.072 feet.) 
VI. There should be at least sixty feet of stray line between the 
float and the zero division, in order that the float may drift out 
beyond the eddies of the vessel before the measurements commence. 
VII. In making an observation on the current, one man holds the 
reel while another notes the time. When the log is cast and the 
stray line is run out, the person holding the reel cries, Now; the 
person with the watch repeats the signal, notes the position of the 
second hand, and when thirty seconds have elapsed, cries out, Stop. 40 
The person at the reel now notices to what division the line has run 
out, and without restraining the log, the experiment is repeated in 
precisely the same manner as that described, except that in this latter 
case some even division of the line is chosen as the initial point, 
instead of the zero, as in the former observation.* 
VIII. A pair of observations for velocity having been taken, the 
log is still suffered to drift until the angle which its line of direction 
makes with some known point on land is carefully measured. 
IX. In selecting the fixed object to which to refer the directions 
of the current, the observer should give the preference to the most 
distant, in order that any error in the determination of his position 
may not so materially affect the accuracy of his observation. 
X. Angles of direction should always be measured with the sextant, 
where its use is possible; but if by reason of darkness or thick 
weather no fixed objects upon the shore can be seen, the observer 
must note the direction by compass in degrees, as N. 43° E., &c. 
Richie's "liquid compass" is the best now in use. 
XI. After using the compass for such purposes, the first opportunity 
should be improved for checking the error of the needle. 
XII. The observations would better commence at slack water, and 
after continuing a half day or a day, terminate with the commencement 
of a flood or an ebb. 
XIII. The velocities and directions should be noted at least every 
thirty minutes while the drift is strong, and more frequently as its 
motion fails, in order that the exact times of cessation and commence- 
ment of current may be obtained. 
XIV. If the current turns without absolutely slacking, the nature 
and progress of this turning should be ascertained by repeated obser- 
vations made at least once in five minutes. In this case, the times of 
most rapid change of direction and the least velocity must be obtained. 
XV. Soundings should be made and the depth of water noted at 
every slack water. 
XVI. The direction and force of the wind, together with the 
appearance of rips, eddies, and any other interesting phenomena, 
should be observed. 
XVII. The observer's watch should be set to mean time each day. 
XVIII. The most convenient form of record is the following: 
* The half minute sand-glass is used in ordinary observations, but the watch is preferable, 
because the former alters its rate with the humidity of the atmosphere. 41 
Current Observations. 
At [No. of station and locality.] (Date.) 
Angles for position, j 
Time. 
Angle of Direction. 
Veloc'y per Hour. 
Wind. 
Remarks. 
Hours. 
Minutes. 
Float 
Right or 
Left. 
of 
o 
Miles. 
Deci- 
mals. 
Direc'n. 
Force. 
§ 49. The current table, if intended to accompany a chart or 
sketch of a harbor, should be of the annexed form. 
Station. 
Cur't Turns 
after Moon's 
Transit. 
Maximum Velocity. 
Duration. 
Slack. 
Remarks. 
No. 
Local- 
ity. 
Flood 
to 
Ebb. 
Ebb 
to 
Flood. 
Flood. 
Direc- 
tion. 
Ebb. 
Direc- 
tion. 
Flood. 
Ebb. 
Flood 
to 
Ebb. 
Ebb 
to 
Flood. 
The time of turning is the middle of slack water, and the intervals 
as well as the durations are reckoned from this time. In the column 
of remarks it should be stated whether the directions are magnetic 
or true bearings; and if the former, the variation of the compass should 
be given. 
As currents are much affected by the winds, it is better to occupy a 
station for a single calm day, and correct the results for mean value 
by tidal comparisons, than to carry the series through a semi-lunation 
of variable weather. The tides in deep channels are but little affected 
by the winds, so that if guage observations are made on the day when 
the current station is occupied, and kept up through not less than a 
semi-lunation, the means are afforded for correcting the current obser- 
vations for half-monthly inequalities. By comparing the tidal inter- 
vals of this particular day with the mean establishment, we shall find 
what to add or subtract from the current intervals and durations. 42 
§ 50. The corrections for velocity may be made by the following 
rule: the velocities of the currents for corresponding hours of different 
tides are to each other as the rates of rise or fall at these hours. With 
the use of this rule the hourly velocities may be computed for tides 
of the mean range, for springs and for neaps, from not less than 
twelve hours' current observations made under the most favorable 
circumstances. The rule is applicable to places where the figure of 
the tide remains essentially the same from point to point. It is not 
applicable to inlets or lagoons, nor to very shallow channels. 
§ 51. Strictly speaking the flood and ebb currents which we observe 
in channels have no direct dependence upon the range of the tide, nor 
upon its rates of rise or fall, but upon the slope of the water's surface. 
They are bodies of water flowing under variable heads, and the observed 
difference of level is that portion of the head which is balanced by 
the resistances of the channel and the inertia of the mass of water.* 
A closer and more generally applicable rule, therefore, for correct- 
ing velocities, is the following: at the same station, the velocities for 
corresponding stages of different tides are to each other as the square roots 
of the observed heads. We use the word stages here to signify depths 
of water; corresponding stages, therefore, give equal sections and 
wetted perimeters. 
§ 52. The heads of the tidal current may be obtained by placing 
two gauges on the border of the channel-one above, the other below 
the current station-and subtracting the observed heights one from 
the other, after referring their zeros to the same datum plane by 
levellings. 
If the channel is deep and straight, mean level may be assumed to 
be a common plane, and the heights referred to this, avoiding thus 
the use of a levelling instrument. If a comparison of the velocities 
is made with heads observed in this way, it will usually be found 
that maximum velocity occurs at a brief interval after the time of 
maximum head, and that motion ceases soon after the head becomes 
zero. If the two gauges are observed upon for at least a semi-luna- 
tion, the means may be obtained for correcting the epochs and veloci- 
ties of a day's current observations in the most exact manner. 
At many ports, even where the rise and fall of the tide is very 
small, the tidal currents are strong, and correct tables of them have 
* A tidal head is not the measure of angular and frictional resistances in the same exact 
sense as the head of a river or canal in permanent motion. The conditions of a tidal stream 
resemble those of a river when a rise js coming down in the form of a swell. 43 
great value. At such places the time of high or low water may be 
shown very indefinitely by the nicest observations, but the time of 
restoration of sea level may be easily and closely ascertained. The 
inlets on our Atlantic coast present cases of this kind. 
§ 53. The relationships between the forms of channels and the 
velocities of the tidal currents which traverse them are so intimate, 
that a knowledge of the tidal range at any place furnishes no adequate 
guide for estimating the velocities or even the times of the currents. 
In ports that are open to the sea the tidal currents are usually feeble, 
while in ports communicating with the sea by narrow avenues they 
are usually strong. At the entrance to the Bay of Fundy, where the 
tides have a range of fifteen feet, they are not so strong as at the en- 
trance to Mobile Bay, where the tidal range is less than two feet. In 
the former case the tide wave moves rapidly and without sudden 
changes of figure, while in the latter it is retarded, and, having to 
pass through a narrow inlet to a broad basin, undergoes a sudden 
degradation which produces a contrast of elevations between the ocean 
without and the bay within. 
§ 54. A degradation of the tide wave is always accompanied by a 
change of epoch in the tidal currents, because the time of restoration 
of sea level is changed. At inlets communicating with extensive 
inland basins the phenomenon of 11 tide and half tide" often occurs. 
The inlet being narrow, the tide wave which enters is scarcely able 
to raise the general surface of the basin, which remains, therefore, 
nearly at the mean level of the sea; so that restoration of level be- 
tween the basin and the ocean occurs at or near time of half tide both 
on the ebb and on the flood. Instead of having slack water at or 
near the times of high and low water stand, we have at these times 
the strongest flood and ebb. The phrase "tide and half tide"' has 
been applied to such inlets by those who conformed the rise and fall 
with the horizontal motion of the water. One may frequently hear 
the remark "that the tides rise earlier on the shore than in the 
channel-waythe more accurate statement is, that the epochs of the 
tide precede those of the dependent currents. 
§ 55. It is usually of more practical importance to the navigator to 
be accurately informed of the currents at inlets than of the tides, be- 
cause the former are usually very strong in the very localities where 
the latter vanish. 
§ 56. If a sound or harbor is visited by two branches of the tide 
wave entering by two different and widely distant avenues, a certain 
intermediate district will be the scene of alternate drifts which cannot 
be characterized by the words ebb and flood, since they are interchanges 44 
of water between two wave systems which have disturbed the sea 
level near their place of meeting by their contrasts of time. These 
interchange currents have peculiar epochs which have no dependence 
upon those of either tide separately considered, but follow the times 
of restoration of level between the two tides. In a case of this kind 
the observer will be presented with three sets of currents; each tide 
will be accompanied for a certain distance by its normal drifts, and 
the intermediate space will be occupied by the interchange drifts. 
Three stations, properly located, will give him the order of luni-cur- 
rent intervals, and perhaps the characteristic velocities; but this will 
not be all the information required to meet the practical wants of 
navigation. The limits of each of the three current districts should 
be ascertained, and if these limits are the scenes of conflicts or stag- 
nant water, full descriptions should be given. At or near each of 
the entrances to the sound or harbor, the best position of current 
station will he found for giving the normal epochs; and the point at 
which the tides meet will be that of which the interchange current is 
most steady and strong. 
The most violent tidal currents which occur in navigable channels 
upon our coast are those of Hell Gate, New York, and those near 
the junction of Vineyard and Nantucket Sounds. In each of these 
cases the drifts are called into activity by the meeting of two branches 
of the tide wave, which, approaching by different paths, after under- 
going different experiences, present to each other a contrast of eleva- 
tions. 
§ 57. In confined and tortuous channels of small depth, the maxi- 
mum velocity of the current does not occur at half ebb or half flood, 
but perhaps a half hour or an hour later, because the water-way is so 
inadequate for the conveyance of the tide that the contrast of elevation 
of the water above and below goes on increasing with the rise or the 
fall till some time after half tide. If this state of things appears at 
a station the current table should contain a remark upon it, or, in an 
additional column, the time of maximum drift should be given in hours 
and minutes after that of turning. 
§ 58. The epochs of the currents upon shoals are usually earlier 
than those of the neighboring channels, partly because large bodies of 
water, by reason of their inertia, continue longer in motion or less 
readily submit to a change, and partly because upon shoals the water 
moves under the more immediate influence of local tides. It is of 
considerable importance to the navigator to be made aware of these 
differences of epoch which cause rips and whirls along the margins of 
shoals at certain times of tide. 45 
§ 59. Very little is known of tidal currents on outside coasts except 
in the immediate neighborhoods of certain dangerous shoals. A 
knowledge of them would, however, be often a great advantage in 
the saving of time in the passages of coasters. To the want of tables 
of the coastwise currents the loss of vessels has in several instances 
been attributed. Coasters frequently lose their reckoning in quiet 
and thick weather by being swept out of their courses by these 
drifts. The coast currents in some places have a velocity of a third 
of a mile per hour in thirty fathoms water. They are, in some 
localities, nearly parallel, in others nearly normal, to the general 
trend of the shore line; and, as far as the few observations we have 
seen may indicate, the directions of ebb and flood are not usually 
opposed, although lying at an oblique angle with each other. The 
epochs of coast currents may be expected to differ widely from those 
of the local tide, especially on a shore where differences of tidal range 
appear from point to point. As as an aid to navigation, we especially 
recommend the observation and tabulation of the currents "o/i 
soundings." 
Thus far we have referred to tidal currents from the navigator's 
point of view, and have described the character of those observations 
and results which prove most useful to him. To the engineer pro- 
jecting the improvement of a harbor, and to the physical enquirer, 
the subject is no less important and interesting. 
§ 60. It is only in rare instances that a knowledge of the currents 
below the surface of the sea can be of practical value to the navigator, 
but the engineer, who looks upon tidal drifts as working agents which 
he may restrain or direct, is especially called upon to measure the 
power and the office of sub-currents. 
To obtain the general motion of a stream of no great depth, a tin 
cylinder a few inches in diameter, and long enough to reach from the 
surface nearly to the bottom, may be used as a log. Tubes of forty- 
eight feet length have been used for this purpose. These were three 
inches in diameter, made in separate sections, air-tight, but with 
stop-cocks for letting in water, that they might be partially filled so 
as to sink to the proper depth. As the tube drifts nearly upright in 
the water, with its top protruding a few inches above the surface, 
its velocity must indicate the mean motion of the stream. leans 
forward or backward, we at once perceive that its foot rests in a 
stratum which has less or greater motion than the surface drift; and 
if its angle of direction differs from that of the surface log, we recog- 
nize the action of a sub-current whose course is at variance with that 
of the surface drift. 46 
§ Gl. To obtain the velocities and directions of the currents of a 
specific water stratum, we may observe upon the motion of two bodies 
connected by a slender cord, the one sunk to the required depth, the 
other floating, and compare the results obtained with those given by 
the surface log. Very good results have been obtained by using two 
hollow copper globes of two feet diameter each, connected by one- 
eighth inch wire rope. The sinking globe is filled with water, but 
the other is loaded only enough to sink nearly to its pole. The upper 
globe has a log-line secured to it, and its motion is recorded at the 
same time that an observation is made with the surface log. 
Let us suppose that the two globes present equal effective areas 
(great circles) to the drifts in which they swim; then their velocity 
will be a true mean of the rates of the surface and sub-currents; 
i. e., | (x + y) where x and y represent, respectively, these rates. 
The velocity of the sub-current may therefore be found by subtracting the 
surface rate f rom twice that of the connected globes. 
The above rule applies to the case where the paths of the drift 
coincide in line of direction, which they usually do. It is sometimes 
the case, however, that the sub-current apparatus takes an opposite 
direction to the surface log; in this case its rate is regarded as having 
an opposite sign to that of the surface in the application of the rule. 
If the surface and sub-currents pursue paths at an angle with each 
other, we have a simple question in the resolution of forces in which 
the motion of the connected globes represents one-half the resultant 
(the diagonal of the parallelogram, of which the surface rate is one 
side) and corresponds with it in direction. 
The wire rope connecting the upper and lower globes should pass 
up through the former and be secured to the portion of the float 
which is above water, so that the observer may alter the draught of 
his apparatus and examine different depths by simply letting out the 
connecting line or drawing it up, without taking the globes from the 
water. 
Although, for several r'easons, spheres are the best forms for use 
in these experiments, the difficulty of managing and connecting the 
floating globe is often so great that the substitution of cylinders is 
resorted to, the floating cylinder being provided with a conical top 
or air chamber which is kept above water. For rough determinations 
barrelsjor boat breakers may be used. 
The surface log would better be a globe or cylinder corresponding 
in dimensions with the float of the sub-current apparatus and equally 
immersed, so that the effects of winds and waves may be equal. 
§ G2. The yawing of a vessel at anchor in the "tide-way" is often 47 
a serious obstacle to the collection of good results, especially from the 
sub-current apparatus; and, again, the graduated log-line used with 
the apparatus, being necessarily of considerable size, is borne away 
by the surface drift so as to deceive the observer. To obviate these 
difficulties, in some cases the use of an anchored vessel is dispensed 
with and the floats set free, but followed by the observer in a boat, 
who frequently determines the position of the surface log, and notes 
the separation of the sub-current apparatus from it. Instead of the 
large log-line referred to above, a very slender graduated cord is 
carried from the upper globe to a reel within the surface log. The 
reeling off of this line measures the difference between the surface 
rate and that of the mean of upper and lower drifts. 
If the surface log is a sphere, the line is suffered to run freely out 
through a hole in the pole. In this manner the currents of great 
depths may be examined. The connecting wire between the upper 
and lower globes may be very slender, so as to offer very little re- 
sistance to the water, and yet be sufficiently strong to bear the 
steady strain to which it is subjected when the apparatus is discon- 
nected from the vessel. Very good results have been obtained from 
a depth of thirty fathoms in the open sea.* 
§ G3. The variation in velocity from surface to bed in the channels 
of tideless rivers and in canals has been the subject of much studv, 
and the curve which this variation usually describes has been found 
to be parabolic, the maximum velocity being a short distance below 
the surface.! The depth at which the maximum velocity may be 
found varies in a remarkable manner, and no satisfactory explanation 
has yet been offered. It has been urged that the friction of the air 
would suffice to render the surface flow preceptibly more tardy. Per- 
haps the same effect would follow from the greater irregularities of the 
shore line near the surface. At contractions the point of maximum 
velocity usually descends, as if the more superficial drifts were con- 
verged less favorably for reinforcing each other than those at greater 
depths. If the contraction is very great, but of short extent, the 
current descending a precipitate slope is sometimes inclined to under- 
run the quiet body of water beyond, the particles preserving, as it 
would seem, their inclined direction for a while. 
See Annual Report of Coast Survey, 1859, for description and sketches of sub-current 
apparatus, together with formulae for reducing particular cases. 
t See Humphrey and Abbot's report on the Mississippi river for the best determination of 
such curves. 48 
In the neighborhood of extensive flats the tidal currents below the 
general level of these flats follow a different law of variation from 
those above, and sometimes differ ten, fifteen, and twenty degrees in 
direction. 
Currents flowing into channels from shallow flats or creeks fre- 
quently continue on as superficial streams for some distance, but ulti- 
mately their motions become merged with the general drift. 
§ 64. In rivers whose beds lie below the level of the sea very 
singular variations may be expected. Sea water frequently flows 
into such a river along its bed during the dry season, even while 
the lighter fresh water upon the surface is flowing towards the 
ocean. In the wet season the same river, acquiring head, is able to 
expel sea water again, and so the bed is alternately traversed by salt 
and fresh waters. The use of a hydrometer in testing specimens of 
water is indispensable in observing such phenomena, because we 
must in this case measure the weight of the head before we can account 
for the degree and kind of activity which it induces. 
The effect of very strong winds blowing up a river is to check the 
surface current, but if the water is deep the outflow below the influence 
of the wind and waves will be greatly increased. 
Rivers flowing into broad basins become superficial and diverge 
rapidly, suffering often a very sudden slacking-up, which permits the 
sediments to fall to the bottom and form bars or mud-banks. 
The waves of the sea break up the currents flowing from rivers 
and harbors, but after long periods of quiet weather, when the waves 
are so small that their influence is not felt at the bottom, the outflow 
in the lower water stratum is often found steady and strong. 
§ 65. From a few observations made along the coast in the midst 
of tidal drifts, it appears that in midsummer cold belts or spots are 
found in certain positions on the surface of the sea, differing ten to 
fifteen degrees from the general surface temperature, but correspond- 
ing with the temperature of the -water near the bottom. Even in 
localities where the water is uniformly deep, and there seem to be no 
shoals or other obstacles calculated to sheer the current upward, this 
phenomenon appears.* The use of a thermometer in connection 
with current observations made in summer is to be recommended. 
If the flows for different depths do not occur upon parallel planes, the 
thermometer will not only detect the general facts, but may follow 
them back to their causes. 
Coast Survey Report, 1361. 49 
§ GG. In the formation of mud-banks or shoals the currents take 
usually a prominent part. In confined portions of their journeys they 
scour away the bed and banks of the channel and carry off the mate- 
rial only to cast it down again in broader sections where their reduced 
velocities disable them from continuing to bear such burdens. The 
following statements, compiled from direct experiments, show the 
power of currents:* 
Clay, fit for pottery, is removed by water having a velocity per 
second of 0.26 feet. 
Fine sand is removed by water having a velocity per second of 
0.53 feet. 
Gravel, about the size of peas, is removed by water having a 
velocity per second of 0. G3 feet. 
Shingle about one inch diameter is removed by water having a 
velocity per second of 2.12 feet.t 
We may divide the material borne along by currents into two 
classes, viz: the suspensible and the rolling. To the suspensible class 
belong muddy sediments composed of vegetable earths, clays, &c., 
which are lifted by the stream and partake fully of its motion; while 
to the rolling class belong sands and gravels, which are not lifted from 
the bottom by currents of ordinary strength, but are urged forward 
at a rate bearing but a small ratio to that, of the water. 
In tidal waters muds are observed to accumulate in sheltered coves 
and in the broader portions of the channel. Wherever they are 
found they indicate inactivity, or at least the recurrence of long 
periods of slack water. It is not so of banks and shoals of sand, 
which are often observed to accumulate in places traversed by strong 
drifts. Sand-banks may accumulate zuherever the resultant of the current 
forces is zero. A particle of sand rolls along so short a distance in a 
single day that the hourly velocities of the currents for the period of 
two floods and two ebbs may be regarded'as forces acting simulta- 
neously. 
§ G7. It is often desirable to ascertain whether a shoal is the result 
of natural operations now in progress, or is the remains of a wasted 
island, or the effect of some past conditions; and it often happens 
that there is in such a shoal little or no geological indication of its 
age or history. If we make a current station upon it, and observe 
the drifts for a tidal day, then compound the velocities according to 
* Dubant's Experiments. 
t More modern experiments reduce these velocities somewhat. 50 
the rules for the composition of forces, we shall be able to determine 
whether, in the event of its being dredged away, it would gradually 
form again or not. 
The formula for the composition of forces lying upon the same plane 
is so simple in its application to this experiment that we may intro- 
duce it here: If the hourly velocities are represented by p2 p} 
&c., their angles by a1 a2 a3 &c., (measured, we may assume, from the 
south round by west,) the resultant by P and its angle of direction 
by <p, we shall have- 
P2 = (j'l sin. + p, sin. a2 + p3 sin. a3 + &c.)2 + (px cos. a, 
+ jo2 cos. a2 + p3 cos a3 + &c.)2 
„ p. sin. a. + p, sin. a, + p3 sin. a3 + &c. 
Lan. <f>-- ! - - 5- 
p3 cos. a1 + p2 cos. a2 4- p3 cos. a3 + &c. 
These forces are to be added with due regard to signs. The angle 
<f> is measured from the same point as those of the component forces 
(from the south we have assumed.) It is acute if the numerator and 
denominator of the above fraction are both positive; it lies between 
one and two right angles when the latter is negative and the former 
is positive; between two and three when both are negative; and 
between three and four when the numerator only is negative. 
§ 68. It will be found most convenient to arrange the hourly veloci- 
ties as in the following table, and to use natural sines and cosines: > 
Tidal 
Hour. 
Angle 
(fl) 
Cosine 
a 
Sine 
a 
Velocity. 
P 
p cos. a 
p sin. a 
o 
00 
00 
• 
00 
I 
311 
656 
-755 
0.65 
426 
-491 
• II 
313 
682 
-731 
1.16 
791 
-848 
The table should commence with slack water and end with slack 
water occurring about twenty-four hours later, if possible, to avoid 
confusion and error in the first and last terms of the series. Veloci- 
ties less than three-tenths of a mile per hour must be reckoned as 
nothing, because, by the table given in § 66, it appears that these are 
too small to move a grain of sand. 
In the first column the number or hour of each observation is given. 
It is well to divide the whole time into twenty-four equal parts, and 51 
call these I, II, III, IV, &c. In the second column are the observed 
angles, (a, a, a3 <fcc., of our formulas;) and in the fifth the observed 
velocities, (pl p2 Ac.-, of our formulae ) 
In this computation we use the two ebbs and the two floods of the 
same tidal day in order to eliminate the diurnal inequality. If the 
tides of the same day are also observed, the velocities may be cor- 
rected for inequalities by method explained in § 50 and 51. 
As repeated trials would have to be made of stations upon a shoal 
to discover the equilibrium point, (supposing one to exist,) and since 
this point would be likely to shift from day to day as the tides vary, 
it will be found more convenient and satisfactory to surround the 
shoal with a cordon of stations, and at each repeat the experiments 
jind computations; then, after plotting the values of p2 upon the 
chart, observe whether they converge towards some point upon the 
shoal or radiate from it; in either case the sand swept along by the 
tidal drift will lodge. We compare the squares of the resultants be- 
cause zoork is to be represented, t. e., scour. 
At the different stations in the cordon about the shoal the observa- 
tions should be made at the same depth. Perhaps the mean velocity 
of a stratum twice the depth on the shoal will answer best; for this 
purpose the tube log should be used. 
This method of examining into the causes of shoals has been suc- 
cessfully used in several of our most important harbors and sounds. 
Bars and shoals exposed to heavy breakers are found to be 
the joint work of the waves and the currents, while banks in quiet 
basins and channel-ways are usually the work of currents only. 
The inside bars or "swashes" of lagoon inlets represent the action 
of tidal currents in a very distinct manner. As the lagoon is less 
affected than the ocean by the tides, there is at a certain stage of the 
ebb and of the flood an inclined surface across the swash, and the 
water pours in or out with great power, the inflow gathering its 
greatest strength on one side, and the outflow on the other, of a cer- 
tain intermediate point. At this intermediate point the inflow and 
outflow are both strong, but in equilibrio, so that here a shoal forms, 
which is usually known by the name " bulkhead.'' These bulkheads 
are curious objects for observation. The sand of which they are com- 
posed may be seen in great-agitation on either ebb or flood; it is in- 
cessantly on the move, continually advancing upon the flood, and re- 
treating upon the ebb, but gaining nothing on the whole. A great 
deal may probably be added to our knowledge of inlets and their bars 
which may have a practical as well as philosophical interest, if data 52 
can be collected from observations upon similar formations under dif- 
ferent conditions of tides, &c. 
§ 69. In the complete gauging of a tidal avenue to obtain the meas- 
ure of its service as a conduit, and its qualifications for this service, 
the following data should be collected: From a close survey profiles 
of characteristic cross sections and a profile of depths along the track 
of greatest depression of channel bed should be plotted. From these 
the mean depth, mean width, mean area of cross section, and mean 
wetted perimeter for each foot of tidal range, may be calculated, also 
the distances measured along the centre of the stream. From simul- 
taneous tidal observations at several points, the form of the tide and 
its changes of figure in its journey, as well as the slope of the water's 
surface for each hour, should be made known. From current obser-' 
vations the curves of velocities from side to side and from top to 
bottom in each characteristic section, and for each tidal hour, should 
be given; also, the directions of motion at many different points and 
the durations of slack water. 
If the avenue communicates with an interior basin, the measure- 
ment of the tidal prism in this basin is advisable, as a check upon 
the results given by the gauging of the channel. 
§ 70. It is conceded that in the currents of rivers the velocities de- 
pend upon the surface slope, and not upon that of the channel bed, 
when other things are equal. The same result seems to be 
reached in a single comparison, which we have made, of ebb and 
flood tidal currents at equal stages; no effect for reverse slope of 
channel bed could be detected.* 
Frequent articles on tidal currents may be met with in the trans- 
actions of the Royal Society and in the U. S. Coast Survey Reports; 
also in the reports of the commissioners on Boston, Portland, and 
New York Harbors, and Cape Cod Canal. Works upon the currents 
of rivers are numerous. The most recent, and the most com- 
plete, is the "Report upbn the Physics and Hydraulics of the Missis- 
sippi River; No. 4 of the professional papers of the Corps of U. S. 
Top. Engineers." The reader is especially referred to this work for 
the laws of running water in large rivers; and for minute inquiries 
in small streams and canals he may consult " Recherches Hy dr antiques, 
Tome XIXd.es memoires presenters par divers savants a V Institut Im- 
perial de France." 
* Tenth Report of U. S. Commission on Boston Harbor. APPENDIX 
TO ARTICLE ON TIDES AND TIDAL PHENOMENA. 
Tide Gauges. 
Simple Staff.-For sheltered and quiet harbors, the most reliable 
gauge that can be used is a simple staff graduated upwards in feet 
and tenths. This staff should be secured in a vertical position with 
its zero below the greatest fall of the tide; and this zero, by levell- 
ings, should be referred to a permanent bench-mark. If very nice ob- 
servations are required, a glass tube of about a half inch diameter with 
a colored float (a red glass bubble, or a few drops of tinted oil) may 
be attached to the staff to enable the observer to read the heights 
more accurately. If this tube extends at all times several feet below 
the surface of the sea, and if its lower end is partly stopped by a cork 
with a tube through it, the float will not be agitated by the surface 
ripples. 
Box Gauge.-There is an inconvenience in the use of the simple 
staff in places where the tidal range is great, since the observer is 
obliged to alter his position frequently to follow the surface of the 
water up or down. To obviate this, an apparatus called the box 
gauge is used. In this gauge the staff has a tin or copper float at 
the foot of it, and moves up and down within a tube or box which is 
secured in a vertical position, with its lower end as far below the 
greatest fall of the tide as practicable. The water is let in at the 
foot of the box at several small holes; and the observer reads the 
relative heights as the figures of the rising or falling staff pass a 
certain mark seen through a convenient opening in the box. This 
mark, which remains fixed throughout the period covered by the entire 
series of observations, is technically called the reading point, and 
should be referred to the bench by levellings. In order, however, to 
furnish the computer with the means of reducing the observed heights 
so as to read from the bench, it is further necessary to record the 
distance from the water line on the float to a stated figure upon the 
staff; and since the float line is subject to change as the staff becomes 
saturated, or otherwise alters its weight, frequent measurements of 
this distance should be made and recorded. 
The chief source of error in the results from this gauge is found in 
the want of sufficient freedom for the ingress and egress of the water 
at the foot of the box. In order to keep out the swells and choppies 
and at the same time to avoid the trouble or inconvenience of sinking 54 
the foot of the gauge box below their influences, the holes are often 
made of the minimum size at the outset, and soon become more or less 
choked by weeds and barnacles, so that the observer unwittingly 
records a distorted tide. The effect of inadequate water holes is to 
increase the intervals, reduce the ranges, alter the magnitudes of all 
the inequalities, and especially to displace the middle portion of the 
wave figure. 
Sef-Registering Gauge.-The bog gauge is sometimes provided with 
a self-registering apparatus, so arranged that the rising or falling 
float carries a pencil with which it describes a curve upon paper 
wrapped about a cylinder revolving by clock-work. In the self-reg- 
istering gauge the graduated rod is replaced by a wire kept taut by 
a counterpoise. In the form of apparatus used in Great Britain the 
paper is secured to a vertical cylinder which makes one revolution in 
twenty-four hours, and the same sheet is suffered to remain for about 
a fortnight, or until some twenty-eight tides are drawn upon it. In 
the Phil. Trans, of 1838 may be found a full description of this form 
of the self-registering tide gauge by Mr. Bunt. 
The self-registering instrument used by our government, devised 
for the U. S. Coast Survey by Joseph Saxton, Esq., differs from the 
British instrument in some essential particulars. The vertical motion 
of the float is converted into horizontal motion for the pencil, and the 
paper unrolls from one cylinder and rolls up upon another so that the 
curves do not repeat. By this improvement a continuous line is 
traced for upwards of thirty days before the paper is exhausted. 
The receiving cylinder makes two revolutions in the course of a day, 
and each half hour is pricked through the paper as it rolls up by 
needle points projecting from either end of the cylinder. 
The Saxton self-registering gauge is portable, and may be set upon 
an ordinary timber wharf or on a platform supported by a half dozen 
piles well tied and braced together. The instrument will work well 
in any situation where the jar is not too great for the proper motion 
of the pendulum attached to the clock. A full description and sketch 
of this instrument is to be found in the Report of the Coast Survey 
for 1853. 
In connection with the self-registering gauge, occasional observa- 
tions upon a staff gauge fixed in the immediate neighborhood are 
requisite, in order to give the absolute heights represented by the 
curve upon the sheet. These staff observations should be made on very 
quiet days, at even hours, and about the time of tidal stand. If at these 
even hours the pencil of the self-registering gauge is drawn back for 
an instant so as to make a vertical mark upon the sheet, and the record 55 
of the true time and height by staff made on this line, a check upon 
the working of the clock and gauge will be given. Such checks can- 
not be made too frequently or too carefully. The curves on the sheet 
of this gauge are read off and converted into figures by the use of scales. 
In order to make the sheet convenient for future reference, it is well 
to write out the dates and hours; also to draw vertical lines connecting 
the corresponding pricks of the upper and lower margins, and equi- 
distant horizontal lines parallel to the lines of pricks, which shall 
divide the verticals into feet and tenths. 
The freezing up of the box in winter may be prevented by pouring 
in a small quantity of kerosene or other oil which remains fluid at low 
temperatures. To insure success under ordinary circumstances, it has 
not been found necessary to displace the water to a great depth in 
the box, but simply to sink its surface about the depth of frost in still 
sea water-a foot, perhaps. 
For stations exposed to the open sea, peculiar modes of securing 
tide gauges have been resorted to, and in some cases the gauges them- 
selves have been made in peculiar forms. In most cases where short 
series of observations are required, a slender iron staff with tube 
attached may be held in place by securing it within a skeleton pyramid 
of iron rods anchored to the bottom. In the form used in the recent 
survey of the Cape Cod canal sites, the base of the pyramid was a 
triangle of one-and-a-half-inch iron rods, to each angle of which a 
small anchor was lashed by its shank. The staff was let down from 
the apex of the pyramid and driven into the bottom, within the base, 
then firmly fixed by screw at top. The observer occupied a boat 
which he could always fasten to the apparatus without disturbing it. 
In the Coast Survey Report of 1854, Appendix 53, a form of tide 
gauge is described which was successfully used at many stations on 
the open coast of Massachusetts. An iron tube with an augur at foot 
is passed through the ring of a heavy anchor and screwed into the 
bottom. It is held in a vertical position by three chain guys secured 
by screw band to the middle of the tube, and extending to three other 
anchors well buried at short distances from the foot. Within the 
tube a rod and float are placed as in the ordinary box gauge. 
Again, in the Coast Survey Report of 1857, a device for observing 
tides at a short distance from the coast is described. It was very 
successfully used in ten-fathom water, but would not probably answer 
so well in less depths, where the waves would offer too great dis- 
turbance. A long pine spar, bolted at the foot, with a universal 
joint, to a block of stone, is lowered down to the bottom of the sea. 
The portion above the surface of the water is slender, so as to be 
less affected by the wind, while the part near the bottom is large, in 56 
order to be as buoyant as possible. To the top of this spar, which 
projects some ten or fifteen feet above the water, is attached the rod 
of a heavy pendulum, which slides over an arc five to eight feet 
below, and determines the inclination of the spar. To this buoy, 
thus arranged, is fastened the gauge, which is a glass tube set into a 
graduated staff, and having a red glass bubble rising and falling 
within. By a small hole in this tube, some eight to ten feet below 
the surface of the sea, the water enters, upon which the bubble 
floats. The observer records at once the height of the bubble and 
the angle of inclination of the gauge. The computer, with the aid 
of a table of natural cosines, readily reduces these observations to 
perpendicular heights. 
"Off-shore Tide and Sounding Meter" is the name given to an ap- 
paratus in which the heights of the water are determined barometri- 
cally. The pressure of the sea upon an elastic air bag lying upon 
the bottom is communicated, through a flexible tube, to a manometer 
above. It was invented by Major E. B. Hunt, U. S. A., as a sounding 
instrument, in 1857, and as such is described in the Coast Survey 
Report of that year. Subsequently this instrument, in an improved 
form, was proposed by John M. Batchelder, Esq., as a tide gauge, 
and has been successfully used during the recent war, by the Coast 
Survey, off Charleston bar. In connection with hydrographic work 
on the open coast and in the neighborhood of exposed shoals, this 
gauge has a decided advantage over many others proposed, because 
a continuous record can be made from the deck of a vessel, the ob- 
server being required simply to note the position of a hand upon a 
dial plate. 
Another ingenious method of measuring tides by weight has re- 
cently been proposed by Mr. Batchelder in a paper read before the 
National Academy of Science, 1865. It has not yet been put to the 
test, but it promises to provide against trouble from ice. 
In the Arctic expeditions of Dr. Kane and Dr. Hayes the ''pulley 
tide gauge" was used. The observations were made by the party 
of Dr. Hayes on a graduated vertical line which was stretched, from 
an anchor on the bottom, over a pulley at the apex of a tripod built 
over a hole in the ice. A counterpoise kept the line taut as the 
tripod rose and fell with the tide, and the observer road off the 
heights as he would have done from a staff gauge. For further 
descriptions we must refer the reader to Dr. Kane's "Arctic Expedi- 
tions," vol. 1, chap, xi, and to the Arctic tidal'articles in the Smith- 
sonian contributions of 1860 and 1865.