water Article Transmission of Water Waves under Multiple Vertical Thin Plates Yifei Yu 1 , Zhiqun Guo 1, * ID and Qingwei Ma 1,2 1 College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; 2015011420@hrbeu.edu.cn (Y.Y.);Q.Ma@city.ac.uk (Q.M.) 2 School of Engineering and Mathematical Sciences, City University London, London EC1V 0HB, UK * Correspondence: guozhiqun@hrbeu.edu.cn; Tel.: +86-451-8258-9204 Received: 17 March 2018; Accepted: 20 April 2018; Published: 20 April 2018   Abstract: The transmission of water waves under vertical thin plates, e.g., offshore floating breakwaters, oscillating water column wave energy converters, and so on, is a crucial feature that dominates the hydrodynamic performance of marine devices. In this paper, the analytical solution to the transmission of water waves under multiple 2D vertical thin plates is firstly derived based on the linear potential theory. The influences of relevant parameters on the wave transmission are discussed, which include the number of plates, the draft of plates, the distance between plates and the water depth. The analytical results suggest that the transmission of progressive waves gradually weakens with the growth of the number and draft of plates, and under the conditions of given number and draft of plates, the distribution of plates has significant influence on the transmission of progressive waves. The results of this paper contribute to the understanding of the transmission of water waves under multiple vertical thin plates, as well as the suggestion on optimal design of complex marine devices, such as a floating breakwater with multiple plates. Keywords: water waves; vertical thin plates; transmission; reflectivity; resonance 1. Introduction There are numerous marine devices that contain vertical thin plates, such as offshore floating breakwaters (OFBs) and oscillating water column wave energy converters (OWC-WECs). It is of great significance to evaluate the transmission of water waves under these vertical thin plates, which dominate the hydrodynamic performance of marine devices. For a long time the research mainly focused on the transmission problem of water waves under one and two vertical plates. As for the wave passing through one thin plate, it is relatively simple and the transmission features are almost clear. As the wavelength increases, more wave energy passes through the plate, while less wave energy is reflected. Assuming the flow to be inviscid and irrotational, Wiegel [1] calculated the transmission coefficient by ignoring the influence of the reflected wave on the flow field, which resulted in a significant error as compared with the experimental results. In order to correct Wiegel’s method, Kriebel and Bollmann [2], based on similar assumptions, employed the analytical method to obtain corrected results with the consideration of the effects of reflected waves. However, their method neglects the disturbance of the wave near the thin plate and, thus, there still exists a deviation in the results. In contrast, Losada et al. [3] proposed a more rigorous mathematical model, in which the control equation is the Laplace equation and all boundary conditions on board and under board are satisfied. Porter and Evans [4] used Galerkin’s method to obtain the same solution as Losada et al. [3]. Water 2018, 10, 517; doi:10.3390/w10040517 www.mdpi.com/journal/water