Long waves approaching the coast: Green’s law generalization

  • Francesco Lalli
  • Matteo PostacchiniEmail author
  • Maurizio Brocchini
Research Article


The original Green’s relationship provides the amplitude of a long wave at a given water depth as a function of the amplitude in deeper waters, accounting for the wave shoaling, taken as a one-dimensional process, and neglecting both diffraction and refraction effects. An analytical application of Green’s law can only be made in case of simple bathymetries, monotonically increasing in the cross-shore direction and being longshore uniform. In the present work, a new formulation is proposed, based on Green’s law, for the direct calculation of the change in amplitude of a long wave that approaches the coast while traveling over a natural bathymetry, characterized by a general shape. Hence, the effects due to the ray curvature provided by the refraction/diffraction phenomena are accounted for. In detail, a generalization of Green’s law is proposed by introducing a numerically computed coefficient. Comparisons have been provided between the wave amplitude evolution, reconstructed using the proposed law, and the results of numerical simulations, run using a solver based on the solution of the shallow water equations. Although local effects due to obstacles are not properly captured, such comparisons reveal that the generalized Green’s law works well in the far field under different wave and complex morphological conditions.


Green’s law Long waves Numerical modeling Shallow water equations 



The Italian Civil Protection is acknowledged for supporting this project, in the framework of the “Italian Tsunami Directive”.


This research has been funded by the Presidenza del Consiglio dei Ministri, Department of Italian Civil Protection.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.ISPRA, Istituto Superiore per la Protezione e la Ricerca AmbientaleRomeItaly
  2. 2.Department of Ingegneria Civile, Edile e ArchitetturaUniversità Politecnica delle MarcheAnconaItaly

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