Propagation of a Tsunami in the Ocean and Its Interaction with the Coast - Physics of Tsunamis - page 201

0.001 0.01 0.1 1,000

Fig. 5.5 Amplitude coefficients for transmission, T , and reflection, R , versus the depth ratio in
the case of wave transformation on a step

The dependences (5.7) and (5.8), calculated within a wide range of depth ra-
tios H 1 / H 2 ,areshowninFig.5.5.If H 1 > H 2 , then the waves that are transmitted
through and reflected from, respectively, a step, will have the same polarity as the in-
cident wave, and the amplitude of the transmitted wave will increase. In the case of
transformation on a step the wave amplitude cannot be more than twice the initial
value. When H 1 < H 2 , the reflected wave changes polarity, and the amplitude of
the wave, reaching deep water, is reduced.
Now, consider the 'classical' problem, akin to the previous one, of transformation
of a long wave above a rectangular obstacle (Fig. 5.4a), of length D and height
| |

. The role of the obstacle can be assumed both by a local elevation of
the bottom and by a depression. We note, right away, that this problem cannot be
reduced to two consecutive independent acts of wave transformation on the front
and back edges of the obstacle, i.e. on two steps. Anyhow, in the case of a solitary
wave, with a length much shorter than the length of the obstacle, such an approach
is quite adequate [Nakoulima et al. (2005)].
In the general case, a correct description of wave transformation above a rect-
angular obstacle requires the examination of a constrained system comprising five
waves. Consider a sine wave incident upon the obstacle and travelling in the positive
direction of axis 0 x . Then, in the regions x < 0 (before the obstacle) and 0 < x < D
(above the obstacle) there exist two wave perturbations, propagating in both the pos-
itive and negative directions, while in the region x > D there is only one perturbation,
running in the positive direction. From the continuity condition for the free-surface
displacement

H 2 − H 1

and the water release ( Hu ) at points x = 0 , D the following expres-
sion is obtained for the amplitude transmission coefficient [Mofjeld et al. (2000)]:

ξ

T min
T min cos 2 T = , (5.9)

+ sin 2

β β