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OCR for page 68
CHAPTEP V
EARTH TIDES
W. D. LAMBERT
U. S. Coast and Geodetic Survey
The ticle-producin~ forces of the sure and moon are not confined to
the surface of the earth but act throughout its mass. Since in nature
there is Do absolutely rigid substance, there must be some yielding to
these forces. This yielding or deformation of the mass of the earth-
considered separately from the water on its surface is called an earth
tide. For brevity let us speak of the "solid" earth, meaning thereb:-
all the rest of the earth except the atmosphere and the hydrosphere, and
not thereby denying the possibility that the substance composing the
central portions of the earth may in many respects act like a very dense
liquid under high pressure.* In addition to the deformation due directly
to the action of the tide-producing forces in the " solid '' earth, there are
obviously deformations transmitted downward from the surface and
caused by the varying, tidal load of the ocean water. These are known as
secondary earth tides and contribute greatly to the complication of the
subject. lTor the sake of simplicity, therefore, we shall consider at first
only the primary earth tides; that is, the deformations of the " solid "
earth as they would be if the hydrosphere were absent; the atmosphere
may be disregarded in this connection.
The oceanic tides of nature are not of the simple kind so often shown
in diagrams illustrating the nature of tidal force; that is, there is no
smooth tidal bulge with its maximum elevations at the two antipodal
points where the tide-producing body is ire the zenith and the nadir.
This fact is expressed by saying that the oceanic tides are not of the
equilibrium type.! One reason for this is that the oceans are too shallow;
their shallowness makes their natural periods of oscillation rather ion,
in con~parison with the- diurnal and semi-diurnal tidal forces. However,
the depth would not have to be increased many fold before it would be
* See The constitution of the interior of the earth, a forthcoming bulletin of this
series.
~ Note the distinction between tides of the equilibrium type, and the equilibrium
tide in the strict sense. (See p. 22.) The height h of the latter is given by
h= V/g, where ~ is the potential of the tidal forces in question and g the ac-
celeration of gravity. The equilibrium height of tide in this strict sense has previ-
ously been used for comparison purposes (see p. 26) and will hereafter be so
used in this chapter.
68
OCR for page 69
EAR TH TIDES
69
sufficient to give ocean tides of the equilibrium type. Now the earth tides
have a certain analogy to tides in a very deep ocean, an ocean extending
clear to the center of the earth. We may therefore expect the primary
earth tides to be of' the equilibrium type, and in fact they are.*
Some idea of the short period required for the earth as a whole to
adjust it,sel:f to equilibrium with deformations of the tidal type may be
obtained from the following: An impulse, such as an earthquake distur-
bance, takes a little over 20 minutes to travel along, a diameter, or say
41 minutes for the round trip. A liquid sphere of the size and mean
density of' the earth would have a natural period of 94 minutes for
deformations of the tidal type. An incompressible elastic sphere of
the size and mean density of' the earth and the rigidity of steel would
have a natural period of 35 minutes. If we disea.rd the simplifying as-
sumption of incompressibility the period is increased by six minutes only
for an assumed rigidity equal to that of steel and by a little less for a
Breather rigidity. Thus from any point of' view it is clear that the earth
has time to adjust itself to the instantaneous field of force of the semi-
diurnal and diurnal tides and a forttori' to the force-fields of the long-
period tides. The primary earth tides are therefore of the equilibrium
type. Friction might conceivably alter the phase, so that the maximum
height of the tide would occur a little alter the moon's meridian pa.ssa¢,e,!
but it will be seen in a later chapter that the lag in phase due to friction
must be very small.
The amount of' tidal rise and fall in the " solid " earth is a question
of' great geophysical interest, for it is closely connected with the properties
of the substance of' which the earth is composed. Of course we cannot
stand outside the earth and measure its expansion and contraction with
~ giant pair of calipers, but we are obliged to infer the rise and fall indi-
reGtly :t by a process presently to be described; for the moment let us
think of the tidal rise and fall in the solid § earth as if it. could be
measured directly.
NVe can easily compute what the tidal rise and fall would be with refer-
e:~:e to the center if the earth were entirely destitute of rigidity, that is,
* The methods by which earth tides are observed will be treated latter.
~ This means that in space the tidal bulge is in advance of the moon, counting
in the direction of the earth's rotation and of the moon's orbital motion. In time
there is a lag in phase.
~ The rise and fall of the ocean tides is observable only relative to the adjacent
land; that is, both the water surface and the land surface change their distances
from the center of the earth and the observed change in the height of the water
is the (a.l,,ebraic) difference of the changes in their distances from. the center of
the earth.
§ From here on the quotation marks enclosing the word solid used in this con-
nect.ion will be omitted, but may be supplied mentally by the reader.
OCR for page 70
JO
FI GURE OF THE EAR TH
if the only resistance due to change of form were that due to gravitation.
This statement rests on two premises: 1) that the ellipticity of the earth
due to rotation represents ~ deformation of' tidal type; 2) that any elastic
stresses due to the rotation that may originally have been present have
so decayed, owing to the lapse of time and to the fact that they often
exceed the resisting, power of the earths substance, that the resulting,
ellipticity due to rotation will be the same as if the earth's substance
had no rigidity whatever; that is, as if it were liquid. The first premise
may be proved mathematically; the second premise seems highly plausible
from what we know of materials under stress and at such high t,empera-
tures as we suppose to prevail within the earth, but it, cannot be asserted
as an absolute, demonstrable truth. The mere rotation of the earth would
impart an ellipticity equal to 1/~80 to the originally spherical equipo-
tential surfaces surrounding a perfectly rigid sphere of mass equal to
that of the earth and having, for the radius of the outer equipotential
surface the mean radius of the earth. This corresponds to the equilibrium
tide in the strict sense of the word. (See footnote, ante.) The actual
ellipt,icit,v is 1/297. This increase is due to the self-attraction of the
equatorial bulge produced by the rotation. The increase would be even
larger if the earth were homogeneous, for the ellipticity would then be
1/232. The ratio of the a.ctualyieldin~, (tide) to the equilibrium tide in
the strict sense is a pure number which we shall denote by h. Here for
the fluid or plastic yielding of' a body like the earth
7~= 1l/397=~.9~.
If, on the other hand, the earth opposes to the deformation elastic resis-
tance as well as gravitational resistance, then the value of h will be much
smaller. If' we treat the earth as of uniform density and incompressible
and ascribe to it the modulus of rigidity A, the value of h will be
5 1
+ ~
Papa
(1)
where a is the radius of tile earth, ,~ gravity at its surface and p its mean
density.:
For ,u = 8.0 x 1011 c.~.s. units, which represents about the rigidity
of steel and corresponds to an estimate formerly much in Vogue, h = 0.783,
for ,u = 20 x 10~t c.g.s. units 7~_ 0.380, which corresponds roughly to
* Note that if ,u = 0, which represents the case of fluid yielding' h= 5/2, which
is the value for a homogeneous earth, i. e., `~
1/232
1/580
OCR for page 71
EAR TH TIDES
?'1
what is actual!: observed. The amplitude of the equilibrium lunar semi-
diurnal tide, 11~`, at the equator is about 24 cm. (10 inches); the actual
amount by which the surface of the earth rises above its mean position
or falls below it is then 0.3x0 inches=a~ inches for ]~2- Since M2
is the; largest of the partial tides. (see p. 42) and since it attains its
maximum range at the equator, it is evident that the actual rise and
fall of the earth's surface due to the primary earth tide is generally a
matter of a few inches.
It is not intended to imply that it is beyond our mathematical power
to dispense with the simplifying assumptions of incompressibility and
of uniform density and elasticity; the assumptions underlying the calcu-
lation can be brought into closer conformity with the presumed actual
conditions prevailing within the earth but only at the expense of an
enormous increase ire the mathematical difficulties of the problem, an
increase attended by a smaller change in the values of h, and of similar
related quantities, than might at first be expected.
Herglotz s made computations that allowed for the decrease in density
from center to surface; the assumptions of uniform modulus of rigidity
and of incompressibility were retained. Schweyclar ''3~'4~'5 made a number
of computations on the subject on various hypotheses of variable density,
and rigidity; he also included the effect of oceanic tides in an ocean
assumed to cover the entire earth, but retained throughout the assumption
of incompressibility. This assumption greatly simplifies the calculation
for a gravitating body as large as the earth, for with so large a com-
pressible body the methods of the ordinary mathematical theory of
elasticity are inadequate. Love i~ showed how to allow for the compressi-
bility of a gravitating, sphere, but did not introduce the complications of
variable density and rigidity. Hosl~ins 9 included compressibility and
variable density and rigidity in his calculations. The numerical calcula-
tions made in most of these papers except Schweydar's-were made
under the influence of the idea that the earth is about as rigid as steel and
the effects of higher rigidities were not as a rule considered. None of the
papers previously cited develops the comparatively modern idea, based
chiefly Ol:1 seismological evidence, of a liquid core with a radius somewhat
greater than half the radius of the earth. This has been done by
Je~reys,~° ii who has, however, made use of various simplifying, assump-
tions. Thus there has been so far no complete mathematical solution
that takes full account of modern ideas regarding the constitution of
the earth.
Historically speaking, the first determination of the earth titles. came
by way of the long-period ocean tides. A tide in the water can be observed
only with reference to the land. This observed quantity may be thought
OCR for page 72
~2
FIGURE OF THE EAR TH
.,
of as the difference between the rise and fall of the water with respect
to the center of the earth and the rise and fall of the land with respect
to the same point. Tides of long, period (a fortnight or a month) were used
because it was assumed that they could be treated theoretically, whereas
the diurnal or semi-diurnal tides were well known to be beyond any
known mathematical treatment at all adequate to the purpose in hand.
The theoretical treatment actually used for the lon~,-period tides was
fairly simple. If TY represents the potential of the lon<,-period tidal
forces, the height of the "equilibrium tide" (referred to the center)
in a non-attra.ctin~ ocean covering an unyielding globe is simply W/q,
~ v ~ v ~ ~ v , v
where g Is the acceleration of gravity. But the earth does yield and its
deformation gives rise to a ~,ra.vitational potential of the same type as
W; this additional potential may therefore be written 1cW where k is ~
pure number like 1-',; its value, like the value of h, depends on the distri-
bution of density within the earth and for the case of homogeneity we
have k=3/a, Tether the yielding be elastic or plastic in character. For
the actual earth we seem to have more nearly 7~=:h.* The " equilibrium
height" of the oceanic tide (referred to the center) instead of bein,
1V/g is (~+k) 1~/g. The tide in the earth (still referred to the center)
is, as we halve seen, h.W/g, so that the observed differential tide of the sea
referred to the land is (1+7c-h) W/g. We know W/g from theory, we
determine the whole expression (~+k-h) W/g from observation and
thence we deduce 1+k-h. A determination from the long-period tides
was published in Thomson and Tait's Natural Philosophy and in revised
form in Sir George Darwin's Collected Scientific Papers.i The observa-
tions of lon~-period tides at all available ports gave for 1+7~-h a value
a little greater than two-thirds. The Indian ports, where long series of
tidal observations were available but which afforded only a small range
of latitude gave a value nearer to unity.! The value 1 + 1~-7~ = 2/3 cor-
responds under the simpli.lyin~, assumptions on which formula ( 1 ),
page 70, was derived to a modulus of rigidity equal to 7.3 x 10~i, or a
trifle less than that of steel.
There are various corrections that might be made or objections that
might be offered to this method. The sel:~-attraction of the water is not
provided for, neither is the effect of the continents on the amplitude and
phase of the tide. Either correction is a simple matter by itself. The
* In this case there is no theoretical reason why the ratio of k to h should be
absolutely identical for elastic and for plastic yielding, but apparently the ratio
is not very different in the two cases.
~ A value of 1 + k-It equal to unity corresponds to an absolutely unyielding
earth, a value equal to zero to an absolutely plastic earth. The nearer the value
is to unity the greater the modulus of rigidity.
OCR for page 73
EAR TH TIDES
~3
self-attraction of the tidal water assumed to cover the entire globe may
be allowed for by means of a simple theorem of the theory of spherical
harmonics; a further mathematical advance has been made by Street '0
in allowing for the pressure of the varying load of tidal water on the
surface of the globe.*
If we omit the self-attraction of the water, the presence of the conti-
nents may be allowed for by the "corrected equilibrium theory" of
Darwin and Turner. Neither of the corrections just mentioned is large,
but unfortunately their joint effect cannot be found by applying the
principle of the superposition of small corrections. The fundamental
difficulty, however, lies in the question of whether the oceanic tides are
really of the equilibrium type. It has been letdown since the time of
Laplace that tides of quite another type (see p. 26) are possible Ott
a rotating globe covered with a frictionless liquid. It had usually been
assumed without much examination that in the case of nature friction
and the presence of the continents would prevent the existence of such
a type of tide and would make the actual tide approximate rather closely
to the equilibrium type, the friction being sufficient to prevent existence
of the non-equilibrium tide just mentioned, without being enough to
introduce any marked lag in the phase. These assumptions, however,
are no longer so generally accented as they were. and so the method based
- - 1 ~ ,
on them tor ctetermlulng the earth tides is no longer considered a
satisfactory. It should be said, however, that the results obtained are ire
general agreement, in spite of these theoretical difficulties, with those
obtained by the method next to be described.
This method consists in determining the earth tides by the changes
they produce in the direction of the vertical. These changes may be re-
ferred to the direction of the earth's axis or to the ground old which the
instrument for detecting them rests.
The latter method is more common. If the changes were larger, the:
might be measured by the deflections of an ordinary plumb line or pendu-
lum, but since they are of the order of only 0'.'0l, and since an ordinary
plumb line or pendulum is disturbed by earth tremors and by the various
thermal effects of the sun's heat, special precautions must be taken.
The so-called " horizontal pendulum " is more sensitive than the orcli-
nary penclulum. A familiar analogue to the horizontal pendulum is an
ordinary gate hinged to an unsteady post. Any slight wobble in the gate
post causes the gate to swing through a large arc. The Darwin brothers,
* Street's published numerical calculations are made with so low a modulus of
rigidity as to have little application to the case of nature. Moreover, his defini-
tion of equilibrium tide is not the one here adopted, which appears to be the pre-
srailing one. This difference in definition is likely to cause confusion.
OCR for page 74
~4
FI GURE OF THE EAR TH
George and I-Iorace, conducted a long series of experiments with one
form of the horizontal pendulum. Their memoirs 3' 4 may be consulted
for references to earlier experiments. Di:lliculties with the constructio
and installation of the apparatus prevented them from obtaining any
very satisfactory results. The first results that could fairly be called
satisfactory were obtained by von Rebeur-Paschwitz.~4 He was followed
by other experimenters, Heeker,` Orloff',~9 and Schweydar,~ who int.ro-
cluced various improvements. In general, the rigidity of the earth
inferred from their results was about that of steel. One peculiarity that
showed to a. ~rea.t.er or less degree in all their experiments was. that the
actual ellipse cleseribed by the plumb line was not similar to the theo-
retical ellipse for the plumb line on a rigid earth, as would naturally
lie expected; the actual ellipse was found to be not only smaller than
the theoretical ellipse, as was to be expected, but flatter also. This was
interpreted as meaning, that the earth was stiffer in one cl.irectio~n than
in another, and various explanations were attempted, all of them more
ingenious than satisfying,; Loved showed that the supposed causes of
this peculiarity were quantitatively insufficient. The most reasonable
explanation is that it is the result of' the secondary effect of the oceanic
tides; these latter of course are not in phase with the earth tides nor do
they vary with latitude and longitude according, to any simple law. NVhat
is known of the tides in the open ocean makes the explanation seem
qualitatively reasonable and the observed effect seems of the right order
of magnitude but no precise quantitative check has been made.
An entirely different kind of apparatus for measuring, the same
quantity was devised by Michelson and Gale. Any small body of water
like a Lyle tends to set itself perpendicular to the plumb line at its middle
point; as the plumb line swings to one side or the other the water at
one side of the Lyle rises; and at the opposite side falls. Tidal oscillations
of this sort have actually been found in Lal~e Superior. To the eye they
are masked by chan<,es of level resulting -from inflow of water, winds and
variations in level due to non-uniform barometric pressure over the lolls.
Harmonic analysis of observations extending over several years was used
to reveal these tidal oscillations. Michelson and Gale, is got rid of'
the various disturbing factors by using horizontal pipes buried in the
,rouncl. Two pipes, each five hundred -feet long,, one in the meridian
ancl one in the prime vertical, were installed in the grounds o:l' the Yerkes
Observatory at NVilliams Bay, Wisconsin. The rise ancl fall of the water
at one end of the pipe due to a change in the direction of the plumb line
is evidently equal to the horizontal motion of a plumb line with a length
of 2~0 feet, hall' that of the pipe; if we combine the observations at
the two ends we have a fluctuation equal to the movement of a plumb
OCR for page 75
~ ~s
bn~ h~e hundreJ feet lo~g. E~cn vitb this len~T1 tbe line~r displ~ce-
ments ~re of conFsc ver~ minute; heing oT tbe order of ~ ver~ Te~ ten-
thous~ndths of ~n incb. Tbe rise ~] f~M of tbe v~ier in tbe cnds of the
pipes ~s re~] ~t f~st ~ith ~ microsco~c ~nd l~tcr h~ me~ns of ~ speci~ll~
devised inte~ferometer ~pp~r~tus. Ibe cur~es derive] from thc otscr~c]
re~dings ~rc surprisi~gly smootb ~] corrcspon] closel~ ~tcr ~plyi~g
suit~tle ~edurtion I~ctor to thc tbeorctic~1 cur~cs Ior ~ rigid c~tb.
In ~ l~ter form of tbe ~r~tus inst~lled ~t F~s~icn~; C~liforni~: it h~s
been IoUnd possitl~ by incre~sing thc scnsitivencss of the intederomcter
~rtus to sborten tbo pTpes use] ~n] st]1 ret~in tbe ~ccur~cy of tbe
. . . .
OIlglO~I 8.ppR.T~lUS.
In ~e e~TEcr rcports on thc work it v~s st~te] tb~t tbe ~eJUction
f~ctor Ior thc e~st-vcst iTspl~cc~cnt di~erc] Irom tb~t Io~ tbe nQTth-
soUtb displ~ccmcnt; in tbis respcct thc rcsUlts ~ppc~rc] to rcsemBlc tbosc
ott~inc] in ~UTOpC vith ~e boTizoGt~1 pCD0UlU~. ~Ut it RppC~TC] l~tcr
tb~t ~n cr~or t~] tccn m~ic i~ thc cslcUl~tion oI thc thcorctic~1 dcHection
~n] tb~t tbe reJUction I~cto1 v~s pr~ctic~ll~ ~c s~e IOT totb directioDs.
Ibc tiJe w~s ~n~l~sc] into its 1Un~r sc~i-dTUr~l; sol~r scmi-diUrn~l,
n~r diUrn~1 ~] 1Un~r Ior~i~Uy bic Ior totb dircctions; thc l~gs
i~ pb~sc ~crc only ~ Ic~ Jegrees ~n] thc v~rio~s rciUction I~ctors vere
^# e-.
Ibe mo~ulUs oI rigidity Jeduce] Irom the preLmin~ry results v~s
8.0 x lOli c. g. s. units; ~ little gre~ter th~n tb~t oI steel ~d tbe coe~rient
of viscosit~ TO.9 x 101° c. g. s. u~its; ~lso ~bo~t th~t of steel. Tbis m~Les
the time of rel~x~tion (see Cb~pter VT) utout ~ dsy ~nd ~ b~lF ~o
corresponding resUlts ~ere given ~hen the deOnitive reJUctions vere
publisbeJ; ~n] tbere ~re re~so~s. Ior helieving (see Ch~pte~ YI on Ii]~1
Friction) tb~t tbe time oI rel~x~tion is mucb too sm~lt tb~t i~ tb~t tbe
soli] p~rt oT tbe e~rtb ss ~ ~bole possesses murb more ne~ll~ perIect
el~stici~ Ior Iorces oI tbe or~in~ry tid~1 periods tb~n tbe coc~cient of
viscosity ~tove given woUl] i~]ic~te. Ibe l~gs in pb~se ~re prot~hl~
m~inly Jue to tbe seconJ~ ehects oI tbe oce~nic tiJes, ~bicb ~re Ielt
to iist~nces Irom tbe co~st tb~t ~t hrst si~t might seem sUrprising.
Ibe IortnigbJ~ tide, ~bicb; bovever; v~s too sm~ll to Be vcry ~ccur~tely
determi~eJ; b~] ~ neg~tive l~g; interprete] solely ~s tbe resUlt of Iriction
tbis ~oUl] be impo~sible. Ibe trUe expl~n~tion is no donh-~p~rt Iro~
errors of otserv~tion-tbe secon]~ry e~ects of tbe oce~nic tides. Tbe
gre~test presc~t nee] in e~rth-tiJe Jetermin~Iions is not ~ more ~ccUr~te
metbod of ~e~sUrin~ th~m th~n is obt~ine] ~itb ~icbelson ~n] O~le~
~ter-l~veI ~n) interIerometer Jpp~l~tUS; tUt ~ hetier kno~leige of the
oce~n tiJea ~n] ~ more ~dequ~te mutbPm~tic~1 tbeoly for tbe eEmin~tion
of tbeir second~> e~ect~ Only ~ben ~e b~ve tbese, sb~G ve Be ~hle
to Jete~mine ~ccU~tely tbe el~stic ~ield1ng of tbe soli] p~lt of tbe os~tb.
OCR for page 76
76
FI GURE OF THE EAR TH
Of course on the coast, and especially on coasts where oceanic tides
are large over extensive areas, these secondary effects are large; they
snag considerably exceed the primary effects of the earth tides. At Pasa-
clena in one component the observed earth-tide is two and one-]:~al£ times
the theoretical value; at a point in Korea about four times.
The opinion has sometimes been expressed that local geological con-
cl.itions may have considerable effect on earth tides; among such conditions
are supposed to be the presence of extensive beds of rock near the surface
having a modulus of rigidity markedly different from the average and
the nearness of faults or volcano vents. Observations of earth tides have
not been sufficient to test this idea in practice. It seems, however, to the
writer that it would be difficult to separate these effects front the secor~clary
effects of the oceanic tides and that their magnitude is probably exag-
gerated. Earth tides are made from the center of the earth outward.
It is generally supposed that to within perhaps 100 kilometers of the
surface the physical properties of the earth-substance are the same at
any given level, whether the point in question be under an ocean or
under a continent. Now each shell of the earth beginning with a small
central core may be considered as receiving from below a certain amount
of swelling or sinking due to the tides below it and as making its own
contribution to the swelling, or sinking and passing on the net result
to the shell next above it.* The earth tide increases from the center
outward,! and when a level of 100 kilometers below the surface is reached
the earth tide has attained nearly the same value as at the surface. The
crust above the 100-kilometer level does little more than ride upon the
earth below it. For these reasons it seems to the writer that conditions
at tl~.e surface can have little effect on earth tides.+
From certain points of view the best determination of 7c and Or is that
of Schweydar.''4 This is not because the horizontal-pendulum apparatus
used by Schweydar gives better results than the apparatus of Michelson
and Gale but because by a combination of ingenuity and favorable cir-
cumsta.nces the secondary effects of the ocean tides appear to have been
snore nearly eliminated. Schwevclar obtains
.. ..
1+7v-16=0.841 §
~ It is equally true of course that the shells above affect those below as well as
those above.
~ The law of increase depends, on the constitution of the earth, both as to density
and to elastic constants.
:~: Perhaps discontinuitics in structure, such as those due to a, fault or ~ volcano
merit may introduce peculiarities into the observed earth tides. The writer has
seen no definite mechanism suggested for an effect of this sort.
§ Schweydar reverses the significance of these symbols. In this article the
original notation of Love and Larmor is adhered to.
OCR for page 77
EAR TH TIDES
The value of 1 + 7~-7~ obtained by Michelson and Gale is 0.690,
which corresponds to a more yielding, earth than Schwe,vdar's Azure.
The writer has made some rough preliminary calculations that tend to
show that an allowance for the secondary effects of the oceanic tides
would tend to raise Michelson and Gale's value to a figure approximating,
Schweydar's. On account of mathematical difficulties and the uncertainty
of the data., namely, the ra.~¢,es and lunitida.l intervals in midocean, the
calculations were not pushed to a definite numerical conclusion.
The value of 1+7e-h is all that observations of earth tides alone
will give. To obtain 7~ and h separate!: or to draw inferences about the
rigidity of the earth, observations of some other nature.must be used
or hypotheses must be made. The value of 7e may be deduced from tile
length of the Eulerian period in the variation of latitude. But even Title
the values of 7~ and A, hypotheses must be made in order to deduce the
elastic constants of the earth.* The modulus of rigidity of S.6xI0'i
deduced from Michelson ancl Gale's work is based on the hypothesis of
an earth composed of incompressible material and uniform as to density
and elasticity.
Schweydar~s values are based on the following, laws for the variation
of density and rigidity:
Density
Here
P=P0~l-3(t)
pO = density: at center of eat th = l O.1,
,8 = 0.764, a pure number,
r = distance from center of earth
a=radius of earth.
The modulus of rigidity ,~ is given by
/ r
a= 30.8 x 10~ c.~,.s. units
(2)
(3)
30.S x 103i is obviously the central clensity. The substance is assumed to
lie incompressible, an assumption that greatly simplifies calculations con-
cerning ~ large gravitating, mass lilts the earth.
These assumptions give
7~ = 0.270
7~=0.429
(zl-)
* The hypothesis. should reproduce the assumed values of k and h, but rather
dissimilar hypotheses will give the same values of k and h, especially if a margin
of observational error be allowed.
OCR for page 78
78
FI CURE OF THE EAR TH
The values of 7~ give an Eulerian period of the latitude variation nearly
agreeing, with the observed one.
If uniform rigidity is assumed throughout, Schweydar finds a rigidity
equal to 17.6 x 10~i c.¢,.s. units, more than twice that of steel. Schwey-
dar's assumptions do not correspond with recent ideas about the internal
constitution of the earth. According to these the nucleus with a radius
equal to more than half the radius of the earth is fluid, or nearly so.
Outside the nucleus lies a shell the rigidity of which increases with the
depth below the outer surface. The transition in physical properties
between the shell and the nucleus is rather abrupt. Schweyclar's formula
makes the rigidity at the center a maximum instead of nearly zero, as it
would be according to the hypothesis of a fluic1 nucleus. Moreover, accord-
ing to seismological evidence the rigidity in the outer shell is rather
higher than Schweyclar's formula makes it.
The quantities k and 1~ are merely measures in some sort of the average
rigidity. As has been said, no computation on earth tides has (so far as
the writer knows) been published that takes adequate account of recent
ideas regarding, the constitution of the earth. The nearest thin<, to such
a computation is the work of Jeffreys is already referred to. This tends
to show that there is no serious conflict between these modern ideas and
the value of 7~ and h determined from earth tides and the variation of
latitude.
Both the horizontal pendulum and the ~ichelson-Gale apparatus
measure the deflection of the vertical with reference to the crust of the
earth. The change in the direction of the vertical might be conceivabl\
referred to the axis of the earth, and this is, in fact, exactly what is clone
when observations for latitude or longitude are made. The effect of any
change in the direction of the instrument due to a tide in the crust of
the earth is removed when the spirit levels of the transit instrument or
zenith telescope are read and the proper correction appliecl. However,
the smallest angular change that can be detected by an observation for
latitude and longitude is much larger than the corresponding quantity
in an observation with the horizontal pendulum or the Michelson-Gale
apparatus, so that trustworthy conclusions can be derived only from long
series of observations. The observations of the International Latitude
Service have been analyzed by Pr%byllok. ~ The yielclin~ of the earth
comes out about the same as from observations especially made to cietect
earth tides.
The quantity determined, however, is not 1+7~-A, as for observa-
tions with the horizontal pendulum or the )Tichelson-Gale apparatus, but
i+7~-1.
OCR for page 79
EARTH TIDES
~9
Here k is the same quantity as before, but I is a quantity dependent on
the linear elastic displacement along the meridian of the place of observa-
tion. Przbyllok apparently overlooks the fact that different quantities
are obtained from the ordinary earth-tide observations and from latitude
observations. ~ + k-h is normally less than unity, whereas ~ + k-I
is greater than unity at least in the case of a homogeneous earth. Ilere
again the secondary effects of the disturbing tides come in and there is
the same difficulty in making the proper correction.
The photographic observations for the variation of latitude made at
Gaithersburg, Maryland, simultaneously with the regular observations
of the International Latitude Service have been discussed by Stetson.~5
He finds a lunar effect, that is, an apparent earth tide, much larger than
that riven by the visual observations at the same place in the hands of
~ it,
~ ~ ~ 1 ~ · ~ ~ ~ 1 ~1 _ 1~ 1 L:~ Lid ~1
l~rzbyllok. The earth tide iounct by Stetson Is several times that pre-
clicted by theory. No adequate explanation has so far been given.
REFERENCES
1. Darwin, G. H. Attempted evaluation of the rigidity of the Earth based on
tides of long period. Thomson and Tait's Natural Philosophy, 2nd ea.,
sect. 848, or Scientific papers, 1: 340. Cambridge, 1907.
2. , and Turner, H. H. On the correction to the equilibrium theory of
tides for the continents. Proc. Roy. Soc., 40:303 (1886); or Scientific
papers, 1: 328. Cambridge, 1907.
3. , Horace, and others (report of a committee). On an instrument for
detecting and measuring small changes in the force of gravity. Brit. Assoc.
Rept. for 1881, p. 93; or Scientific papers, 1: 389. Cambridge, 1907.
, and others (Second Committee Report). The lunar disturbance of
gravity; variation in the vertical due to elasticity of the earth's surface.
Brit. Assoc. Rept. for 1882, p. 95; or Scientific papers, 1: 430. Cambridge,
1907.
5. Ehlert, R. Die Horizontalpendelbeobachtungen im Meridian zu Strassburg
i. e., Gerlands Beitr. Geophysik., III:(1898), p. 131.
6. Haid, M. Gezeiten und Starrheitskoeffizient n der festen Erde. Verhandl. der
17ten Allgememeinen Konferenz der Internationalen Erdmessung. I Tell.
Hamburg, 1912.
7. Decker, O., and Meissner, O. Beobachtungen an Horizontalpendeln uber die
Deformation des Erdl~orpers. II Heft. Veroffentl. d. Konigl. Preuszischcn
Geodatischen Inst. not. 49. Potsdam, 1911.
8. Herglotz, G. i:;ber die Elastizitat der Erde bei Berucksichtigung ihrer variablen
Dichte. Z. Math. Physik., 52: 275 (1905).
9. Hoskins, L. M. The strain of gravitating sphere of variable density and elas-
ticity. Trans. Am. Math. Soc., 21: 1 (1920).
10. Jeffreys, Harold. The rigidity of the Earth's central core. Monthly Notices
Roy. Astron. Soc. Geophysical Supplement, 1: 371 (1926).
11. . The Earth, its origin, history and physical constitution. 2nd ed.
Cambridge (Eng.), 1929.
~.
OCR for page 80
80
FIGURE OF THE EARTI-I
12. Lambert, W. D. Les marees de l'ecorce terrestre et leur relations avec let
autres branches de la geophysique. Bull. geodesique? No. 5: 45 (1925).
13. . Earth tides. Travaux de la Section de geodesic. Rapports generaux
etablis a ['occasion de la 3'l'e assembler generate a Prague. Paris, 1929.
14. Larmor, J. The relation of the Earth's free precessional notation to its resis-
tance against tidal deformation. Proc. Roy. Soc., London' 82 A: 89 (1909).
15. Love, A. E. H. The yielding of the Earth to disturbing forces. Proc. Roy.
Soc. London, 82 A: 73 (1909) .
16. . Some problems of geodynamics. (Adams Prize Essay), Cambridge,
1911.
17. Michelson, A. A. Preliminary results of measurements of the rigidity of the
Earth. Astrophys. J., 39: 105 (1914).
18. , and Gale, H. G. The rigidity of the Earth. Astrophys. J., 50: 330
(1919) .
19. Orloff, A. Beobachtungen uber die Deformation des Erdkorpers. Astro-
nomische Nachrichten, 186: 80 (1910).
20. Przbyllok, E. Uber die M~-Tide der Lotbewegung. Astronomische Nachrich-
ten, 218: 85 (1923).
21. von Rebeur-Paschwitz, E. Horizontalpendel-Beobachtungen auf der haiser-
lichen Universitats-Sternwarte zu Strassburg, 1892-1894. Gerlands Beitr.
Geophysik, 2: 2;11 (1895) .
22. Schweydar, Mr. Ein Beitrag Or Bestimmung des Starrheitsl;oeffizienten der
Erde. Gerlands Beitr. Geophysil;? 9: 41 (1908).
23. . Untersuchungen uber die Gezeiten der festen Erde. Veroffentlichung
des Koniglich. Pireuszischen Geodatischen Institutes, No. 54, Potsdam,
1912.
24. . Lotschwankung und Deformation der Erde durch Flutlfrafte. Zen
tralbureau der International Erdmessung. No. 38. Berlin, 1921.
25. Stetson, H. T. The variation of latitude and the moon's position. Nature,
123: 127 (1929); or Science, 19: 17 (1929).
26. Street, R. O. Oceanic tides as modified by a yielding Earth. Monthly Notices,
Roy. Astron. Soc, 1: 292 (1925) .
Representative terms from entire chapter:
oceanic tides
