According to a standard, idealized analysis, the Moon would produce a 54 cm equilibrium tidal bulge in the Earth's oceans. This analysis omits many factors (beyond the scope of the simple idealized model) that dramatically influence the actual height and timing of the tides at different locations, but it is nevertheless an important foundation for more detailed studies. Here, we show that the standard analysis also omits another factor—the gravitational interaction of the tidal bulge with itself—which is entirely compatible with the simple, idealized equilibrium model and which produces a surprisingly non-trivial correction to the predicted size of the tidal bulge. Our analysis uses ideas and techniques that are familiar from electrostatics, and should thus be of interest to teachers and students of undergraduate E&M, Classical Mechanics (and/or other courses that cover the tides), and geophysics courses that cover the closely related topic of Earth's equatorial bulge.

1.
John R.
Taylor
,
Classical Mechanics
(
University Science Books
,
Mill Valley, California
,
2005
). (The tides are discussed in Chapter 9, “Mechanics in Noninertial Frames.”)
2.
The ratio of the height of the solid-earth tide to that of the equilibrium ocean tide is given by the Love number, h. For the actual Earth, this is estimated to have a value of about 0.6, approximately midway between the values for a hypothetical perfectly rigid (h = 0) and fluid (h = 1) planet. See
Frank D.
Stacey
,
Physics of the Earth
(
John Wiley & Sons
,
New York
,
1969
), pp.
59
64
.
3.
See, for example, some discussion of astrophysical applications in
Bradley W.
Carroll
and
Dale A.
Ostlie
,
An Introduction to Modern Astrophysics
(
Addison-Wesley
,
New York
,
1996
).
4.
Eugene
Butikov
, “
A dynamical picture of the oceanic tides
,”
Am. J. Phys.
70
(
9
),
1
11
(
2002
).
5.
Steacy
Dopp Hicks
, “
Understanding tides
,” NOAA Center for Operational Oceanographic Products and Services,
2006
, <https://tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf>.
6.
David
Morin
,
Introduction to Classical Mechanics
(
Cambridge U.P.
, Cambridge, UK,
2007
). Morin discusses in Chapter 10 both the tides, in Sec. 10.3, and the Earth's equatorial bulge, in Problem 10.12 and its solution. It is interesting that Morin explicitly steps the student through including the gravitational self-interaction correction in the analysis of the equatorial bulge, but never mentions the parallel correction in the case of the tides. Note also that while Morin introduces the same model for the bulge that we use in Sec. IV, we find his suggested method of finding VB (namely, by direct and, as it turns out, numerical integration using d V B = G σ d A / s ) to be less elegant than the method we utilize here.
7.
David J.
Griffiths
,
Introduction to Electrodynamics
, 2nd ed. (
Prentice Hall
,
New Jersey
,
1989
), see especially Chap. 3.
8.
J. D.
Jackson
,
Classical Electrodynamics
, 2nd ed. (
John Wiley & Sons
,
New York
,
1975
), see especially Sec. 1.6 and Chap. 3.
9.
Hans C.
Ohanian
and
Remo
Ruffini
,
Gravitation and Spacetime
, 2nd ed. (
W.W. Norton & Co.
,
New York
,
1994
), p.
45
.
10.
Special thanks to Howard Wiseman for raising the questions that we address in this section.
11.
Frank D.
Stacey
,
Physics of the Earth
(
John Wiley & Sons
,
New York
,
1969
).
12.
Donald L.
Turcotte
and
Gerald
Schubert
,
Geodynamics
, 2nd ed. (
Cambridge U.P.
,
Cambridge, UK
,
2002
).
13.
Paul A.
Tipler
,
Physics for Scientists and Engineers
, 6th ed., Vol. III (
W. H. Freeman
,
New York
,
2007
).
14.
Wikipedia entry on “Equatorial bulge,” <https://en.wikipedia.org/wiki/Equatorial_bulge> (accessed on January 6,
2017
).
15.
Jean le Rond
d'Alambert
,
Réflexions sur la Cause Générale des Vents
(
David l'Ainé
,
Paris
,
1747
).
16.
Isaac
Newton
,
The Principia
, translated by I. B. Cohen and Anne Whitman (
University of California Press
,
Berkeley, CA
,
1999
), see especially Book 3, Proposition 37, pp.
875
878
.
17.
Vincent
Deparis
,
Hilaire
Legros
, and
Jean
Souchay
, “
Investigations of tides from antiquity to Laplace
,”
Tides in Astronomy and Astrophysics
, Lecture Notes in Physics 861, edited by
J.
Souchay
 et al.
(
Springer-Verlag
,
Berlin
,
2013
).
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