Resonance of an object floating within a surface wavefield

TL;DR

We investigate the resonance of a buoyant cylinder in a gravity-wave field, focusing on its natural heave frequency and how it depends on radius $R$, height $h$, and the density ratio $\rho/\rho_w$. Using impulsive vertical perturbations and surface-field reconstruction with Fast Checkerboard Demodulation, the authors extract a Lorentzian resonance peak at frequency $f_0$ and a damping parameter $\mu$, with $f_0$ approximately following $f_0 \approx \frac{1}{2\pi}\sqrt{\frac{\rho_w g}{\rho h}}$ in the simple limit and showing $f_0 \propto (hR)^{-0.22}$ experimentally. In an external wavefield, the floater radiates and diffracts incident waves, and the transverse emission is minimized when the forcing frequency is near $f_0$, with $f_{\min} \approx f_0$. The results provide a framework for understanding energy redistribution in larger systems such as sea-ice floes and inform modeling of wave–structure interactions in coastal engineering.

Abstract

We examine the interaction between floating cylindrical objects and surface waves in the gravity regime. Since the impact of resonance phenomena associated with floating bodies, particularly at laboratory scales, remains underexplored, we focus on the influence of the floats' resonance frequency on wave emission. First, we study the response of floating rigid cylinders to external mechanical perturbations. Using an optical reconstruction technique to measure surface wave fields in both space and time, we study the natural resonance frequency of floats with different sizes. The results indicate that the resonance frequency is influenced by the interplay between the cylinder geometry and the solid-to-fluid density ratio. Second, these floating objects are placed in an incoming wave field. These experiments demonstrate that floats diffract incoming waves, while radiating secondary waves that interfere with the incident wavefield. Minimal wave generation is observed at resonance frequencies. These findings can provide insights for elucidating the behavior of larger structures, such as sea ice floes, in natural wave fields.

Figures (9)

• Figure 1: Experimental setup for measuring the natural frequency $f_0$ of a floating body. (a) Side-view schematic (not to scale) and (b) body geometry.
• Figure 2: Wave field and resonance frequency of floating cylinders experiencing instantaneous vertical impacts. (a) Top view of a floating cylinder ($R = 30 \, \rm mm$, $h = 15 \, \rm mm$) displaced by a rod (as depicted in the setup of Fig. \ref{['fig:Schematic_resonance_frequency_setup']}a) over a checkerboard pattern with square size $a_{c} = 4.0 \, \rm mm$. (b) Water surface topography at time $t = 1.39 \, \rm s$ post impact. (c) Fourier spectrum $|\langle \hat{\zeta} \rangle_{x,y}| (f)$ of the wave field generated by the object perturbation shown in (b). The inset in (c) depicts the same Fourier spectrum in log-log scale. The main peak represented by red dots is fitted with a Lorentzian function, plotted as a red line. (d) Measured resonance frequency as a function of the floaters' height $h$ and for different radii $R$. Each data point corresponds to an individual experiment. The black dotted line represents the prediction $f_0 = \frac{1}{2\pi}\sqrt{\frac{\rho_w g}{\rho h}}$ obtained without considering the floater added mass.
• Figure 3: Schematics for the model. (a) Floating cylinder. In the high-frequency limit, the heaving added mass of the shape in panel (b) is equal to half the translatory added mass, $M_{22}$, of the shape in panel (c) in an infinite fluid. The shape in (c) is the union of the submerged portion of the body in (b) and its reflection about the free surface.
• Figure 4: Experimental data compared to our theoretical model, using (\ref{['eq:true']}). Each point in the plot represents a single experiment. Cylinder radii are given by color code. For a given radius, cylinder heights are, from right to left in the plot: $h = 5 \, \rm{mm}$, $h = 10 \, \rm{mm}$, $h = 15 \, \rm{mm}$, $h = 20 \, \rm{mm}$.
• Figure 5: Experimental setups and sample data for characterizing the wave field in the vicinity of a floating object: configuration without (a,c,e) and with (b,d,f) cylindrical floater. (a, b) Schematics of the experimental setups (not to scale), in which a wavemaker generates harmonic waves in the tank. (c, d) Sample instantaneous height fields $\zeta(x,y)$ with wavemaker frequency $f = 6.0 \, \rm Hz$. In (d) the floating cylinder has radius $R = 20 \, \rm mm$ and height $h = 10 \, \rm mm$. (e) Processed spatio-temporal profile along the dashed line at $y=y_c$ in (c). (f) Processed spatio-temporal profile along the dashed line at $x=x_o$ in (d), passing through the center of the floating object. Blue lines show the wave height at a given time, while red lines are the envelopes of the processed signal.