where the separation distance *r*MX = [(*x*m - *x*)2 + (*y*m - *y*)2 + (*z*m - *z*)2]1/2 with an equivalent expression for *r*SX.

(d) The attractive force of the moon and sun at any point *X *is defined as

P

(II-5-10)

where L is the vector gradient operator defined as

P

%M

%M

L ' (M

)

(II-5-11)

M*x*

M*y*

M*z*

(e) From Figure II-5-4, the attractive force at the center of the earth (centripetal) *b*O is balanced by the

centrifugal force *-b*O (i.e., equal in magnitude but opposite in direction). Because any point on the earth

experiences the same centrifugal force as that at *O*, the resultant force at any point *X *will be equal to *b*X - bO.

This resultant force difference is the tide generating force, the force that causes the oceans to deform in order

to balance the sum of external forces. Therefore, the difference between the tidal potential at point *O *and at

point *X *becomes the tidal potential responsible for the tide-producing forces.

(f) The moon's tide-generating potential can be written as

1

1

(II-5-12)

&

&

with the tide potential for the sun written as

Water Levels and Long Waves

II-5-7

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