Geoscience Reference
In-Depth Information
the width of the obstacle is commensurable with the wavelength, interference effects
start to become apparent. When the phase difference
(Fig. 5.6b) changes, the val-
ues of coefficient
T
change periodically from
T
min
up to 1. These changes are ex-
plained as follows. The interference between the waves, reflected from the front
and rear boundaries of the obstacle, leads to mutual cancelling of the waves and,
consequently, to amplification of the transmitted wave intensity. We recall that this
effect is widely applied for producing antireflection optics. As to applications to
the tsunami problem, we are, first of all, interested in wave transformation on ob-
stacles of small sizes,
D
β
. The point is that the transformation of waves by
large-scale bottom irregularities,
D
λ
, is automatically taken into account in
numerical simulations. Therefore, it is practically important to estimate the con-
tribution of small-scale (several kilometers and less) or so-called sub-net inho-
mogeneities, the size of which turns out to be smaller than the distance between
the nodes of the mesh. We shall return to this estimate at the end of the section.
Small-scale inhomogeneities of the open ocean bottom exhibit heights signifi-
cantly smaller than the thickness of the water column. Therefore, it is reasonable to
introduce the relative height of an obstacle,
λ
H
1
−
H
2
α
≡
,
H
1
that is a small quantity. When
α
1 and
β
1, from formula (5.9) we obtain
the simple approximate relation
)
2
(
αβ
T
≈
1
−
.
(5.10)
8
The expression obtained permits to assert that in the case of transformation of
a wave passing above an obstacle, the decrease in amplitude is proportional to
the square area of the obstacle.
We have hitherto considered influence of the bottom relief on tsunami waves
within the framework of one-dimensional problems. Actually, tsunami propagation
takes place in two-dimensional space: on a plane or on the surface of a sphere. Cer-
tain two-dimensional peculiarities of the bottom relief, such as underwater oceanic
mountain ridges, the shelf, are capable, for example, of effectively capturing waves,
thus creating priority directions for tsunami propagation and providing for pro-
longed 'sounding' of tsunamis at a coast. Phenomena of such kind are readily
tracked making use of beam theory, which is also called an approximation of ge-
ometrical optics. Ray theory provides an effective instrument for operative calcu-
lation of tsunami arrival times. Its application permits to determine the contours of
a tsunami source from data of the network of mareograph stations. Ray theory is
extremely illustrative and permits to judge about the directions of tsunami energy
propagation. Computational methods have been developed for calculating wave am-
plitudes on the basis of equations written 'along the ray' [Pelinovsky (1996)].
The velocity of a wave
c
being a function of two coordinates,
x
and
y
,theray
equations for nondispersive waves are written as follows [Lighthill (1978)]:
