Part A | 4.3
80 Part A Fundamentals
4.3 Properties of Small Amplitude Gravity Waves
In the absence of a body in the fluid, a periodic twodimensional (long crested) progressive small amplitude
(linear) wave solution to the linearized problem (as formulated above) can be written as
D
H
2
g
!
cosh k.z C h/
cosh kh
sin .kx !t/ ;
Á D
H
2
cos .kx !t/ ;
(4.12)
where k Á 2=L denotes the wave number, H the wave
height (which the vertical distance between the crest
and trough), and h the mean water depth (taken here to
be constant). In the case of a deep water wave, i. e., in
the limit kh ! 1 with z 2 Œ0; h the above solution
becomes
D
H
2
g
!
e
kz sin .kx !t/ ;
Á D
H
2
cos .kx !t/ :
(4.13)
For the above velocity potential, the velocity field (u D
r) is given by
u D
H
2
gk!
cosh k.z C h/
cosh kh
cos .kx !t/ ;
v D 0 ;
w D
H
2
gk!
sinh k.z C h/
cosh kh
sin .kx !t/ :
From above expressions one finds that the horizontal
velocity field is in phase with the wave elevation. Its
slip on the bottom is to be interpreted as the velocity at
the outer edge of a bottom boundary layer. Using the
Euler integral one can obtain the following expression
for the pressure
p D Dgz C g
H
2
cosh k.z C h/
cosh kh
cos .kx !t/
D Dgz C gÁ
cosh k.z C h/
cosh kh
:
In the case of shallow water waves – i. e., for kh ! 0
with z 2 Œ0; h (or approximately, kh Ä =10) –, the
expression for pressure reduces to
p D Dg.z Á/:
The above expression looks very similar to the familiar
hydrostatic pressure equation for homogeneous water.
The static pressure corresponds to the depth measured
from the calm surface while total pressure to the depth
measured from the actual free surface with the difference contributes to the dynamic pressure.
4.3.1 Linear Dispersion Relation
As per the linear solution obtained above, the wave
number k Á 2=L and wave frequency ! Á 2=T are
related by the dispersion relation
!
2
D gk tanh kh ;
(4.14)
which approximates to
!
2
D gk; for kh ;
(4.15)
and to
!
2
D gk
2 h; for kh Ä =10 :
(4.16)
The dispersion relation thus provides a basis for defining deep water, shallow water, and intermediate waves.
Waves satisfying kh , which is the same as h L=2,
are referred to as deep water (or short) waves; that satisfying kh Ä =10, which is same as L 20h, are referred
to as shallow water (or long) waves and that in between
(i. e., =10 < kh < <) as the intermediate waves.
4.3.2 Phase Speed
The phase speed (also known as the wave speed or wave
celerity) for linear waves becomes
C p Á
L
T
D
!
k
D
r
g
k
tanh kh ;
(4.17)
which reduces to
C p D
r
g
k
tanh kh; for =10 < kh < <
(intermediate waves)
D
r
g
k
; for kh
(deep water or short waves)
D
p
gh; for kh Ä =10
(shallow water or long waves) :
(4.18)
As can be observed in the above expressions, longer
deep water waves travel faster. In the case of shallow
water waves the wave speed depends only on the local
depth; the phase speed decreases with decreasing depth.
4.3.3 Group Speed
Often it is the group speed that govern many important
wave properties, including the speed of energy propa-
80 Part A Fundamentals
4.3 Properties of Small Amplitude Gravity Waves
In the absence of a body in the fluid, a periodic twodimensional (long crested) progressive small amplitude
(linear) wave solution to the linearized problem (as formulated above) can be written as
D
H
2
g
!
cosh k.z C h/
cosh kh
sin .kx !t/ ;
Á D
H
2
cos .kx !t/ ;
(4.12)
where k Á 2=L denotes the wave number, H the wave
height (which the vertical distance between the crest
and trough), and h the mean water depth (taken here to
be constant). In the case of a deep water wave, i. e., in
the limit kh ! 1 with z 2 Œ0; h the above solution
becomes
D
H
2
g
!
e
kz sin .kx !t/ ;
Á D
H
2
cos .kx !t/ :
(4.13)
For the above velocity potential, the velocity field (u D
r) is given by
u D
H
2
gk!
cosh k.z C h/
cosh kh
cos .kx !t/ ;
v D 0 ;
w D
H
2
gk!
sinh k.z C h/
cosh kh
sin .kx !t/ :
From above expressions one finds that the horizontal
velocity field is in phase with the wave elevation. Its
slip on the bottom is to be interpreted as the velocity at
the outer edge of a bottom boundary layer. Using the
Euler integral one can obtain the following expression
for the pressure
p D Dgz C g
H
2
cosh k.z C h/
cosh kh
cos .kx !t/
D Dgz C gÁ
cosh k.z C h/
cosh kh
:
In the case of shallow water waves – i. e., for kh ! 0
with z 2 Œ0; h (or approximately, kh Ä =10) –, the
expression for pressure reduces to
p D Dg.z Á/:
The above expression looks very similar to the familiar
hydrostatic pressure equation for homogeneous water.
The static pressure corresponds to the depth measured
from the calm surface while total pressure to the depth
measured from the actual free surface with the difference contributes to the dynamic pressure.
4.3.1 Linear Dispersion Relation
As per the linear solution obtained above, the wave
number k Á 2=L and wave frequency ! Á 2=T are
related by the dispersion relation
!
2
D gk tanh kh ;
(4.14)
which approximates to
!
2
D gk; for kh ;
(4.15)
and to
!
2
D gk
2 h; for kh Ä =10 :
(4.16)
The dispersion relation thus provides a basis for defining deep water, shallow water, and intermediate waves.
Waves satisfying kh , which is the same as h L=2,
are referred to as deep water (or short) waves; that satisfying kh Ä =10, which is same as L 20h, are referred
to as shallow water (or long) waves and that in between
(i. e., =10 < kh < <) as the intermediate waves.
4.3.2 Phase Speed
The phase speed (also known as the wave speed or wave
celerity) for linear waves becomes
C p Á
L
T
D
!
k
D
r
g
k
tanh kh ;
(4.17)
which reduces to
C p D
r
g
k
tanh kh; for =10 < kh < <
(intermediate waves)
D
r
g
k
; for kh
(deep water or short waves)
D
p
gh; for kh Ä =10
(shallow water or long waves) :
(4.18)
As can be observed in the above expressions, longer
deep water waves travel faster. In the case of shallow
water waves the wave speed depends only on the local
depth; the phase speed decreases with decreasing depth.
4.3.3 Group Speed
Often it is the group speed that govern many important
wave properties, including the speed of energy propa-