The interaction of water waves with multiple composite flexible membranes is studied based on the assumption of linear wave theory and small membrane response. The eigenfunction expansion method and the wide spacing approximation are used to obtain the reflection and transmission coefficients. The characteristics of Bragg reflection are investigated by changing various factors including the relative group spacing, the end condition, the number, the tension, the draft and the protrusion of membranes. It is found that the multiple composite flexible membranes can enlarge the bandwidth of Bragg reflection. The study can provide guidance on design of multiple composite flexible membranes as effective floating breakwaters by taking advantage of Bragg reflection.
Floating breakwaters offer an alternative to protect harbors and shorelines from wave attack when conventional fixed breakwaters are inappropriate due to large costs. Floating breakwaters often consist of all kinds of structures, such as boxes, pontoons, mats, tires and flexible membranes. Among them, flexible membranes, generally made of synthetic fiber, rubber or polymeric materials, have been thought as prospective floating breakwaters due to the merits of light, cheap and rapidly deployable. Early typical researches on the wave interaction with flexible membranes can be found in Williams (1996), Kim and Kee (1996) and Cho and Kim (1998). Lee and Lo (2002) analyzed the efficiency of wave attenuation of single and double surface-penetrating flexible membranes of finite draft. Hsiao et al. (2007) proposed a composite breakwater consisting of a flexible membrane and a submerged breakwater, and found that it had better reflection characteristics than a single submerged breakwater for long period waves. Recently, Karmakar et al. (2012) investigated the interaction between water waves and multiple vertically moored surface-piercing membranes with the spacing between adjacent membranes keeping all the same in finite water depth.
Multiple structures usually consist of a number of isolated structures. When the spacing between adjacent isolated structures keeps the same, the multiple structures can be defined as periodical structures, and the periodical length is equal to the spacing. In the field of water wave interaction with periodical structures, Bragg reflection is a phenomenon that the peak reflection coefficient could be achieved when the wavelength of the incident wave is approximately twice of the periodical length of the structures. Earliest experimental results on the Bragg reflection of submerged sand bars were obtained by Heathershaw (1982). After that, the phenomenon of Bragg reflection in the wave interaction with submerged structures has been widely studied by experimental, theoretical and numerical methods (Davies and Heathershaw, 1984; Mei, 1985; Hsu et al., 2007). Later, the phenomenon of Bragg reflection of floating structures with periodicity has also been investigated. For example, Linton (2011) analyzed the phenomenon of Bragg reflection of water waves propagating over arrays of horizontal cylinders floating in the water. When multiple flexible membranes with equal spacing serve as floating breakwaters, their periodicity can definitely induce the phenomenon of Bragg reflection and attenuate the incident wave with corresponding wavelength. The multiple flexible membranes are usually arranged vertically in the water, and fixed or moored through buoys, cables or springs. The phenomenon of Bragg reflection of the multiple flexible membranes was simply indicated by Karmakar et al. (2012) and no more details were provided. Moreover, the bandwidth of Bragg reflection by the traditional multiple flexible membranes is often narrow because they have only one periodical length, corresponding to one primary Bragg reflection.