Time domain modelling of a Helmholtz resonator analogue for water waves

In the context of water waves, we consider a resonator with deep subwavelength resonance, analogue to the Helmholtz resonator in acoustics. In the shallow water regime, using asymptotic analysis, a one-dimensional model is derived in which the effect of the resonator is reduced to effective transmission conditions. These conditions clearly highlight two contributions. The first is associated with the dock on its own and it is responsible for a jump of the potential at the free surface. The second is due to the resonant cavity and it is responsible for a jump in the horizontal velocity. It involves as well the uniform amplitude within the resonant cavity with a transient dynamics explicitly given by the equation of a damped oscillator forced by the incident waves. The one-dimensional model is validated in the harmonic regime by comparison to direct two-dimensional numerics. It is shown to reproduce accurately the scattering coefficients and the amplitude within the resonator; interestingly, this remains broadly true for finite water depths. We further inspect the spatio-temporal behaviour of different types of wave packets interacting with the resonating and radiating cavity.