1/2 to 4

π to 4

.1

Deep water

Transitional

1/20 to 1/2

π/10 to π

tanh (kd)

. kd

Shallow water

0 to 1/20

0 to π/10

(i) In deep water, tanh (kd) approaches unity, Equations II-1-7 and II-1-8 reduce to

(II-1-12)

'

2π

and Equation II-1-9 becomes

(II-1-13)

2π

(j) Although *deep water *actually occurs at an infinite depth, *tanh (kd)*, for most practical purposes,

approaches unity at a much smaller *d/L*. For a relative depth of one-half (i.e., when the depth is one-half the

wavelength), tanh (2π*d/L*) = 0.9964.

(k) When the relative depth *d/L *is greater than one-half, the wave characteristics are virtually

independent of depth. Deepwater conditions are indicated by the subscript *0 *as in *L*o and *C*o except that the

period *T *remains constant and independent of depth for oscillatory waves, and therefore, the subscript for

wave period is omitted (Ippen 1966). In the SI system (System International or metric system of units) where

units of meters and seconds are used, the constant *g/2π *is equal to 1.56 m/s2 , and

9.8

(II-1-14)

'

2π

2π

and

9.8 2

(II-1-15)

'

2π

2π

(l) If units of feet and seconds are specified, the constant *g/2π *is equal to 5.12 ft/s2 , and

(II-1-16)

' 5.12 *T ft*/*s*

2π

and

' 5.12 *T * 2 ft

(II-1-17)

2π

(m) If Equations II-1-14 and II-1-15 are used to compute wave celerity when the relative depth is *d/L *=

0.25, the resulting error will be about 9 percent. It is evident that a relative depth of 0.5 is a satisfactory

boundary separating deepwater waves from waves in water of *transitional depth*. If a wave is traveling in

Equation II-1-8 and II-1-9 must be used when the relative depth is between 0.5 and 0.04.

II-1-8

Water Wave Mechanics

Integrated Publishing, Inc. |