Chapter 1: INTRODUCTION
For deep-water waves,
1
b
kh !! , and expression (1.101) reduces to
1
exp
sin
k a
kz
k x
t
I
Z
Z
˜
G G
,
(1.103)
with dispersion relationship
1/ 2
3 /
s
k
gk
k
Z
V
U
.
(1.104)
For very short surface waves, where
2
s k
g
V
U
!!
, gravity becomes
negligible compared to surface tension; waves of this type are known as
capillary waves. On the other hand, when
2
s k
g
V
U
, surface tension is
negligible and the dynamics are dominated by gravity.
The phase speed of deep-water waves following from dispersion
relationship (1.104) is as follows
/
/
/
s
c
k
g k
k
Z
V U
.
(1.105)
Analysis of (1.105) shows that the phase speed has a minimum
1/ 4
4
/
0.23
m
s
c c
gV U
|
m s
-1 at
1/ 2
/
3 6 0
m
s
k k
gU V
|
m
-1 where
gravity and surface tension are equally important. The phase speed of gravity
waves (
m
k
k
) increases with wavelength
2 /k
O S
or with decreasing
wavenumber k. The phase speed of capillary waves (
m
k
k
!! ) decreases with
1.6.3 Nonlinear waves
A solution for the nonlinear system (1.94)-(1.97) is the plane steady
G
with a constant speed c. Stokes (1880) showed that the surface wave
elevation of a plane wave train in deep water can be expanded in powers of a
small parameter H = ak << 1. The third order result is as follows:
2
2 3
1
3
cos
cos 2
cos3 ...
2
8
a
k a
k a
K
T
T
T
,
(1.106)
where the phase
kx
t
T
Z
and
45
wavelength O or with decreasing wavenumber k.
nonlinear wave in the form K
K
x, t
x ct propagating along axis x