Surface gravity water waves 137
The water depth is also classified as
• Deep for h
L o
> 2
with celerity: c
gT
o = 2π
• Shallow for L > 10h with celerity: c
gh
o =
• Intermediate for all other cases with celerity given by Equation 6.4
The bed friction acting on a propagating wave can be described by a
quadratic formula as
τ
ρ
bw
w b
f u
=
1
2
2
(6.6)
where f w is a friction coefficient and u b the amplitude of the near-the-bed
velocity:
u
H
T
kh
b =
π
sinh( )
(6.7)
6.2 ONE-DIMENSIONAL GRAVITY LONG WAVES
By making the same assumptions as those made for the derivation of the
unsteady free surface flow equations (see Chapter 5, Section 5.2.2.1) the
wave equations for a unit-width, frictionless domain can be written (based
on the continuity [Equation 5.26] and momentum [Equation 5.24]) as
follows:
∂
∂
+
∂
∂
=
y
t
uy
x
( ) 0
(6.8)
∂
∂
+
∂
∂
= −
∂
∂
+
u
t
u
u
x
g
y
x
gS o
(6.9)
By re-defining the total depth y(x,t) (see Chapter 5, Figure 5.2) as
y(x,t) = ζ(x,t) + h(x)
(6.10)
the system of the wave Equations 6.8 and 6.9 can be modified as
∂
∂
+
∂
+
 
 
∂
=
ζ
ζ
t
h
u
x
(
)
0
(6.11)