Cette thèse est divisée en quatre parties. Dans la première, je vais présenter notre travail sur le run-up des vagues longues et sur les phénomènes d’amplification par résonance. Grâce à des simulations numériques basées sur les équations en eau peu profonde non-linéaires, nous montrons que dans le cas des vagues monochromatiques d’incidence normale sur une plage inclinée, une amplification résonante du run-up se produit lorsque la longueur de la vague d’entrée est 5.2 fois plus grande que la longueur de la plage. Nous montrons également que cette amplification résonante de run-up peut être observée à partir de plusieurs profils de vagues. Cependant, l’amplification résonante du run-up n’est pas limitée aux plages inclinées infinies. En faisant varier le profil bathymétrique, la résonance est également présente dans le cas de bathymétries linéaires par morceaux et pour des bathymétries réalistes. Dans la deuxième partie, je présente une nouvelle solution analytique pour étudier la propagation des tsunamis générés par une source non ponctuelle sur une profondeur constante en utilisant la théorie des vagues en eau peu profonde linéaires. La solution, qui repose sur la séparation des variables et sur une double transformée de Fourier dans l’espace, est exacte, facile à mettre en œuvre et permet l’étude d’ondes de formes réalistes comme les ondes en forme de N (N–waves). Dans la troisième partie, j'étudie l’effet de protubérances localisées sur la génération de vagues longues. Même lorsque le déplacement final est connu grâce à l’analyse sismique, le plancher océanique qui se déforme peut avoir du relief comme des montagnes et des failles. On étudie analytiquement l’effet de la bathymétrie sur la génération des vagues de surface, en résolvant les équations en eau peu profonde linéaires avec for. Nous constatons que quand la hauteur du rebord augmente, le piégeage partiel de la vague permet de réduire la hauteur des vagues dans le champ lointain, tout en l’amplifiant au-dessus du rebord. Je vais aussi présenter brièvement une solution de la même équation forcée au-dessus d’un cône. Enfin, dans la dernière partie, nous verrons si les petites îles peuvent protéger les côtes proches de tsunamis comme il est largement admis par les communautés locales. Des découvertes récentes sur le tsunami des îles Mentawai en 2010 montrent un run-up amplifié sur les zones côtières derrière de petites îles, par rapport au run-up sur les lieux adjacents, qui ne sont pas influencés par la présence des îles. Nous allons étudier les conditions de cette amplification du run-up en résolvant numériquement les équations en eau peu profonde non-linaires. Le dispositif expérimental est régi par cinq paramètres physiques. L’objectif est double: Trouver l’amplification maximale du run-up avec un nombre minimum de simulations. Nous présentons un plan d’expériences actif, récemment mis au point et basé sur les processus Gaussiens, qui réduit considérablement le coût de calcul. Après exécution de deux cents simulations, nous constatons que dans aucun des cas considérés l’île n’offre une protection à la zone côtière derrière elle. Au contraire, nous avons mesuré une amplification du run-up sur la plage derrière elle par rapport à une position latérale sur la plage non directement affectée par la présence de l’île. Cette amplification a atteint un facteur maximal de 1.7. Ainsi, les petites îles à proximité du territoire continental agissent comme des amplificateurs des vagues longues dans la région directement derrière elles et non comme des obstacles naturels comme il était communément admis jusqu’ici. / This thesis is divided in four parts. In the first one I will present our work on long wave run-up and some resonant amplification phenomena. With the use of numerical simulations for the nonlinear shallow water equations, we show that in the case of monochromatic waves normally incident on a plane beach, resonant run-up amplification occurs when the incoming wavelength is 5.2 times larger the beach length. We also show that this resonant run-up amplification can be observed for several wave profiles such as bichromatic, polychromatic and cnoidal. However, resonant run-up amplification is not restricted to infinitely sloping beaches. We varied the bathymetric profile, and we saw that resonance is present in the case of piecewise linear and real bathymetries. In the second part I will present a new analytical solution to study the propagation of tsunamis from a finite strip source over constant depth using linear shallow-water wave theory. The solution, which is based on separation of variables and a double Fourier transform in space, is exact, easy to implement and allows the study of realistic waveforms such as N-waves. In the third part I will explore the effect of localized bathymetric features on long wave generation. Even when the final displacement is known from seismic analysis, the deforming seafloor includes relief features such as mounts and trenches. We investigate analytically the effect of bathymetry on the surface wave generation, by solving the forced linear shallow water equation. Our model for bathymetry consists of a cylindrical sill on a flat bottom, to help understand the effect of seamounts on tsunami generation. We derive the same solution by applying both the Laplace and the Fourier transforms in time. We find that as the sill height increases, partial wave trapping reduces the wave height in the far field, while amplifying it above the sill. Finally, in the last part I will try to explore whether small islands can protect nearby coasts from tsunamis as it is widely believed by local communities. Recent findings for the 2010 Mentawai Islands tsunami show amplified run-up on coastal areas behind small islands, compared with the run-up on adjacent locations, not influenced by the presence of the islands. We will investigate the conditions for this run-up amplification by numerically solving the nonlinear shallow water equations. Our bathymetric setup consists of a conical island sitting on a flat bed in front of a plane beach and we send normally incident single waves. The experimental setup is governed by five physical parameters. The objective is twofold: Find the maximum run-up amplification with the least number of simulations. Given that our input space is five-dimensional and a normal grid approach would be prohibitively computationally expensive, we present a recently developed active experimental design strategy, based on Gaussian Processes, which significantly reduces the computational cost. After running two hundred simulations, we find that in none of the cases considered the island did offer protection to the coastal area behind it. On the contrary, we have measured run-up amplification on the beach behind it compared to a lateral location on the beach, not directly affected by the presence of the island, which reached a maximum factor of 1.7. Thus, small islands in the vicinity of the mainland will act as amplifiers of long wave severity at the region directly behind them and not as natural barriers as it was commonly believed so far.

This thesis is divided in four parts. In the first one I will present our work on long wave run-up and some resonant amplification phenomena. With the use of numerical simulations for the nonlinear shallow water equations, we show that in the case of monochromatic waves normally incident on a plane beach, resonant run-up amplification occurs when the incoming wavelength is 5.2 times larger the beach length. We also show that this resonant run-up amplification can be observed for several wave profiles such as bichromatic, polychromatic and cnoidal. However, resonant run-up amplification is not restricted to infinitely sloping beaches. We varied the bathymetric profile, and we saw that resonance is present in the case of piecewise linear and real bathymetries. In the second part I will present a new analytical solution to study the propagation of tsunamis from a finite strip source over constant depth using linear shallow-water wave theory. The solution, which is based on separation of variables and a double Fourier transform in space, is exact, easy to implement and allows the study of realistic waveforms such as N-waves. In the third part I will explore the effect of localized bathymetric features on long wave generation. Even when the final displacement is known from seismic analysis, the deforming seafloor includes relief features such as mounts and trenches. We investigate analytically the effect of bathymetry on the surface wave generation, by solving the forced linear shallow water equation. Our model for bathymetry consists of a cylindrical sill on a flat bottom, to help understand the effect of seamounts on tsunami generation. We derive the same solution by applying both the Laplace and the Fourier transforms in time. We find that as the sill height increases, partial wave trapping reduces the wave height in the far field, while amplifying it above the sill. Finally, in the last part I will try to explore whether small islands can protect nearby coasts from tsunamis as it is widely believed by local communities. Recent findings for the 2010 Mentawai Islands tsunami show amplified run-up on coastal areas behind small islands, compared with the run-up on adjacent locations, not influenced by the presence of the islands. We will investigate the conditions for this run-up amplification by numerically solving the nonlinear shallow water equations. Our bathymetric setup consists of a conical island sitting on a flat bed in front of a plane beach and we send normally incident single waves. The experimental setup is governed by five physical parameters. The objective is twofold: Find the maximum run-up amplification with the least number of simulations. Given that our input space is five-dimensional and a normal grid approach would be prohibitively computationally expensive, we present a recently developed active experimental design strategy, based on Gaussian Processes, which significantly reduces the computational cost. After running two hundred simulations, we find that in none of the cases considered the island did offer protection to the coastal area behind it. On the contrary, we have measured run-up amplification on the beach behind it compared to a lateral location on the beach, not directly affected by the presence of the island, which reached a maximum factor of 1.7. Thus, small islands in the vicinity of the mainland will act as amplifiers of long wave severity at the region directly behind them and not as natural barriers as it was commonly believed so far.