Part A | 4.3
82 Part A Fundamentals
4.3.7 Water Particle Trajectory
For a propagating wave with velocity known (from
above) one can determine the trajectory of water particles through time integration. For small amplitude
waves, the trajectory is elliptical and closed
.X X o /
2
A 2
C
.Z Z o /
2
B 2
D 1 ;
where .X o ; Z o / denote the coordinates of the center of
the ellipse, A the major semi-axis
A Á
H
2
gk
! 2
cosh k.Z o C h/
cosh kh
;
and B the minor semi-axis
B Á
H
2
gk
! 2
sinh k.Z o C h/
cosh kh
:
As to be expected, near the bottom the particle will oscillate back and forth parallel to the bottom. In the case
of deep water waves (kh ! 1, or for practical calculations kh ), above expressions for semi-axes reduce
to
A D B D
H
2
gk
! 2 e
kZo
D
H
2
e
kZo
:
In other words, the trajectory is a circle for small amplitude deep water waves. As the particle paths are closed,
the average mass transport is zero. However, in the case
of finite amplitude waves, this is not the case. In the
case of finite amplitude waves, the trajectory is not quite
closed, meaning that a particle will not return to the
original position after one wave period but to a point
slightly ahead of the original point. This is referred to
as the Stokes drift. More on finite amplitude wave properties are given in later sections of this chapter.
4.3.8 Spatio-Temporal Evolution of Waves
Consider the generation of waves by (for example)
a mudslide into water or by a seismic activity and the resulting waves propagating over water of varying depth.
Here both wave numbers and frequency will vary with
space and time. Let the direction of the wave propagation be along the x direction. Here the wave number
and frequency may vary both in x and in time. In other
words, the phase function will be
D .k.x; t/x !.x; t/t/ :
With the so-called slowly varying wave assumption, the
wave number and wave frequency can be written as
k
@@
@x
I !
@@
@t
:
Cross-differentiating the above expressions we obtain
@k
@t
C
@!
@x
D
@
2
‚
@x@t
@
2
‚
@t@x
D 0 :
(4.22)
In the special case of the steady-state limit, i. e.,
k D k.x/ and ! D !.x/, from above we find ! to be
constant. Thus, in the case of steady-state evolution
of waves, the wave number (hence wave length) may
change spatially in x, but the frequency will be the same
everywhere.
The above equation for k and ! can be re-written as
@k
@t
CC g
@k
@x
D 0; where C g D Group Speed D
d!
dk
:
(4.23)
The above equation is called the wave number conservation equation; its solution implies that k will appear
constant to an observer moving at the group speed. One
could draw several other important conclusions using
this equation. For example, assume waves are propagating in deep water (i. e., so that kh ). At a fixed
field point away from the wave source, as per the wave
number conservation equation, the wave frequency and
wave number will increase with respect to time, meaning that as time progresses shorter and shorter waves
will reach the field point. To deduce this fact from the
wave number conservation equation, consider the time
that will be taken by a wave of wave number k generated at the source to reach the field point
t D
d
C g
! C g D
d
t
;
where d denotes the distance between the source and the
field point. Since C g D 0:5 g=! for deep water gravity
waves, the above becomes
1
2
g
!
D
d
t
! ! D
1
2
gt
d
:
In other words, the frequency of waves reaching the
field point increases as the time increases. With !
2
D
gk for deep water waves, the above equation can be
written as
k D
1
4
gt
2
d 2 ;
meaning that the number of waves reaching the field
point will also increase with time.
82 Part A Fundamentals
4.3.7 Water Particle Trajectory
For a propagating wave with velocity known (from
above) one can determine the trajectory of water particles through time integration. For small amplitude
waves, the trajectory is elliptical and closed
.X X o /
2
A 2
C
.Z Z o /
2
B 2
D 1 ;
where .X o ; Z o / denote the coordinates of the center of
the ellipse, A the major semi-axis
A Á
H
2
gk
! 2
cosh k.Z o C h/
cosh kh
;
and B the minor semi-axis
B Á
H
2
gk
! 2
sinh k.Z o C h/
cosh kh
:
As to be expected, near the bottom the particle will oscillate back and forth parallel to the bottom. In the case
of deep water waves (kh ! 1, or for practical calculations kh ), above expressions for semi-axes reduce
to
A D B D
H
2
gk
! 2 e
kZo
D
H
2
e
kZo
:
In other words, the trajectory is a circle for small amplitude deep water waves. As the particle paths are closed,
the average mass transport is zero. However, in the case
of finite amplitude waves, this is not the case. In the
case of finite amplitude waves, the trajectory is not quite
closed, meaning that a particle will not return to the
original position after one wave period but to a point
slightly ahead of the original point. This is referred to
as the Stokes drift. More on finite amplitude wave properties are given in later sections of this chapter.
4.3.8 Spatio-Temporal Evolution of Waves
Consider the generation of waves by (for example)
a mudslide into water or by a seismic activity and the resulting waves propagating over water of varying depth.
Here both wave numbers and frequency will vary with
space and time. Let the direction of the wave propagation be along the x direction. Here the wave number
and frequency may vary both in x and in time. In other
words, the phase function will be
D .k.x; t/x !.x; t/t/ :
With the so-called slowly varying wave assumption, the
wave number and wave frequency can be written as
k
@@
@x
I !
@@
@t
:
Cross-differentiating the above expressions we obtain
@k
@t
C
@!
@x
D
@
2
‚
@x@t
@
2
‚
@t@x
D 0 :
(4.22)
In the special case of the steady-state limit, i. e.,
k D k.x/ and ! D !.x/, from above we find ! to be
constant. Thus, in the case of steady-state evolution
of waves, the wave number (hence wave length) may
change spatially in x, but the frequency will be the same
everywhere.
The above equation for k and ! can be re-written as
@k
@t
CC g
@k
@x
D 0; where C g D Group Speed D
d!
dk
:
(4.23)
The above equation is called the wave number conservation equation; its solution implies that k will appear
constant to an observer moving at the group speed. One
could draw several other important conclusions using
this equation. For example, assume waves are propagating in deep water (i. e., so that kh ). At a fixed
field point away from the wave source, as per the wave
number conservation equation, the wave frequency and
wave number will increase with respect to time, meaning that as time progresses shorter and shorter waves
will reach the field point. To deduce this fact from the
wave number conservation equation, consider the time
that will be taken by a wave of wave number k generated at the source to reach the field point
t D
d
C g
! C g D
d
t
;
where d denotes the distance between the source and the
field point. Since C g D 0:5 g=! for deep water gravity
waves, the above becomes
1
2
g
!
D
d
t
! ! D
1
2
gt
d
:
In other words, the frequency of waves reaching the
field point increases as the time increases. With !
2
D
gk for deep water waves, the above equation can be
written as
k D
1
4
gt
2
d 2 ;
meaning that the number of waves reaching the field
point will also increase with time.