subscript is omitted.
seconds are specified,
and
(Ippen, 1966b, pp 21-24.) If units of feet and
the constant g/2ir is equal to 5.12 ft/sec and
C = — = 5.12 T (ft/sec) ,
°
2ir
L = — = 5.12 T2 (ft).
°
2ir
(2-7)
(2-8)
If Equation 2-7 is used to compute wave celerity when the relative depth
is d/L = 0.25, the resulting error will be about 9 percent. It is évident 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 transitioncil depths, Equations 2-2 and 2-3 must be used without simplification. Care should be exercised to use Equations 2-2 and 2-3
when necessary, that is, when the relative depth is between 1/2 and 1/25.
When the relative water depth becomes shallow, i.e., 27id/L <1/4 or
d/L < 1/25, Equation 2-2 can be simplified to
C = v'gd'.
(2-9)
This relation, attributed to Lagrange, is of importance when dealing with
long-period waves, often referred to as long waves. Thus, when a wave
travels in shallow water, wave celerity dépends only on water depth.
2.232 The Sinusoidal Wave Profile. The équation describing the free
surface as a function of time t, and horizontal distance x, for a
simple sinusoidal wave can be shown to be
[2iïx
27rt\
H
Ærrx
2îrt\
i? = a cos ----- — ---- = — cos ---- — ---- |,
(2-10)
\ L
T/2
\ L
T )
where ri is the élévation of the water surface relative to stillwater
level, and H/2 is one-half the wave height equal to the wave amplitude
a. This expression represents a periodic, sinusoidal, progressive wave
traveling in the positive x-direction. For a wave moving in the négative
x-direction, one need only replace the minus sign before 2irt/T with a
plus sign. When (2ttx/L - 2irt/T) equals 0, tt/2, tt, 3tt/2, the corresponding
values of n are H/2, 0, - H/2, and 0, respectively.
2*233. Some Useful Functions. It can be shown by dividing Equation 2-3 by
Equation 2-6, and by dividing Equation 2-4 by Equation 2-8 that
C
L
/2nd\
c~ = F = tanh T •
(2-n)
C?
O
\
/
2-10
seconds are specified,
and
(Ippen, 1966b, pp 21-24.) If units of feet and
the constant g/2ir is equal to 5.12 ft/sec and
C = — = 5.12 T (ft/sec) ,
°
2ir
L = — = 5.12 T2 (ft).
°
2ir
(2-7)
(2-8)
If Equation 2-7 is used to compute wave celerity when the relative depth
is d/L = 0.25, the resulting error will be about 9 percent. It is évident 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 transitioncil depths, Equations 2-2 and 2-3 must be used without simplification. Care should be exercised to use Equations 2-2 and 2-3
when necessary, that is, when the relative depth is between 1/2 and 1/25.
When the relative water depth becomes shallow, i.e., 27id/L <1/4 or
d/L < 1/25, Equation 2-2 can be simplified to
C = v'gd'.
(2-9)
This relation, attributed to Lagrange, is of importance when dealing with
long-period waves, often referred to as long waves. Thus, when a wave
travels in shallow water, wave celerity dépends only on water depth.
2.232 The Sinusoidal Wave Profile. The équation describing the free
surface as a function of time t, and horizontal distance x, for a
simple sinusoidal wave can be shown to be
[2iïx
27rt\
H
Ærrx
2îrt\
i? = a cos ----- — ---- = — cos ---- — ---- |,
(2-10)
\ L
T/2
\ L
T )
where ri is the élévation of the water surface relative to stillwater
level, and H/2 is one-half the wave height equal to the wave amplitude
a. This expression represents a periodic, sinusoidal, progressive wave
traveling in the positive x-direction. For a wave moving in the négative
x-direction, one need only replace the minus sign before 2irt/T with a
plus sign. When (2ttx/L - 2irt/T) equals 0, tt/2, tt, 3tt/2, the corresponding
values of n are H/2, 0, - H/2, and 0, respectively.
2*233. Some Useful Functions. It can be shown by dividing Equation 2-3 by
Equation 2-6, and by dividing Equation 2-4 by Equation 2-8 that
C
L
/2nd\
c~ = F = tanh T •
(2-n)
C?
O
\
/
2-10