Rules of application of algorithms for tsunami waves kinematic computations based on the Huygence principle

The process of tsunami waves propagation can be well described by the shallow-water equations. One of the corollaries of this model is the independence of a wave-front velocity from the amplitude of a tsunami wave. This velocity depends only on the depth.

Methods of wave-front computation, that are based on a determination of trajectories of wave rays have some certain shortcomings. At first, as the result of divergence (during computations), the distance between the neighboring wave rays becomes too long. It means that we need to insert a new wave rays or wave-front points. In order to make such a procedure correctly (without distortion of an actual segment of a front motion), the complicated enough algorithm is required. It can cause the significant growth of calculations volume. The simplified procedure of an insertion of new points of front (or rays) can consist in "installation" of a new point of a wave-front to the middle of a segment connecting neighboring points that moved away from each other. However, this method decelerates the movement of a wave-front. The same concerns to the wave rays.

The second significant shortcoming of the indicated methods is the practical impossibility of a prolongation of the wave-front kinematic calculation in those places of an area, where the wave passes through narrow straits. In this situation, 1-2 rays can pass through a strait (or 1-2 points of a wave-front set), and, then the continuation of further movement of this part of a front is impossible, using standard algorithm. The minimum three points of a wave-front set is required to find the next position of only one central point. The further movement of two edge points of this set remains undefined. Therefore, adding new points of the wave-front to the edges of safely past through a narrow strait segment is necessary.

As far as wave rays are concern, in such situation they have a full autonomy (do not depend on neighboring rays), but the sudden reduction of their quantity after passing a narrow strait can cause the situation that in a water area behind this strait we will observe only narrow bunch of wave rays. In a reality, after passing through a strait, the wave rays due to diffraction will propagate practically in all directions. Thus, during computations of wave-front sets by the indicated methods in areas with complicated topology (narrow straits, long narrow islands) significant areas of "shadow" can remain. In such places at the coastline or in water areas, tsunami travel-time can not be defined. As an example, we want to show a fragment of a tsunami travel-time chart which is produced in the Krasnoyarsk Computing Center under the contract with IOC UNESCO.