The spatial separation of frequency of any kind of wave occurs due to the presence of obstructions in the medium where the wave is propagating. This phenomenon is termed Rainbow trapping and has proven to be very useful in devising innovative technologies in the field of optics and acoustics. The spatial localization of the individual frequencies from the whole spectrum of a broadband wave opens up possibilities to harness each frequency individually for frequency specific purposes. We will be investigating this separation phenomenon by using an array of uniform surface piercing vertical barriers with graded spacing for water waves. The effectiveness of this arrangement as breakwaters will also be tested. The 2D linear water wave theory will be used to model our problem, and the solution will be derived by using the boundary element method (BEM). Further, we will also look into the water wave energy amplification caused due to the Rainbow reflection phenomenon.
Waves are an integral part of our physical world. Therefore, it is not surprising that many fields of physical and engineering studies deal with understanding the nature and behavior of waves. One such property of waves is the rainbow reflection phenomenon, which was first observed in the study of light waves (Tsakmakidis et al. 2007). This phenomenon involves the presence of periodic obstacles or structures in the medium of propagation, which causes spatial separation of the broadband wave into its spectral components followed by amplification of the localized wave energy (He et al. 2012). Later, similar observations were made in the studies of acoustic waves (Zhu et al. 2013) and seismic waves (Colombi et al. 2016). The trapping of these spatially separated waves have led to the creation of many metamaterial structures (Zhao and Zhou, 2019) that have the ability to harness the broadband waves for frequency specific purposes like filtering, etc. These kinds of phenomena like the resonance (Bragg resonance) and rainbow reflection is generally observed when waves propagate through a periodic array of structures (Xu et al. 2023). Usually Bragg scattering occurs in the presence of uniform (with respect to physical parameters) array of structures in the propagating medium (Davies, 1982), while rainbow reflection occurs when there is some kind of grading in one or more physical parameters of the structures involved (Wilks et al. 2022). Successfully harnessing these phenomena for the purpose of technological advancements in the fields of optics and acoustics has motivated researchers to focus their attention on water waves (Bennetts et al. 2018). The main purpose behind spatially separating the broadband water waves is the accessibility to individual frequency wave components in physical domains where they are localized. These monochrome waves are then easily harnessed by different mechanisms like wave energy converters to extract energy. Also, it is possible to achieve perfect absorption from a monochromatic wave. To maximize energy extraction, we use the property of amplification by adjusting various parameters of the barriers and the array, where wave energy is amplified before getting reflected. Arrays of barriers in shallow and deep waters have been studied for a long time to understand the associated properties like scattering, resonance, structural stability and so on (Ha et al. 2002; Kakuno et al. 1996; Walker and Taylor, 2005). These arrays of barriers are useful for various purposes like breakwaters for protection of coastal structures (Dalrymple and martin, 1990; Ji et al. 2016), supporting marine structures like oil rigs and bridges (Evans and porter, 1999; Fu et al. 2020) and harvesting energy from wave energy parks (Göteman, 2017; Giassi and Göteman, 2018), to name a few. These types of arrays give rise to rainbow trapping and rainbow reflection phenomenon when the structures are graded, usually in their diameters, submergence or the spacing in between them. Several studies have been done with the help of graded arrays of cylinders to demonstrate these phenomena in water waves. Hu et al. (2017) used C-shaped cylinders to study the bandgaps associated with such arrays. Bennetts et al. (2018) further studied these C-shaped cylinders with graded diameters and observed rainbow reflection in the array. The studies were further extended for surface piercing rectangular barriers with graded drafts and graded inter barrier spacing (Wilks et al. 2022). All these studies involve grading in the spacing and/or grading in the dimensions of the barriers. Archer et al. (2020) conducted an experimental study with uniform surface piercing cylinders in a wave flume with graded spacing and got successful observations for rainbow reflection. Recent studies were done over a uniform array of submerged bars with graded spacing in increasing order, which depicts similar results regarding energy separation and amplification (Xu et al. 2023). Once, several different arrays are studied and understood, we can move forward with further optimization of the parameters so as to obtain optimal results. This motivates us to study different configurations of arrays of barriers that might show similar results. So, here, we will consider a finite number of uniform surface-piercing barriers with rectangular cross-sections and graded spacing to check the occurrence of rainbow reflection.