Abstract. Resonance has recently been proposed as the fundamental underlying mechanism that shapes the amplification in coastal run-up for storm surges and surf beats, which are long-wavelength disturbances created by fluid velocity differences between the wave groups and the regions outside the wave groups. It is without doubt that the resonance plays a role in run-up phenomena of various kinds; however, we think that the extent to which it plays its role has not been completely understood. For incident waves, which we assume to be linear, the best approach to investigate the role played by the resonance would be to calculate the normal modes by taking radiation damping into account and then testing how those modes are excited by the incident waves. Such modes diverge offshore, but they can still be used to calculate the run-up. There are a small number of previous works that attempt to calculate the resonant frequencies, but they do not relate the amplitudes of the normal modes to those of the incident wave. This is because, by not including radiation damping, they automatically induce a resonance that leads to infinite amplitudes, thus preventing them from predicting the exact contribution of the resonance to coastal run-up. In this study we consider two different coastal geometries: an infinitely wide beach with a constant slope connecting to a flat-bottomed deep ocean and a bay with sloping bottom, again, connected to a deep ocean. For the fully 1-D problem we find significant resonance if the bathymetric discontinuity is large.The linearisation of the seaward boundary condition leads to slightly smaller run-ups. For the 2-D ocean case the analysis shows that the wave confinement is very effective when the bay is narrow. The bay aspect ratio is the determining factor for the radiation damping. One reason why we include a bathymetric discontinuity is to mimic some natural settings where bays and gulfs may lead to abrupt depth gradients such as the Tokyo Bay. The other reason is, as mentioned above, to test the role played by the depth discontinuity for resonance.