A wide-spacing approximation model for the reflection and transmission of water waves over an array of vertical obstacles

Abstract

Abstract

Modelling the far-field effect of drag-induced dissipation in wave–structure interaction: a numerical and experimental study

“…In general, the four complex amplitude coefficients η− → , η− ← , η+ → and η+ ← are related to R and T as follows: However, as already documented by Mérigaud et al (2021), the phase of R from (B5) is highly sensitive to the exact distances between the probes and the barrier, while that of T is highly sensitive to the exact value of the wavelength (which, in experiments, may slightly deviate from the theoretical dispersion relation, due to the presence of a mean surface current, for example). To represent the two coefficients in the complex plane, we thus rather use the fact that R + T = 1 (Mérigaud et al 2021), to determine R and T based solely on the magnitude of (B5), which is a procedure almost insensitive to the probe location and wavelength. More specifically, the law of cosines is used for the triangle of side lengths 1, | R| exp and | T| exp , as illustrated in figure 13.…”

“…To do so, we adopt the geometric representation of reflection and transmission coefficients proposed by Mérigaud, Thiria & Godoy-Diana (2021) and further generalised and detailed by Mérigaud et al (2023). Since the barrier is infinitely thin, it can be shown (Mérigaud et al 2021) that R + T = 1.…”

“…(3.1) Thus, it suffices to represent T to also represent R, which can be read as 1 − T. Note that, for other shapes of obstacles, one would still have the property | R + T| = 1 (Mérigaud et al 2023). Without dissipation, the incident wave energy is preserved in transmitted and reflected waves, which translates mathematically as | R| 2 + | T| 2 = 1.…”

“…When dissipation is introduced, as explained in § 2.2, or as will be the case with physical experiments, the sum of transmitted and reflected energy coefficients decreases below unity, and the transmission coefficient location thus moves towards the interior of the circle, as in cases 1 , 2 and 3 of figure 3. More precisely, the dissipated power P d , relative to the incident wave power P wave , can be read through the distance r from T to the centre (1/2, 0), as P d /P wave = 2(1/4 − r 2 ) (Mérigaud et al 2023).…”

“…In such a set-up, the far-field representation reduces to incident, reflected and transmitted waves, which greatly simplifies the analysis. In particular, we shall make use of the geometrical framework proposed by Mérigaud, Thiria & Godoy-Diana (2023), which consists of representing the transmission coefficient in the complex plane, and allows for a straightforward visualisation of the energy budget between reflected waves, transmitted waves and power dissipation near the structure.…”