Encyclopedia of Environmental Science and Engineering, Volume I and II

OCEANOGRAPHY 799

becomes: v 2 
 gk. The wave characteristics (e.g., wavelength,

phase speed, etc.) are therefore independent of depth.

For very shallow (long) waves, we have (kh) much less

than one, so that tanh(kh) goes to (kh) and the dispersion

relation becomes: v 2 
 gk^2 h. The wave characteristics are

now strongly dependent on water depth. A further observa-

tion is that shallow water waves all move at the same phase

speed, C 
 (gh) 1/2.

The two-dimensional fluid motion beneath a surface

wave can be described by the following relations:

U

agk cosh k(z h)

cosh(kh)

cos(kx t),

W

agk sinh k(z h)

c







v

v

v oosh(kh)

sin(kxvt),

where U and W are the horizontal and vertical components,

respectively, of the water motion; z is the vertical position of

interest, decreasing from z 
 0 at the surface to z 
 h at

the bottom; and a is the wave amplitude, measured from the

equilibrium, or still water level, to the wave crest.

Note that the wave-induced water velocities, U and W,

are 90 out of phase, indicating that the fluid particle trajec-

tories beneath surface waves are elliptical in shape. Since the

value of sinh k(z  h) decreases to zero while cosh k(z  h)

decreases to unity as one approaches the bottom (z 
 h),

the fluid motion becomes more and more horizontal with

depth. Theoretically, at the bottom the wave-induced motion

is purely horizontal.

Once again, if we examine the limits of shallow water

and deep water waves, we can make some interesting obser-

vations. In the case of deep water (short) waves, we have very

large (kh), and, noting that sinh(x) 
 cosh(x) for large x, the

particle trajectories are circular in shape. For shallow water,

or long waves, we have very small (kh) so that the vertical

velocity, W, is much smaller than the horizontal velocity, U,

and the particle motions are almost exclusively horizontal.

We should caution that the equations noted here are

derived from the linear form of the Navier-Stokes equations.

That is, the non-linear, or convective acceleration terms have

been assumed very small. This assumption is satisfied when

the ratio of the wave height, H 
 2a, to the wavelength, L,

is very small: H/L << 1, or in physical terms, when the wave

steepness is very small. For this reason, the equations listed

here constitute a description of “linear,” or “small ampli-

tude” water waves. It follows that in regions where the wave

steepness is relatively large (in the breaking wave zone, for

example) our theory loses some of its accuracy, and one

should be careful in its application. A classic treatment of

the derivation of the governing equations for both linear and

non-linear water waves can be found in Whitham (1974).

TIDES

Ocean tides can be described as the periodic rise and fall of

the ocean surface. The forcing responsible for this motion is

the gravitational attraction of the moon, sun and (to a much

lesser extent) the other planets. Because of its close proxim-

ity to the earth, however, the moon plays the dominant role

in tidal forcing.

A simple explanation for the generation of tides can be

obtained from an examination of the earth-moon system. We

know that the attractive force due to gravity is inversely pro-

portional to the square of the separation distance between two

bodies: F  1/d^2. For this reason, the ocean surface closest to,

or facing the moon experiences a much higher attractive force

than does the ocean surface on the opposite side of the earth.

The water surface on the opposite side of the earth. The water

surface facing the moon is therefore “pulled” toward the moon.

At the same time, the attractive force of the moon “pulls” the

entire earth from under the ocean surface on the opposite side

of the earth. The net result, therefore, is an increase in water

surface elevation on both the side of the earth facing the moon

and the side opposite. Two peak tides are therefore created at

the same time on opposite sides of the earth. Since the earth

rotates once in 24 hours, we expect two high tides and two

low tides each day. However, the cycle, or lunar day is actually

approximately 24 hours, 50 minutes, so that each day the high

and low tides occur approximately 50 minutes later.

Of course, the above description of the generation of

ocean tides is rather simplistic. In actuality, we have the

h

Rigid form

H

L

C

x

FIGURE 6 Definition sketch-surface water characteristics.

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