Subharmonic resonance of Venice gates in waves. Part 1. Evolution equation and uniform incident waves

Abstract

For flood protection against storm tides, barriers of box-like gates hinged along a bottom axis have been designed to span the three inlets of the Venice Lagoon. While on calm days the gates are ballasted to rest horizontally on the seabed, in stormy weather they are raised by buoyancy to act as a dam which is expected to swing to and fro in unison in response to the normally incident sea waves. Previous laboratory experiments with sinusoidal waves have revealed however that neighbouring gates oscillate out of phase, at one half the wave frequency, in a variety of ways, and hence would reduce the effectiveness of the barrier. Extending the linear theory of trapped waves by Mei et al. (1994), we present here a nonlinear theory for subharmonic resonance of mobile gates allowed to oscillate about a vertical plane of symmetry. In this part (1) the evolution equation of the Landau–Stuart type is first derived for the gate amplitude. The effects of gate geometries on the coefficients in the equation are examined. After accounting for dissipation effects semi-empirically the theoretical results on the equilibrium amplitude excited by uniform incident waves are compared with laboratory experiments.

Weakly nonlinear theory for oscillating wave surge converters in a channel

“…With the introduction of these two small parameters the formulation of the problem becomes algebraically simpler to Sammarco et al (1997a). Indeed, let Ω + (Ω − ) denote the fluid regions to the right (left) of the gates and distinguish physical quantities in Ω ± through ±.…”

“…The latter forcing terms are algebraically simpler than those of Sammarco et al (1997a). Indeed, the second-order boundary problem of Sammarco et al (1997a) is forced on the free surface by the first-order solution.…”

“…Indeed, the second-order boundary problem of Sammarco et al (1997a) is forced on the free surface by the first-order solution. In the present case instead, the forcing term for the second-order free-surface mixed boundary condition F 2 is null, as well as F 1 .…”

“…In this section we analyse the subharmonic resonance of a single trapped natural mode of an array of Q gates. Performing harmonic expansion of the governing equations, we obtain that the zeroth harmonic problem at the first and second order is unforced while the boundary value problem governing the first harmonic at the second order is identical to that governing the first harmonic at the first order (Sammarco et al 1997a). For these reasons, we return in physical variables, omit the primes for brevity and assume the following solution:…”

“…Laboratory experiments have revealed that, for certain frequencies, the incident waves resonate subharmonically the trapped modes of the barrier (Mei et al 1994). The weakly nonlinear theory which explains this resonance mechanism has been developed by Sammarco, Tran & Mei (1997a), Sammarco et al (1997b) for uniform and modulated waves. A simplified model for the subharmonic resonance of sliding gates in shallow waters has been developed by Vittori, Blondeaux & Seminara (1996).…”