Doubling of Group Velocity near the Coast due to Storm Surge, Tides or Tsunami: An Analytical Study

Tsunami may cause due to storm surge, earthquake at coastal region or at under the sea (obviously Hypocenter of earthquake must be within the earth’s crust or below). Each cases a wave packet which develops a set of simple harmonic motion of different wave lengths. These simple harmonic waves have been analyzed analytically. The velocity of propagation of waves depends upon the wave lengths. So the waves of nearly equal wave lengths can be considered as a group. This group of waves will propagate with nearly equal velocity which is known as group velocity. On considering two consecutive simple harmonic waves of same amplitude, we can find two equations describing simple harmonic motions having slightly different wave lengths and time periods. The combination of these two said SHM’s we can find another resultant SHM with different amplitude than of the previous two SHM’s. This new SHM has a slight variation in wave length and time period than that of that of the said earlier two SHM’s. Then using the perturbation technique on this resultant equation of SHM, we can find a new wave velocity (group velocity) in differential notation of wave velocity. This differential notation of wave velocity has been eliminated from the relation between the wave velocity on the surface of water and the depth of the sea. Then we get a relation between group velocity and wave velocity depending with depth of the sea. The application of boundary conditions on depth of deep sea and that of at the sea shore, we can find the group velocity at each region respectively. It is seen from the analysis that the group velocity at the sea shore is as much as double that of at the deep sea. Several recent cases of tsunamis or storm surges have been studied and it is found that the results depicted the same implementation through devastation that established by analytical study.