Wave propagation through periodic arrays of freely floating rectangular floes

Abstract

The two-dimensional propagation of small-amplitude waves through an infinite periodic array of freely-floating rectangular floes is considered under the assumptions of inviscid linearised wave theory. Fluid gaps between adjacent floes allow a complex interaction of the fluid with heave, surge and pitch motions. In particular, the presence of fluid resonance in the vertical channels between floes has a significant influence on wave propagation around certain critical frequencies. Bloch-Floquet theory is used and encodes the wavenumber for propagating waves into periodic boundary conditions. Solutions of the resulting boundary-value problem posed in a fundamental cell are formulated in terms of integral equations in which the three rigid body modes of the problem are treated individually. The dispersion relationship between frequency and wavenumber is expressed in terms of the vanishing of a 3 x 3 determinant which encodes the hydrodynamic coupling between the modes. Accurate numerical solutions are determined using Galerkin's method to approximate solutions to the integral equations. A particular focus of the paper is determining simple explicit approximations for the dispersion relation by assuming the gap between adjacent floes is small compared to the submerged draft of the floe. Approximations are shown to compare well to numerical results for a large range of gap sizes and some surprising results emerge for low-frequency wave propagation. This is particularly relevant to the application area that motivates this study: the modelling of wave propagation through broken ice. Supplementary Material: https://github.com/LloydDafydd/pre-prints/blob/caa889f1ee1eb51980785271bd8d27ce5213bda7/wave-propogation-fff-supp-mat.pdf

Figures (14)

• Figure 1: Definition of the coordinate system and variables used
• Figure 2: Dispersion curves ($kL$ versus $K \hat{\rho} d$) for heave-constrained motions in the case $\hat{\rho}=0.9$, $L/d = 1 + \epsilon$ (square ice floes). Exact results (blue solid curves), explicit $O(\epsilon)$ small-gap approximation (orange, dotted),leading order small-gap approximation (yellow, dashed) and small gap root-finding (purple, dot-dashed) for gap sizes: (a) $\epsilon = 0.01$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.08$, (d) $\epsilon = 0.12$.
• Figure 3: Dispersion curves ($kL$ versus $K \hat{\rho} d$) for surge-constrained motions in the case $\hat{\rho}=0.9$, $L/d = 1 + \epsilon$ (square ice floes). Exact results (blue solid curves) and small-gap approximations (orange, dashed) for gap sizes: (a) $\epsilon = 0.01$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.08$, (d) $\epsilon = 0.12$.
• Figure 4: Dispersion curves ($kL$ versus $K \hat{\rho} d$) for heave and surge-constrained motions in the case $\hat{\rho}=0.9$, $L/d = 1 + \epsilon$ (square ice floes). Exact results (blue solid curves) and small-gap approximations (orange, dashed) for gap sizes: (a) $\epsilon = 0.01$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.08$, (d) $\epsilon = 0.12$.
• Figure 5: Dispersion curves ($kL$ versus $K \hat{\rho} d$) for pitch-constrained motions in the case $\hat{\rho}=0.9$, $L/d = 1 + \epsilon$ (square ice floes). Exact results (blue solid curves) and small-gap approximations (orange, dashed) for gap sizes: (a) $\epsilon = 0.01$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.08$, (d) $\epsilon = 0.12$.