[1] The role of free modes of oscillation of coastal areas in tsunami amplification at the coast is investigated here. A finite element numerical model for modal analysis was applied and the numerically calculated natural frequencies were compared to those resulting from the spectra of measured sea level time series. Two case studies have been selected: that of Poverty Bay (New Zealand); and that of Kuluk Bay (Adak Island, Alaska, USA). The natural modes of the sea areas that extend in front of these locations are shown to play an important role in tsunami amplification at both the considered bays. In fact, the enhancement of wave height is found to be related to both the small-scale resonance controlled by the coastal shape, and the large-scale one governed by the continental shelf bathymetry. In particular, the model application to Poverty Bay reveals that some of the continental shelf modes are more energetic and occur at frequencies higher than the bay fundamental one. These modes are identified as both cross-shelf modes and trapped edge waves. On the other hand, the application to Kuluk Bay shows that geographical entrapment can be relevant for chain islands, making the bay and the continental shelf modes almost coincident.Citation: Bellotti, G., R. Briganti, and G. M. Beltrami (2012), The combined role of bay and shelf modes in tsunami amplification along the coast,

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This paper presents a novel technique for the computation of eigenvalues and eigenvectors of partially\ud enclosed basins such as harbours and bays. The procedure makes use of the finite element approximation\ud of the linear shallow water equations, and converts the time-depending problem into an eigenvalues one.\ud The main point of novelty of this research is the mathematical condition used at the boundary that separates\ud the computational domain from the open sea. While classical techniques impose a zero surface elevation (i.e.\ud a nodal line), here an approximate radiation condition is applied. The use of a radiation condition at the open\ud boundary gives a quadratic eigenvalue problem that admits as solution complex eigenvalues and eigenvectors,\ud thus describing the flow in terms of both standing and progressive waves. The new method is applied\ud to an idealized long and narrow harbour, for which an analytical solution of long wave resonance exists,\ud and to the harbour of Marina di Carrara (Italy), for which measurements and previous numerical computation\ud results are available. In both cases the results show good agreement with the available data

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