In fluid dynamics, Green's law, named for 19th-century British mathematician George Green, is a conservation law describing the evolution of non-breaking, surface gravity waves propagating in shallow water of gradually varying depth and width. In its simplest form, for wavefronts and depth contours parallel to each other, it states: or
Article
In fluid dynamics, Green's law, named for 19th-century British mathematician George Green, is a conservation law describing the evolution of non-breaking, surface gravity waves propagating in shallow water of gradually varying depth and width. In its simplest form, for wavefronts and depth contours parallel to each other (and the coast), it states:
are the wave heights at two different locations – 1 and 2 respectively – where the wave passes, and
h
1
{\displaystyle h_{1}}
and
h
2
{\displaystyle h_{2}}
are the mean water depths at the same two locations.
Green's law is often used in coastal engineering for the modelling of long shoaling waves on a beach, with "long" meaning wavelengths in excess of about twenty times the mean water depth. Tsunamis shoal (change their height) in accordance with this law, as they propagate – governed by refraction and diffraction – through the ocean and up the continental shelf. Very close to (and running up) the coast, nonlinear effects become important and Green's law no longer applies.
Contents
Description
According to this law, which is based on linearized shallow water equations, the spatial variations of the wave height
H
{\displaystyle H}
(twice the amplitude
a
{\displaystyle a}
for sine waves, equal to the amplitude for a solitary wave) for travelling waves in water of mean depth
along a wave ray. The oscillation period (and therefore also the frequency) of shoaling waves does not change, according to Green's linear theory.
Derivation
Green derived his shoaling law for water waves by use of what is now known as the Liouville–Green method, applicable to gradual variations in depth
h
{\displaystyle h}
and width
b
{\displaystyle b}
along the path of wave propagation.
Wave equation for an open channel
Starting point are the linearized one-dimensional Saint-Venant equations for an open channel with a rectangular cross section (vertical side walls). These equations describe the evolution of a wave with free surface elevation
η
(
x
,
t
)
{\displaystyle \eta (x,t)}
and horizontal flow velocity
u
(
x
,
t
)
,
{\displaystyle u(x,t),}
with
x
{\displaystyle x}
the horizontal coordinate along the channel axis and
t
{\displaystyle t}
the time:
b
∂
η
∂
t
+
Transformation to the wave phase as independent variable
The next step is to apply a coordinate transformation, introducing the travel time (or wave phase)
with the subscripts 1 and 2 denoting quantities in the associated cross section. So, when the depth has decreased by a factor sixteen, the waves become twice as high. And the wave height doubles after the channel width has gradually been reduced by a factor four. For wave propagation perpendicular towards a straight coast with depth contours parallel to the coastline, take
b
{\displaystyle b}
a constant, say 1 metre or yard.
For refracting long waves in the ocean or near the coast, the width
b
{\displaystyle b}
can be interpreted as the distance between wave rays. The rays (and the changes in spacing between them) follow from the geometrical optics approximation to the linear wave propagation. In case of straight parallel depth contours this simplifies to the use of Snell's law.
Green published his results in 1838, based on a method – the Liouville–Green method – which would evolve into what is now known as the WKB approximation. Green's law also corresponds to constancy of the mean horizontal wave energy flux for long waves:
are denoting partial derivatives with respect to space and time. The slow variation of width
b
{\displaystyle b}
and depth
h
{\displaystyle h}
with distance
x
{\displaystyle x}
along the channel axis is brought into account by denoting them as
b
(
μ
x
)
{\displaystyle b(\mu x)}
and
h
(
μ
x
)
,
{\displaystyle h(\mu x),}
where
μ
{\displaystyle \mu }
is a small parameter:
μ
≪
1.
{\displaystyle \mu \ll 1.}
The above two equations can be combined into one wave equation for the surface elevation:
In the Liouville–Green method, the approach is to convert the above wave equation with non-homogeneous coefficients into a homogeneous one (neglecting some small remainders in terms of
μ
{\displaystyle \mu }
).
x
{\displaystyle x}
are related through the celerity
c
=
g
h
.
{\displaystyle c={\sqrt {gh}}.}
Introducing the slow variable
X
=
μ
x
{\displaystyle X=\mu x}
and denoting derivatives of
b
(
X
)
{\displaystyle b(X)}
and
h
(
X
)
{\displaystyle h(X)}
with respect to
X
{\displaystyle X}
with a prime, e.g.
b
′
=
d
b
/
d
X
,
{\displaystyle b'=\mathrm {d} b/\mathrm {d} X,}
the
x
{\displaystyle x}
-derivatives in the wave equation, Eq. (1), become:
The next step is transform the equation in such a way that only deviations from homogeneity in the second order of approximation remain, i.e. proportional to
