Tide-producing forces.

EM 1110-2-1100 (Part II)

30 Apr 02

approximately two tides per solar day (referred to as semidiurnal), with a maximum tide occurring

approximately every 12 hr 25 min. However, differences in the relationship of the moon and sun in

conjunction with local conditions can result in tides that exhibit only one tidal cycle per day. These are

referred to as diurnal tides. Mixed tides exhibit characteristics of both semidiurnal and diurnal tides. At

certain times in the lunar month, two peaks per day are produced, while at other times the tide is diurnal. The

distinction is explained in the following paragraphs.

(c) The description of typical tidal variability begins with a brief background description of tide-

producing forces, those gravitational forces responsible for tidal motion, and the descriptive tidal envelope

that results from those forces. This sub-section will be followed by more qualitative descriptions of how the

tidal envelope is influenced by the position of the moon and sun. Once this basic pattern is established,

measured tidal elevations can, in part, be shown to be a function of the influence of the continental shelf and

the coastal boundary on the propagating tide.

(2) Tide-producing forces.

(a) The law of universal gravitation was first published by Newton in 1686. Newton's law of gravitation

states that every particle of matter in the universe attracts every other particle with a force that is directly

proportional to the product of the masses of the particles and inversely proportional to the square of the

distance between them (Sears and Zemansky 1963). Quantitative aspects of the law of gravitational attraction

between two bodies of mass m1 and m2 can be written as follows:

m1 m2

Fg ' f

(II-5-7)

2

r

where Fg is the gravitational force on either particle, r is separation of distance between the centers of mass

of the two bodies, and f is the universal constant with a value of 6.67 10-8 cm3/gm sec2. The gravitational

force of the earth on particle m1 can be determined from Equation II-5-7. Let Fg = m1 g where g is the

acceleration of gravity (980.6 cm/sec2) on the surface of the earth, and m2 equal the mass of the earth E. By

substitution, an expression for the gravitational constant can be written in terms of the radius of the earth a

and the acceleration of gravity g.

a2

f'g

(II-5-8)

E

(b) Development of the tidal potential follows directly from the above relationship. The following

variables are referenced to Figure II-5-4 (although Figure II-5-4 refers to the moon, an analogous figure can

be drawn for the sun). Let M and S be the mass of the moon and sun, respectively. rm and rs are the distances

from the center of the earth O to the center of the moon and sun. Let rmx and rsx be the distances of a point

X(x,y,z) located on the surface of the earth to the center of the moon and sun. The following relationships

define the tidal potential at some arbitrary point X as a function of the relative position of the moon and sun.

(c) The attractive force potentials per unit mass for the moon and sun can be written as

fM

fS

VM '

(II-5-9)

,

VS '

rMX

rSX

II-5-6

Water Levels and Long Waves