Separation of regular waves by a transfer function method

Abstract

This paper presents a transfer function method (TFM) which can separate a regular wave field into incident and reflected waves based on the linear wave theory. The TFM uses specific transfer functions and corresponding convolution integrals to separate time series data measured in a combined partial standing wave system into incident and reflected waves. After this separation, estimation of the reflection coefficient becomes very easy. All manipulations have been performed in time domain. Furthermore, this method does not involve the calculation of wave heights and/or phase differences. The present method is demonstrated through numerical sample and physical model experiments carried out in a wave flume. Compared with other methods, the TFM gives much better estimates of the incident wave heights for physical model experiments in this study.

Introduction

In experimental studies on physical models of coastal structures, the general case is that the incident waves are generated by a wavemaker in a wave flume, propagate towards a physical model and impinge on it. Part of the incident wave energy is dissipated by the structure while the rest reflected. Incident and reflected waves form a partial standing wave system in the flume. In order to explore the reflection performance of a physical model, estimation of wave reflection is often needed. However, we have no direct means to measure the reflected waves. The earliest method of reflection measurement in a wave flume is to traverse a wave gauge along the wave propagation to measure the maximum and minimum wave heights H_max and H_min. The incident wave height H_I and the reflection coefficient K_R can be obtained as H_I=(H_max+H_min)/2 and K_R=(H_max−H_min)/(H_max+H_min). Unfortunately, it is very difficult to keep the measured data reliable while traversing a wave gauge for a wide range. Additionally, this method is time-consuming and subject to human errors. Hence, it has little use in extensive experimental studies. In order to overcome the shortcomings, several methods that use fixed wave gauges have been developed.

The first method was presented by Thornton and Calhoun (1972) and by Goda and Suzuki (1976). In their method, two fixed wave gauges at different locations are used to obtain wave signals. Two wave heights corresponding to the wave gauges and the phase difference between them are needed. The incident and reflected wave heights can be estimated as

$$\text{H}{}_{\mathrm{<mtext>I</mtext>}}\text{=}\text{H}{}^2{}_1\text{+H}{}^2{}_2\text{−2H}{}_1\text{H}{}_2\text{cos}\text{(Δ+δ)}\text{/|}\text{sin}\text{Δ|}$$

$$\text{H}{}_{\mathrm{<mtext>R</mtext>}}\text{=}\text{H}{}^2{}_1\text{+H}{}^2{}_2\text{−2H}{}_1\text{H}{}_2\text{cos}\text{(Δ−δ)}\text{/|}\text{sin}\text{Δ|}$$

where H_j=1,2=wave height of local wave signals recorded by gauge Gj, Δ=non-dimensional distance between wave gauges G1 and G2 and δ=phase difference between two wave signals.

The second method was developed by Mansard and Funke (1980). Three fixed wave gauges at different locations are used to obtain wave signals. The incident and reflected wave heights are estimated based on a least-square technique

$$\text{H}{}_{\mathrm{<mtext>I</mtext>}}\text{=|S}{}_2\text{S}{}_3\text{−3S}{}_4\text{|/|S}{}_5\text{|}$$

$$\text{H}{}_{\mathrm{<mtext>R</mtext>}}\text{=|S}{}_1\text{S}{}_4\text{−3S}{}_3\text{|/|S}{}_5\text{|}$$

where

H_j=1,2,3=wave height of local wave signals recorded by wave gauges Gj, Δ_j=1,2,3=non-dimensional distance between wave gauges G1 and Gj and δ_j=1,2,3=phase difference between gauges G1 and Gj.

The third method was presented by Isaacson (1991). It also uses three fixed wave gauges to measure wave signals. Only three wave heights corresponding to the wave gauges are needed. The incident and reflected wave heights are estimated as

$$\text{H}{}_{\mathrm{<mtext>I</mtext>}}\text{=(}\text{P+Q}\text{+}\text{P−Q}\text{)/2}$$

$$\text{H}{}_{\mathrm{<mtext>R</mtext>}}\text{=(}\text{P+Q}\text{−}\text{P−Q}\text{)/2}$$

where

$$\text{P=}\text{H}{}^2{}_1\text{sin}\text{[2(Δ}{}_3\text{−Δ}{}_2\text{)]−H}{}^2{}_2\text{sin}\text{(2Δ}{}_3\text{)+H}{}^2{}_3\text{sin}\text{(2Δ}{}_2\text{)}\text{sin}\text{[2(Δ}{}_3\text{−Δ}{}_2\text{)]−}\text{sin}\text{(2Δ}{}_3\text{)+}\text{sin}\text{(2Δ}{}_2\text{)}$$

$$\text{Q=}\text{1}\text{2}\text{H}{}^2{}_1\text{+H}{}^2{}_3\text{−2P}\text{cos}\text{(Δ}{}_3\text{)}{}^2\text{+}\text{H}{}^2{}_1\text{−H}{}^2{}_3\text{sin}\text{(Δ}{}_3\text{)}{}^2$$

H_j=1,2,3=wave height of local wave signals recorded by wave gauge Gj and Δ_j=1,2,3=non-dimensional distance between gauges G1 and Gj.

These methods have been widely adopted to estimate wave reflection from the experimental data obtained in the laboratory wave flume. Recently, brief comparison and concise comment on the aforementioned three methods were made by Nallayarasu et al. (1995). In all these methods, a hard nut has to be cracked, that is, how to obtain the precise wave heights and/or phase difference(s) from measured data that are not strictly sinusoidal signals.

In this paper, a new method is developed to separate the composite field of regular waves into incident and reflected waves without calculating wave heights and phase difference. The present method introduces specific transfer functions to perform convolution integrals with wave signals and give the separation of incident and reflected waves. By implementing a series of mathematical manipulations to composite waves, two sets of transfer functions are drawn. One is used to separate incident waves, while the other is used to separate reflected waves. Using these transfer functions, we obtain corresponding impulse response functions, perform the convolution integrals with measured signals, and derive complex expressions of incident and reflected waves. The reflection coefficient is obtained from the separated waves. Numerical sample and physical model experiments are used to demonstrate the validity of the present method. Comparison also has been made with the aforementioned three methods. The present method gives much better estimates of incident wave heights for physical model experiments.