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OCR for page 19
CHAPTER II
TIDAL THEORY
A. T. DOODSON
Liverpool Observatory trod Tidal Institute
The ultimate state of knowledge of the tides will exist when the tides
can be explained quantitatively without reference to observation. Such
an ideal is still very far from attainment, though steady progress is
being definitely made towards it. So far as the forces which give rise
to tides are concerned there is no difficulty in stating these with extremely
great accuracy, since the distances, masses, and general motions of the
sun and moon are well determined and the mathematical expression of
these forces is not complicated by the physical properties of the particle,
nor by its relations to other particles Which the forces act. It is not
very easy,: though, without recourse to mathematical formula or compli
cated diagrams to give a satisfactory notion of the way in which the
tidegenerating forces are determined; it is sufficient to say that these
forces can be expressed for any given locality in formulae involving the
masses of the Sun, Loon, and Earth; the distances of the Sun and Moon
from the Earth; the astronomical latitudes and longitudes of Sun and
Moon; the geographic coordinates of the locality in question, and the
time. These forces are periodic, since the positions of the attractive
bodies, with respect to the particle considered, are periodic; the perio
dicity is naturally somewhat complex, and under such circumstances the
expressions for the forces are best resolved into a large number of simply
periodic terms, proportional to cosines of angles which increase uniformly
with the time. Also, it is usual to deal with the "potential" of the
forces rather than with the forces themselvesthe potential, of course,
is related to the work done against the forces by bringing the particle
from infinity to its present position. The actual forces in any specified
directions are derived from the potential by ordinary processes of difEer
entiation in the required directions. Such expansions have been given,
in increasing detail and accuracy, by Kelvin, Ferret, Darwin, and
Doodson; the lastmentioned expansion is more than sufficient for all
preseIlt demands of the subject.
* For explanations of tides in popular language, see textbooks by Darwin and
Marmer; the latter is probably the best textbook extant.
19
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20
FIGURE OF THE EAR TH
The peculiar advantage o:l such expansions is really associated Title
a principle formulated by Wallace, who stated that any periodic variation
in the forces must inevitably cause in the tide a variation with exactly
the same period. All modern methods of tidal analysis are based upon
this principle. The result is that we can consider separately the various
terms of the potential, and, to a first approximation at least, we can
consider their effects in the tide as independent. It is usual, therefore,
in mathematical investigations to consider the effects of a simplyvaryin<,
f'oree, represented by a cosine term whose phase increases with the time.
The expansion for the tidegenera/in<, potential indicates that the
principal terms have periods of about hall' a solar day; other terms, in
decreasing importance, have periods of about one day, a third of a day,
and a quarter of a day. Consequently, we expect tidal variations to he
resolvable into terms having similar charaeteristies, and the potential
gives us a good idea of the more important terms to be looked for. Indeed,
from it, we can readily deduce the main eharaeteristies of tidal motion.
This development, also, is used as a standard of reference for tidal
analysis; the actual amplitude of the tidal term is usually stated but
the phase is related to the phase of the corresponding term of' the potential
by a constant appropriate to the place (the phasela,; the amplitude
and phaselag, as determined by analysis, are the " harmonic constants "
for the term in question.
Several terms have " speeds " (rates of' movements of phase) so nearly
equal that it is found desirable to combine them into one constituent
whose amplitude and phase are the amplitude and phase of the pre
dominant term but modified by slowly varying quantities which can be
considered as constants within a period of' about a year. This is per
missible provided that we can assume the same relative amplitudes of
these tidal terms as holds for the corresponding terms of the potential,
and that the phaselags of' the tidal terms so combined are identical.
Experienee has shown that in general the relative amplitudes and the
phaselags change very slowly. with the speeds of the terms, so that I'or
terms whose speeds are nearly equal the assumptions made are quite
probably valid; there are exceptions to the rule, however.:
Having an exact kllowlecl~,e of' the external forces, the next requirement
is ail exact lolowlecl~,e Ott' the hchaviour oi' ~ single particle ot' water under
the action of' 1) these l:'orces, 2) gravity, varying, with position, 3) the
rotation of the earth and 4) intrinsic i'orces characteristic of the fluid
thus the particle suffers pressure from other particles, it may expand
* See paper by Doodson, Perturbations of harmonic tidal constants. Proc. Roy.
Soc. A. 106: 513 ( 1924) .
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TIDAL THEORY
21
or contract, it suffers friction with other particles, and so forth. Fortu
nately, we can assume water to be incompressible, with subsequent simpli
fication of the problem. Now these forces can be expressed mathematically
with great exactitude so that differential equations can be formulated;
normally, there would be three dynamical equations of motion required,
on account of possible motions in three independent directions, together
with an equation, called the equation of continuity, which expresses the
invariability of fluid volume within a given space. Such equations were
first given by Laplace and they have been critically examined by Lamb
and others. The exact equations, however, are seldom used by mathe
~naticians because of the great mathematical difficulties involved; the
subject has developed, as is usual, by easy stages involving the assumption
o:t' conditions which are not always even rough approximations to those
actually existing. Fortunately, the equations of motion can be simplified
very considerably, especially if the vertical motion is known to be small
relatively to the depth; this is always the case except in estuaries and
shallow bays, whose influence on the main oscillations in the large oceans
must necessarily be small; even in these apparent exceptions the simplified
equations, on a priors grounds alone, are quite good approximations.
Thus we can usually ignore in the equations of motion all terms (includ
ing, of course, frictional terms) depending upon the squares and products
of small quantities. Further, since the vertical acceleration is small
compared with the horizontal accelerations we can neglect it, in conse
quence of which we regard the fluid pressure at any point as equal to
the statical pressure due to the depth below the surface; this implies
a "long wave "; that is, the wavelength is large compared with the
depth, and herein is the distinction between surface oscillations and
tidal oscillations.
The tidal motion is a ":forced oscillation " in that it is maintained
by the external forces, but if we omit from the equations of motion all
reference to the external forces, then we have the equations of "free
motion." Clearly, if the fluid in a limited basin is suitably disturbed
and then left free, it will oscillate in a manner peculiar to the basin,
and the motion will persist, in the absence of friction, with a characteristic
period; there will be several periods, as a matter of fact, just as a violin
string will give out a note of a definite period with overtones of periods
onehalf, onethird, onequarter of the principal period; in the case
of a complex basin, however, there may not be such a simple relation
as this between the various periods. The importance of these free oscilla
tions consists in the liability of the tidal oscillations to become larvae if
the period of any constituent of the impressed force is nearly equal to
that of a free period. In case of exact agreement of periods it is necessary
v
. . ~
, . . ~
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22
FI GURE OF THE EAR TH
to rely on friction to prevent infinitely large oscillations, so that the
approximate equations of motion are no longer appropriate, because
:frietion has been neglected and vertiea.1 motion has been supposed to be
small compared with horizontal motion. Our knowledge of tidal motion
over the earth is su:Fieient to show, however, that it is only in a few
cases that such resonance is su~eiently great to require us to abandon
the use of the approximate equations of motion.
The application of these equations to tidal motions in specified basins,
with the earth rotating as in actuality, is very difficult and only in a
few cases has it been possible to obtain solutions of the problems, some
of which we shall proceed to discuss.
The most famous of these solutions yields the "equilibrium tide,"
so called because the resulting spheroid of liquid, when the fluid covers
the whole earth, is the equilibrium form that the free surface of the water
could take if the moon remained over one meridian of the earth. To
obtain this solution, under actual conditions of motion of earth and.
moon, it is necessary to assume that the water loses its inertia while it
retains its gravitational properties. The resulting free surface is a " level
surface " under the combined action of the earth's gravity, centrifugal
force, and the disturbing forces; on this surface the sum of the potentials
of the forces must be constant, for no work would be done on a particle
in moving it along the surface. If g is the height of tide, g the acceleration
due to gravity (supposed constant over the earth for all tidal purposes),
V the potential of the external forces, then
the lunar equilibrium tide = g = V/gC,
where C is a constant if the earth and moon are fixed relatively to one
another One other condition remains to be fulfilled, that the volume of
water has remained unchanged by the deformation. Clearly the mean
value of ~ over the basin ea.n be taken as zero, so that C represents the
mean value of V/g taken over the basin. It can be shown that if a
harmonic term of V/g is E cos +, where E is constant and ~ is. a function
of the time, then the eorresponding.term of V/gC is qE eos (+cry
where q and a are constants depending on the basin and varying from
place to place. Now this result will be shown to be of very great im
portanee in the exposition of the subject of tidal theory, for we are
intimately concerned with the effects of boundaries upon tidal motion;
the "eorreetion," as it is called, may have an important influecee on
the time of high water, which is no longer synchronous with the time of
maximum of the potential.
The corrections have been evaluated by Darwin and Turner for the
equilibrium tide as a whole, using an approximate representation of the
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TIDAL THEORY Y
23
land surface of the globe; they found that the corrections were of no
importance for the oceans as a whole, as we know them. Possibly a much
more important correction to the equilibrium tide is that which takes
account of the selfattraction of the tidal spheroid; this has been evalu
ated ~ and it appears that the expressions for the tides are thereby in
crea.sed in the ratio 1 12. It is extremely difficult, however, to correct
the equilibrium tide for both land.a.nd selfattraction simultaneously.
The equilibrium theory has received a degree of attention out of pro
portion to its merits, simply because of the facility with which it is
obtained. No results can be deduced from it, in general, which cannot
be deduced from the attractive forces (or their potential) with equal
ease and with greater generality. However, its present importance is
that it is a convenient standard of reference in tidal matters; the contri
butions of the tidegenerating forces to the dynamical equations are
conveniently expressed in terms of the equilibrium tide; and in theoretical
investigations the results are compared with the equilibrium tide, as is
also the case with tidal analysis, so far as the phases of the constituents
are concerned.
The failure of the equ,~t,ibr~n,m theory to express quantitatively the tides
of the main oceans is due, of course, to the untenable assumption that
the inertia of the water can he neglected.
In small basks, however, it is reasonable to suppose that we can neglect
the motion of the water. The dynamical equations of motion describe
the relationship between the components of current on the one part and
the difference between the actual tide and the equilibrium tide on the
other. If the water is deep then the currents must in any case be small;
also, the zero values of currents transverse to the boundaries must also
tend to keep down the strengths of the currents. Hence it appears that
in small seas we may expect that the actual tide will approximate to the
equilibrium tide, suitably corrected it should be noted that this expecta
tion has not been rigidly proved. NVe have also to determine the possible
effects of resonance, and simple calculations give the following free periods
for a nonrotatin~, rectangular basin:
Depth 100 feet
Length = 100 miles 5.2 hours
Length = 500 miles 25.9 hours
1~000 feet
1.6 hours
8.2 hours
10,000 feet
0.5 hours
2.6 hours
These will serve as a useful, though very rough, guide to the free
periods to be expected. It is clear that for a basin 500 miles long the
mean depth must be somewhat greater than i,000 feet before we can
feel reasonably sure that marked resonance of the semidiurnal tide is
* See Thomson and Tait, Natural philosophy, 2nd ea., Pt. II, p. 815.
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24
FIGURE OF THE EARTH
unlikely to occur. Supposing resonance to be absent, then the general
result sketched out above is applicable to any shape of small sea and to
any law of depth provided that the mean depth is not small, and the
problem is simply to determine the correction, C, referred to above.
For a rectangular basin it is a simple matter to show that the small
portion of the equilibrium spheroid may be taken as a plane and that
the condition for continuity of volume yields the result that this plane
must always oscillate about its central point. The results are suf~cient.ly
interesting and important to justify the formula for the motion being
quoted; if we take axes Ox, Oy eastwards along the parallel of latitude
and northwards along the meridian respectively, with A as the latitude
at the central point O; ~ the speed of the tidal constituent,; t the time,
measured from the moment of maximum tide~,enerating potential at O;
then the elevation is given by
I=A cos A(y sin A cos crt+x sin at)
where A is a positive constant. The formula does not hold near the Pole.
When t=0 we have high water at all points south of 0, and low water
at all points north of 0, on the meridian. Three hours later, ~rt=,r/2, we
have high water along Ox to the west of 0, and low water east of 0. Also
for any given time high water occurs simultaneously along a. straight line
radiating from O; and if' this straight line is produced through O it
joins all the points at which low water is then being experienced. A line
joining points at which high water is being experienced simultaneously
is called a cordial line; it is not usually a straight line. It is very con
venient to describe the tidal motion in any basin by means of cotidal
lines. In the present instance the cotidal lines all radiate from one point
at which no tide is ever. experienced; such a system is called an amph~
drom~c system and the notide point is called an amphidrom~c point.
Again, ire this instance, the times of' high water are progressively late as
we move around the a.mphidromic point in the negative sense (that. of
the hands ol' a watch) and the rotation is said to be negative; in the
southern hemisphere the rotation is reversed. It is impossible to over
est.imate the importance which we must attach to amphidromic systems
such as we have described, for such systems are the rule rather than the
exception, not only in small seas but also in the large oceans.
Further interesting, and typical, conditions can be deduced from the
simple formula for the rectangular basin; the cotida,1 lines are not, equally
spaced out but spread out from the center lily two fans symmetrically
Placed, the fans being almost closed near the equator; in the limit we
have the tidal motion near the equator (A=O) given by
I=Ax sin at
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TIDAL THEOR Y
20
so that high water occurs simultaneously throughout the western half,
and low water simultaneously throughout the eastern half, when at=~/2.
We thus have two cot~dal areas and the water oscillates about the central
meridian in a seesaw fashion; at t=0 the whole surface is horizontal.
Such cotidal areas are also experienced in the oceans.
The formula quoted above is true for any small flat sea, if O is the
centroid in the mathematical sense, and similar results have been obtained
for a circular and also for a serr~icircular sea. In the latter case an exact
solution has been obtained by J. ProuUman and this indicates that for a
sea comparable with the Black Sea the solution is only just appreciably
different from the corrected equilibrium solution.
The problems of oscillations in narrow bays are very varied and they
are quite different in character from the problems just. discussed, for
such tidal oscillations as exist are maintained by the oscillations generated
in the main ocean, and the variation of the equilibrium tide within the
bay is relatively negligible. Also, the rotational effects of the earth are
usually negligible for a narrow bay; the rotation of the earth, of course,
tends to throw the Hater to the right o:t' its motion, but the result is
inappreciable except in a broad basin. A number of simple cases have
been worked out, such as
a) depth constant, breadth varying with distance from the closed end;
b) breadth constant, depth varying with distance from the closed end;
c) breadth and depth varying with distance from the closed end.
These solutions show that tides tend to increase in range towards the
closed end of' the bay, owing to the restriction of the cross sectional area.
Ultimately, frictional effects become prominent, but it is remarkable
how these simple approximations account for the la.r'~,e tide in an estuary
such as the Bristol Channel.
Considering now the subject of' broad seas which carrot be considered
small enough for the corrected equilibrium theory to be applicable, then,
even when rotation is absent, the problem is a difficult one. Certain
cases have been evaluated on the supposition that the earth does not
revolve and the fluid is bounded by two meridians; the solutions for
these special boundaries show an intricate series of amphidromic points.
In order to gain some conception of the tidal motion in enclosed oceans
a considerable degree of attention has been paid to motion in long canals;
thus G. B. Airy in 1846 discussed the case of' a canal of varying, rectan~u
lar section lying, alone, any complete great or small circle of the earth's
surface, and his most important cases were treated by Lamb (1895~.
In 1910 H. Poi~:~care discussed the general principles for the treatment
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26
FIGURE OF THE EARTH
of a network of canals, and H. Lamb and L. Swain in 1915 elaborated the
solution for a canal of uniform rectangular section occupying only a
portion of the earth's equator. Until quite recently no solution was
available for a large ocean except that given by Laplace in 1776 for an
ocean of constant depth covering the whole earth, or bounded only by
the equator. This solution has given rise to much discussion, as it was
disputed by Airy and by W. Ferret, but was vindicated by W. Thomson
(Lord Kelvin) in a very remarkable paper.
Generally speaking, a system of linear differential equations can be
satisfied in an infinite number of ways, and the problem is to combine
these solutions so that they satisfy all the conditions at the boundaries;
there is only one such combination appropriate to the given conditions
but the same independent " complementary functions " can be combined
in different ways with a " particular integral " to satisfy the conditions
for a large number of different conditions. In the tidal problem a large
number of functions can be expressed, either algebraically or numerically
which formally satisfy the equations, and these functions have to' be
appropriately combined to satisfy the conditions of zero velocity trans
versely to the boundaries.
Returning to Laplace's problem, there is only one " boundaryeondi
tion '' and that is at the equator; considerations of symmetry show that
the velocity transverse to the equator must be zero. Now Laplaee, without
adequate explanation, essentially combined a particular integral and a
complementary function so as to satisfy this boundarycondition in one
operation. Thomson's explanation, to many, only made the matter more
mysterious than it really is, but the method of combining two separate
solutions, as indicated, is very simple and the full solution has been
elaborated in this way by Doodson (1928) for the semidiurnal tide.
Laplaee's solution has been elaborated by Hough, who took also into
account the selfattraction of the water. The results obtained by Laplace
and Hough may be summarized as follows:
R.\T:IO OF TIl)E lo CORRESPON1)IN(! EQLT1LIBRIUM TIDE
On
the
Equator
Depth of ocean
f. Longperiod Laplace 0.455
tides 1 Hough 0.426
'`,SGO foot
0.551
14 r,)0 feet 29,040 feet .SS.0£0 feet
0.708
0.681
0.817
0.796
Semidiurnal I Laplace 7.4341.821 11.259 1.924
tides 1 Hough 7.9551.502 234.9 2.139
At the J Longperiod (Laplace 0.154
Pole :` tides Though 0.140
0.266
0.470
0.443
0.651
0.628
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TIDAL THEOR Y
27
Also, at the Pole, the semicliurllal tides are practically Vilely by the
Equilibrium Tide, but in any case the t:icles there are smalls
The diurnal ticl.es are rather remarkable, in that their ranges are
everywhere very small indeecl. On the whole, the lon~period tides are
" direct " ill that the:: have hi~,h~ater s:nchrol~ousl: faith tl~e equilib
rinn~ titles; the equilibrium tide has noclal lines (lines o:t zero range of
tide) and these are slightly shifted in the solutions obtainecl by Palace
v ,,
, ~
4 ,} ,)
AT ~ ~1 · ~ · ~ ~ · ~ · ~ I · I
ancl llou~h. l he sem~d~urnal t:rcLcs prov:~cLe most interest; for large
depths they are direct and tend to be represented by the equilibrium
values as the depth increases; but when the depth is about 29,040 feet
the range of tide on the equator becomes very great. Laplace's results
inclicate that a depth between 14,320; feet and 29,040 feet is " critical "
in the sense that resonance occurs at that depth for squaller depths
the tides are "inverted" (high water occurring, when the equilibrium
tide would <,ive low water) and for larger depths they are " clirect." A
seconcl critical depth is indicated below 7,260 feet. By allowing for the
mutual attraction o:t the riveter llou~,h obtained slightly clifferent values
for the critical depths; also, IIou~h iou~cl that when the depths are
29.040 feet and rY,260 feet then the periods of free oscillations are i?
hours ~ minute, and 12 hours ~ minutes, respectively, anal he concluded
that the tides will not become excessively large unless there is very close
agreement between the periods o:t the tidal constituent and free oscillation.
The case of a POINTS Ocean iS readily treated in the sane manner as
that of the ocean bounded by: the equator, and a solution has been pub
lished be G. It. Golclsboro~,l:~ (~)13; see also Doc~aso~' 1998~. His
results Cain be s~narizecl as follows:
~. , . . ~. . . .
RATIO OF TIDE IN POLAR OCEANS, BOUNDED NT LATITUDE 60, TO EQUTI IBRTUN! TIDE
At the pole At the boundary:
~, ~
Lo7~period Lolls per iod Diurnal Semidi~rnal
tides tides tides tides
.. 0.100  0.216  2.86 1.0~0
. 0.133  0.2a=7  1.16 1.030
. 0.158  0.269  0.194 1.014
. 0.176  0.2~6 ~.386 1.003
]]el)tl,
~ 260 feet ........
14.520 feet
29.040 feet
58,080 feet
For the lon~period
ticles, if He c onsiclered the ocean as s~nall and
computed the correc tion as explainecl previously, the ratio of tide to
equilibriumtide would be 0.192 at the pole and0.293 at the bounclary,
so that if the depth is very great the corrected equilibriumtide is a
close approximation.
The se~nidiurnal tides are practically equal to the equil:ibriu~ticles
of the same species, the " correction " being, zero. The diurnal tides are
of most interest in. view of the fact that these are very small incleed when
3
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28
FI CURE OF THE EAR TH
the water covers the whole earth; we have a further illustration here
of the remarkable effects of boundaries.
Goldsborough finds that the largest critical depth for resonance of the
semidiurnal tide is about 526 feet when the boundary is in latitude 60°
and about 38 feet when the boundary is in latitude 7~° 3O'; the smallness
of these depths precludes the possibility of resonance in Polar oceans
of depths comparable with actual average depths of the oceans.
An ocean bounded by two parallels of latitude can also be similarly
discussed. The results obtained by Goldsborough (1914) are as follows:
ZONAL OCEAN BOUNDED AT LATITUDES 30 AND 14 30 : RATIO OF TIDE TO
EQUILIBRIUM TIDE
..0O :30 °
T,atitude T4° 30'
~. ~
Sergei Lon~ Semi
Lon~period Diurnaldiurnal period Diurnal diurnal
Depth tides tidestides tides tides tides
7,260 feet  1.10  0.100.48 0.31  0.5~3 0.39
14,520 feet  1.20  0.39 0.58 0.31  0.99  0 07
29,040 feet  1 .23  0.84 1.46 0.31  1.63  0.78
58,080 feet  1.24  42.61.81 0.31  31.0 0.37
ZONAL OCE.~N BOUNDED AT LATITUDES 30 AND  14° 30 : RATIO OF TIDE TO
EQUILIBRIUM TIDE
Latitude 30° [Latitude _14 ° '30'
_k ~, ~
Semi Lon~ Semi
Lon~rperiod Diurnal diurnal period Diurnal diurnal
Depth tides tides tides tides tides tides
7,260 feet  2.04 1.95  0.59  0.018 3.48  0.46
14,520 feet  2.14 1.21 4.49  0~.010 2.16  3.85
29,040 feet  2.20 0.45  1.11  0 004 2 19  0~70
58,080 feet  2 21  3 00  6.65  0.002 0.69  3.44
Again it is found that the longperiod equilibrium tide corrected for
boundaries gives a close approximation to the actual tides, but for the
other species of tides the corrected equilibrium tide offers no useful
indication of the true tidal motion. The effects of the boundaries on
the diurnal tides may again be noted, extremely large values being ob
t.ained in one instance. Goldsborough determined the critical depths for
semidiurnal tides and for diurnal tides.
The solution of the problem of tides in an ocean e~tendq~r~g front Pole
to Pole arid bounded by two meridians is much more di~eult than any
yet considered, for it is theoretically necessary to deal with an infinite
number of functions in order to make the velocity zero a.eross the equator
(for the semidiurnal tides) and across the bounding meridians. The
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TIDAL THEORY
2'3
problem can be formally solved but the actual arithmetical solution is
exceedingly difDcult. A solution has recently been published by Golds
borough for ail ocean whose depth varies as the square of the cosine of
the latitude, in which case the tidal equations are somewhat simplified.
The form of the solution does not permit of the determination of the
free periods of oscillation but it gives, what is almost as useful, the
critical depth at which resonance takes place. By taking the bounding
meridians 60° apart and the mean depth as 16,980 feet, results roughly
applicable to the Atlantic Ocean were obtained. Previous papers by the
same author had demonstrated that large semidiurnal tides cannot be
generated in the polar oceans with the depths known to occur. IIe now
remarks that this points conclusively to tile impossibility of large semi
diurna.1 tides in the Southern Ocean, even though some allowance be
made for the Antarctic Continent. Consequently, large oscillations in
the Atlantic Ocean must be~ due to resonance and not to propagation
northwards. His results indicate that the critical depth. in the cited
case is 1~o,500 feet; the mean depth of the Atlantic Ocean is 12,?00 feet
and for resonance to occur with this mean depth the bounding meridians
should be about 53° apart. Allowing for varying contours and the form
of depth, this seems strong, evidence that the Atlantic semidiurnal tides
are due to partial resonance; in the case Dollied out. the tides were 50
to 60 times the equilibrium tides.
Shell further elaborated, and fully illustrated, this solution should
give a fairly satisfactory, and extremely instructive, representation of
the tidal motion in such an ocean as the Atlantic Ocean.
The theoretical studies outlined in the preceding paragraphs are ex
tremely valuable in spite of their artificiality, but the important problems
of the future will be concerned with coccal seas arid oceans. The ideal
solution is one which tallies account of all irregularities of form of basin,
and only in a fear instances has it been found possible to bring all the
irregularities of a natural body of water, just as they exist, into a
dynamical treatment. Apart from the investigation of the periods and
elevationform of seiches, which are essentially free oscillations, in Lal~e
Geneva, the only instances of complete treatment in this respect are due
to Sterneck and Decant, who have dealt with narrow seas and. channels
such as the Adriatic Sea and the lied Sea. These examples deal with
forced oscillations, the motions at the mouths of the seas being taken
from observations. Such methocls, however, have been applied to seas
and channels so narrow that transverse motions have been negligible,
and the incorporation of all geographical details for a broad sea such as
the North Sea has not yet been achieved. In general, therefore, the de
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30
FI CURE OF THE EA R TH
tailed explanation of tides is very incomplete, and attention has been
largely `~,;ven in the past to the <
TIDAL THEORY
31
The importance of resonance was much. emphasized by P. A. Harris,
and he attempted to select special portions of the watersurface of the
earth in which the free tides have speeds nearly equal to that of the
tidegenerating constituent; he also assumed that the free tides have a
general character which can be cleterm.ined without taking into account
the earth's rotation, and that the reaction of the water in other parts
of the oceans can be ne~lect,ed. These latter assumptions malice it ex
to
W.?
(~e
In'\

...:
~:'~''~\W ~
'.'...2'.''..'.'.2.'.'.''~ ~
4'~'~'.'.~', ~,~
3 //
~ ::
~/ )~43/
4'~"'~':~1:
FIG. 1. Cotidal lines for the North Sea according to
W. Whewell, 1836.
tlemely doubtful whether Harris' hypothesis can lead to a satisfact,ory
explanat.ion of' the tides in any of' the oceans, but undoubtedly the
importance which he attached to resonance has made for progress in
these matters.
In 1~)20 If.. Sternest proclucec~ a. set of coticlal lines for the se~idiurnal
constituents for all the oceans, and in 1922 a corresponding, setf'or the
~l.iurnal constituents. The orals, principle ut:ilizecl appears to lie that co
tidal lines for phases cli~'erin~, by ~ shall be as nearly parallel as is con
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32
PI GURE OF THE EAR TH
sistent with observations, and there may be theoretical justification for
this principle.
It will be most useful at this stage to consider the tides in the North
Sea. Omitting all frictional consiclerations and, for the present, the
rotation of the earth, then any tidal oscillation in the ~'\tlant.ic Ocean
would maintain a "standing oscillation" in the North Sea with two
9/
'~41\ /~~
it'
rat
::~ ~
.',,.,,,,.,,,,, ~
1~\\\~\
FIG. 2. Cotidal lines for the North Sea according to
R. A. Harris 1904.
nodal lines about the lat.itucles oo° id. and .~8° IT, as may readily be
shown 'iron considerations of' emotion in a nonrotatin~, trough. Along
these lines would be zero range of' tide, but st.ron~, tidal currents across
the lines would exist three hours before or after the hours of high water
in acljoinin~, regions. The effect of the earth7s rotation on these currents
would be a surface ~,raclient along, the line, and so there would be a tidal
oscillation of level on. the line, the range now being zero at only one
point of it. The phase of this superposed oscillation will thus change
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TIDAL THEORY
33
b.y 180° as we pass through this point, and similarly the phase of the
original standing oscillation changes by' 180° as we pass through the
point transversely to the nodal line. NATe have, in fact., an amph.idromic
point. An explanation of this sort agrees fairly well with Sterneck's
chart.
~ purely mathematical investigation of the tides in a rot.atin~ rectangu
lar gulf of constant depth and with approximately the same dimensions
W~
..; .: . . ~ ~ ~ ~ . ~
. .,: I, :2. :~J
'I,/
;~
A'''.' '...,'..2~ ~ it,
FIG. 3. Cotida1 lines for the North Sea according to
R. Sternecl;, 1920.
as the North Sea has been ~,~iven by O. I. Taylor, and his results show
two amphidromic points s:~n~Letricallv placed in the gulf.
Sternecl<'s map is thus an advance on previous maps and it gives a
good idea of the actual conditions. Ii we include the effect of friction,
however, it becomes apparent that the range of the tide decreases north
wards along the continental shore of the North Sea, and thus there is a
bias ot the amphiclromic point towards this coast, the bias increasing,
northwarcls; the northerly amphiclromic point should thus be much nearer
than the southerly point is to the coast.
, ~,. ,. ~
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34
FI GURE OF THE EAR TH
Theoretically, the position of' an amphidromic point depends upon the
speed of the tidal motion, and each harmonic constituent of the tides
has its own amphiclromic Anoint, though all constituents of' one species
will have their amphiclromi.c pouts clear to one another. Thus, actually,
the range of tide will not always vanish, though it will be small near the
amphidromic point of the principal constituent. At and near the amphi
dromic point for the principal lunar semidiurnal constituent the tides
of other species may become relatively prominent.
In the actual drawing of coticla1 lines much use can be made of the
known effects of capes, bays and islands. These have been investigated
bar Harris and by ProuUman. An island may have very remarkable effects
upon the tides in its vicinity and in general cotidal lines converge on a
cape and diverge into a bay.
It is somewhat doubtful whether satisfactory cotidal charts for the
larger oceans yet exist, as there is still a large element of uncertainty as
regards the validity of' the hypotheses. Where approximately exact maps
are required special methods must be used. Of course, observations of'
the rise and fall of tide at a large number of' places throughout a sea
1 V
will provide ample data for drawn such charts, but direct observation
is very difficult and, even when practicable, is very expensive, so that
extremely little valuable information has been obtained in this mariner.
Observations of' tidal currents are more easily made, and the relation
between current and elevation can be exactly expressed in the dynamical
tidal equations rel'errecl to above. From the currents it is possible to
compute the spacerate of' increase of' elevation in any direction and if'
sufficient observations are available then methods old numerical integration
can be applied to these =~raclient.s of elevation from coast to coast. For
the North Sea such a method was carried out at the Tidal Institute,
and use was also made of' much miscellaneous observation, with a satis
:t'actory measure of inte~a=~reement. The resulting chart is illustrated
in Figure 4 and a slightly different form of it was adopted by the
British Admiralty. The position of the northerly amphiclromic point is a
~ . . . ~ . · . . · · 1 ~ ~ 1 ~ 1 ~ 1 1 _ _ __ _ _ _. `` ~ _ ~ ~ _~ _ an._ 1 ~ '' / 1~ ~ ~
little uncertain; it is possibly the case that It becomes Degenerate ~ final
is, the actual point to which the coticlal lines converge may be inland)
but, as the tides near Norway are very small, it does not matter appreciably
where the point is situat.ecl. The chart is appropriate only to the principal
lunar constituent M.>; the time of' high water at a given place, in mean
solar hours after the transit of the moon at Greenwich, is equal, on the
average, to the phasel divided by 29. It may be noted that the method
outlined Eves also coran~e lines; on the chart illustrated the semiranges
O
are given in centimetres.
OCR for page 19
[ recopy
35
TO this Outbox no Use bus been Ague Of the equation of continuity
which expresses the elevations in terms of the sp~cegr~]ients oT the
currents; ant Detest has given ~ chart utbising oI(y this principle. Obese
FIG. 4. CotidulEDeG for the~ortbSe~uCco~ingtotbeTid~llDstitute, 1923.
Condo. Cowl.
~ ~ gener~1 agreement betwceD his chart RD] Abut of the li4~1 IDsthUte
but his lines Ace gore curve], especially Bear the coast. Since even
the best current information is not very exact it ma re~son~b~ {e SUg
geste] that the use of the continuity equation, involving DUme~ic~1 di~er
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36
FI CURE OF THE EAR TH
entiations, is interior to the use of the dynamical equations, involving,
numerical integrations. From the point of view of tidal theory, however,
it is very interesting that two such dissimilar methods should give similar
results.
Returning, to the general problem of the construction of maps of
cotidal lines for 13road Seas and Oceans, a method has been devised by
Proudman, based upon exact clynamical equations, which theoretically
determines the tidal motion, at any one point of a sea, from a knowledge
only of the coastal observations, ~eograpl~ical deposits of the basin, and
of the tidegenerating forces. It requires the evaluation of certain func
tions over the sea and it has not yet been exploited.
The ideal result of the dynamical theory of the tides will be achieved
when it is possible to dispense even with the coastal observations, and
indeed Poincare reduced the complete clynamical explanation of the tides
to a sequence of direct mathematical operations, but he fully admitted
that the amount of computation required to apply his process to the
actual oceanic basin was entirely prohibitive. A mathematical method
of the same general nature, though entirely different in detail, has. been
devised by Proudman, and, though the amount of computation involved
ir1 its application is large, it is believed to be not entirely prohibitive.
It is quite likely that the next great advances in tidal theory will be
corrected with methods of numerical integration applied to a.ctua.1 oceans,
for it is in the interpretation of the mathematical results that. the chief
difficulties lie. Generally speaking,, therefore, we ma: close our survey
of the subject of tidal theory with the remark that of late years much
definite progress has been made and that the general lines of development
~ ~ _
~ ~ ~ ~ ~ { ~ 1 ~ 1 ~
of the subject are quite clear; It IS a mailer OI labor and patience to
achieve the ultimate state of knowledge referred to at the outset.
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l'lDAL THEORY
37
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