Surface gravity-wave lensing

Ryan B. Elandt, Mostafa Shakeri, and Mohammad-Reza Alam

Phys. Rev. E 89, 023012 – Published 19 February 2014

Abstract

Here we show that a nonlinear resonance between oceanic surface waves caused by small seabed features (the so-called Bragg resonance) can be utilized to create the equivalent of lenses and curved mirrors for surface gravity waves. Such gravity wave lenses, which are merely small changes to the seafloor topography and therefore are surface noninvasive, can focus or defocus the energy of incident waves toward or away from any desired focal point. We further show that for a broadband incident wave spectrum (i.e., a wave group composed of a multitude of different-frequency waves), a polychromatic topography (occupying no more than the area required for a monochromatic lens) can achieve a broadband lensing effect. Gravity wave lenses can be utilized to create localized high-energy wave zones (e.g., for wave energy harvesting or creating artificial surf zones) as well as to disperse waves in order to create protected areas (e.g., harbors or areas near important offshore facilities). In reverse, lensing of oceanic waves may be caused by natural seabed features and may explain the frequent appearance of very high amplitude waves in certain bodies of water.

Received 14 September 2013

DOI:https://doi.org/10.1103/PhysRevE.89.023012

Authors & Affiliations

Ryan B. Elandt, Mostafa Shakeri, and Mohammad-Reza Alam^*

Department of Mechanical Engineering, University of California, Berkeley, California 94720, USA

^*Corresponding author: reza.alam@berkeley.edu

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Issue

Vol. 89, Iss. 2 — February 2014

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Images

Figure 1
(a) Geometric construction of the class I (triad) Bragg resonant waves. (b) Physical implication of class I Bragg resonance: an incident wave with the wavenumber vector $k_{i}$ (assumed to move along the positive $x$ axis) resonates the resonant wave $k_{r}$ upon the interaction with the topography with the wavenumber $k_{b}$ .
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Figure 2
(a) Schematic representation of the gravity wave lensing: Incident waves enter from left and upon resonance interaction with the (properly designed) seabed ripples get focused at a desirable focal point. The focal point can be designed to be on the upstream as well as the downstream side of the patch. In analogy to optics, we call these configurations, respectively, a concave mirror and a convex lens. (b) Direct simulation of a concave mirror for gravity waves. In this case, a monochromatic surface wave of $ε_{i} = k_{i} a_{i} =$ 0.080 and $k_{i} h =$ 0.84 enters from left and upon interaction with the rippled patch (maximum of $ε_{b} = k_{b} a_{b} =$ 0.640) its energy gets focused at the designated focal point at $x / λ_{i} =$ 0 ( $λ_{i}$ is the wavelength of the incident wave). In the snapshot shown ( $t / T_{i} =$ 14.7, $T_{i}$ is the period of the incident wave), at the focal point $η_{f} / a_{i}$ =6.3, i.e., surface elevation at the focal point is 6.3 times the amplitude of the incident wave. Simulation parameters are $N x = N y =$ 256, $δ t / T_{i} =$ 30, and $M = 3$ for which the simulation is converged. (c) Direct simulation of a convex lens for gravity waves. Parameters are the same as in panel (b) except $ε_{b, \max} =$ 0.450. For the snapshot shown ( $t / T_{i} =$ 9.6), $η_{f} / a_{i} =$ 2.7. In both cases, the amplification factor increases by the increase in the amplitude and number of ripples.
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Figure 3
Experimental investigation of the gravity wave lensing. (a) Experimental setup. The photograph shows the middle section of a 60 m $\times$ 2.4 m (width) $\times$ 1.8 m tank at the U.C. Berkeley's Richmond Field Station. To measure the time evolution of the water surface profile, a laser-induced fluorescence (LIF) technique was used (greenish color is due to fluorescent dye). Ripple crests [cf. panel (c)] can be seen through the water and are marked by dashed lines on the left side of the figure. (b) The 2.4 m $\times$ 2.4 m rippled patch before it is colored in black and placed inside the wave tank. (c) A side-by-side comparison of the experimental (left) and direct simulation (right) results. The experiment is designed for a monochromatic incident wave of $ε_{i} =$ 0.016, $k_{i} h =$ 1.57, and $ε_{b, \max} =$ 1.26. In terms of physical variables, these parameters correspond to a water depth of 15 cm, incident wave amplitude of 1.5 mm, and wavelength of 60 cm, topography wavelength along the centerline of 30 cm, and topography amplitude of 6 cm (see Supplemental Material movies 1 and 2 in Ref. [20]).
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Figure 4
Gravity wave lensing for a broadband incident spectrum. To achieve focusing in this case, the topography needs to be polychromatic. (a) A water-surface snapshot from direct simulation of focusing of a broadband (Gaussian) incident spectrum. Colors show the surface elevation normalized by the significant wave amplitude ( $a_{s} = H_{s} / 2$ , where $H_{s}$ is the significant wave height). In the snapshot shown, the waveheight of the wave seen at the focal point exceeds 4 $a_{s} =$ 2 $H_{s}$ and therefore is a rogue wave by definition. (b) Steady-state spectra at the upstream (blue dashed line) and downstream (black dash-dotted line) of the lensing area compared with that at the focal point (red solid line). Clearly, the spectrum at the focal point (F) has a much higher energy (about four times the energy of the incident spectrum). The downstream (shadow zone) spectrum, as expected, has a lower level of energy. In this figure, $ω_{p}$ is the peak frequency of the spectrum, and normalized spectral density function $S^{*} (ω)$ is defined as $\int S^{*} d ω = \sum_{j} 1 / 2 {(a_{j} / a_{s})}^{2}$ . Simulation parameters are $N x = N y = 512$ , $δ t / T_{p} = 30$ , and $M = 4$ , for which the simulation is converged.
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