On tsunami and the regularized solitary-wave theory
For ideal hydrodynamic modeling of earthquake-generated tsunamis, the principal features of tsunamis occuring in nature are abstracted to provide a fundamental case of a one-dimensional solitary wave of height a, propagating in a layer of water of uniform rest depth h for modeling the tsunami progressing in the open ocean over long range, with height down to a/h ≃ 10^(−4) as commonly known. The Euler model is adopted for evaluating the irrotational flow in an incompressible and inviscid fluid to attain exact solutions so that the effects of nonlinearity and wave dispersion can both be fully accounted for with maximum relative error of O(10^(−6)) or less. Such high accuracy is needed to predict the wave-energy distribution as the wave magnifies to deliver any devastating attack on coastal destinations. The present UIFE method, successful in giving the maximum wave of height (a/h = 0.8331990) down to low ones (e.g. a/h = 0.01), becomes, however, impractical for similar evaluations of the dwarf waves (a/h < 0.01) due to the algebraic branch singularities rising too high to be accurately resolved. Here, these singularities are all removed by introducing regularized coordinates under conformal mapping to establish the regularized solitary-wave theory. This theory is ideal to differentiate between the nonlinear and dispersive effects in various premises for producing an optimal tsunami model, with new computations all regular uniformly down to such low tsunamis as that of height a/h = 10^(−4).
© 2010 Springer Science+Business Media B.V. Received: 12 April 2010; Accepted: 22 September 2010; Published online: 19 October 2010. The authors are very pleased to have this good opportunity of contributing this paper to the Special Issue of the Journal of Engineering Mathematics in honorary commemoration of Professor Ernest O. Tuck. One of them (Wu) wishes to express a token of his deep appreciation for knowing and admiring Ernie Tuck as a shining star of mathematics, and especially for the fond memory on the precious two-year visits Professor Tuck spent with Mrs. Helen Tuck and Family at Caltech in the early sixties for fruitful scientific collaborations in Wu's research group. Warm thanks are also due from the authors to Michelle H. Teng for valuable discussions on the historical survey of the 1964 tsunami, to Yue Yang for useful assistance, and to Chin-Hua S. Wu for strong support and encouragements. Part of this work is sponsored by The American–Chinese Scholarship Foundation.