The Reciprocity Principle for the Shallow Water Waves, Taking into Account the Earth’s Rotation

Korolev Yu. P.

The reciprocity principle in acoustics is known for a long time and has a practical application. The reciprocity relationship is asymptotic one and is true for linear processes in heterogeneous media in the presence reflecting, absorbing or impedance boundaries. Long wave ocean processes (in particular, tsunamis) have features that are not essential in acoustics. These include a significant inhomogeneity (long wave velocities differ by more than 10 times: from 10 m/s on water of 10 m in depth to 200 m/s on water of 4000 m in depth), as well as the possibility of the Coriolis effect manifestation. The reciprocity principle in detail, as well as taking into account the Earth’s rotation has not been researched with respect to long waves. The purpose of this paper is to analyze the reciprocity principle for long waves on shallow water propagating on the rotating Earth: the evaluation of limits of the applicability of reciprocity relationship, taking into account the media inhomogeneity and evaluation of the possible influence of the Coriolis effect. The similarity criterion was derived, which links the horizontal sizes of sources and water depths in their epicenters, under which waveforms in reciprocal points coincide. On the basis of numerical experiments using real bathymetry it is found that the reciprocity relationship is valid for distances between sources, comparable to their horizontal sizes (with travel time, comparable with the characteristic wave periods). It is shown that in general case taking into account the Earth’s rotation, the reciprocity principle is not satisfied; however, in the special case of symmetric sources it is fair. The reciprocity relationship for long waves in this case coincides with the reciprocity relationship in acoustics. Waveforms from reciprocal sources, located in areas near the Kuril Islands and off the coast of Chile, are in a good agreement with each other. The reciprocity principle (relationship) can be applied to solve problems related to the tsunami problem, and for other applications.

Download original text