Geoscience Reference
In-Depth Information
where
Direction of propagation
Sinusoidal wave
H
o
= crest-to-trough wave height at the source point (m)
K
r
= refraction coefficient (dimensionless)
mean
sea level
K
s
= shoaling coefficient (dimensionless) (Green's Law)
b
o
= distance between wave orthogonals at a source point
water (m)
b
i
= distance between wave orthogonals at any shore-
ward point (m)
Stokes wave
d
o
= water depth at a source point (m)
d
i
= water depth at any shoreward point (m)
Note that there is a plethora of definitions of wave height
in the tsunami literature. These include wave height at the
source region, wave height above mean water level, wave
height at shore, and wave run-up height above present sea
level. The distinctions between these expressions are pre-
sented in Fig.
2.2
. The expression for shoaling—Eq.
2.6
—
is known as Green's Law (Geist
1997
). For example, if a
tsunami with an initial height of 0.6 m is generated in a
water depth of 4000 m, then its height in 10 m depth of
water on some distant shore can be raised 4.5 times to
2.7 m. Because tsunami are shallow-water waves, they feel
the ocean bottom at any depth and their crests undergo
refraction or bending around higher seabed topography. The
degree of refraction can be measured by constructing a set
of equally spaced lines perpendicular to the wave crest.
These lines are called wave orthogonals or rays (Fig.
2.5
).
As the wave crest bends around topography, the distance, b,
between any two lines will change. Refraction is measured
by the ratio b
o
:b
i
. Simple geometry indicates that the ratio
b
o
:b
i
is equivalent to the ratio cosa
o
:cosa
i
, where a is the
angle that the tsunami wave crest makes to the bottom
contours as the wave travels shoreward (Fig.
2.5
). Once this
angle is known, it is possible to determine the angle at any
other location using Snell's Law as follows:
Solitary wave
N-waves
Simple
Double
Fig. 2.4
Idealized forms characterizing the cross-section of a tsunami
wave. Based on Geist (
1997
). Note that the vertical dimension is
greatly exaggerated
sin a
o
C
1
¼
sin a
i
C
1
ð
2
:
7
Þ
o
i
where
Equation
2.3
holds for linear, sinusoidal waves and is not
appropriate for calculating the wavelength of a tsunami as it
moves into shallow water. Linear theory can be used as a
first approximation to calculate changes in tsunami wave
height as the wave moves across an ocean and undergoes
wave
a
o
= the angle a wave crest makes to the bottom contours
at a source point (degrees)
a
i
= the angle a wave crest makes to the bottom contours
at any shoreward point (degrees)
C
o
m s
1
= wave speed at a source point
ð
Þ
shoaling
and
refraction.
The
following
formulae
= wave speed at any shoreward point (m s
-1
)
C
i
apply:
For a tsunami wave traveling from a distant source—
such as occurs often in the Pacific Ocean—the wave path or
ray must also be corrected for geometrical spreading on a
spherical surface (Okal
1988
). Equation
2.4
can be rewrit-
ten to incorporate this spreading as follows:
H
¼
K
r
K
s
H
o
ð
2
:
4
Þ
0
:
5
K
r
¼
b
o
b
1
ð
2
:
5
Þ
i
0
:
25
K
s
¼
d
o
d
1
ð
2
:
6
Þ
H
¼
K
r
K
s
K
sp
H
o
ð
2
:
8
Þ
i
