Chapter 8 Decomposition & Superposition of Irregular Waves 8.1 Introduction 8.2 Decomposition 8.3 Superposition 8.4 Uni-Directional Hybrid Wave Model (HWM) 8.5 Directional Hybrid Wave Model (HWM) 8.6 Numerical verification of HWMs

8.1 Introduction Most studies focus on nonlinear wave evolution if the information of FWs in an irregular wave field is given. The resultant wave characteristics = those of free waves + their nonlinear interaction (bound waves). It is straight-forward, if one knows how to compute bound waves. Practical problems: measured wave records ( wave elevation, dynamic pressure and kinematics) do not directly provide the information of FWs, instead of the resultant of all free waves and their interactions (BWs). Insufficient studies on nonlinear wave decomposition. Nonlinear wave decomposition does not have direct solution

Linear Decomposition (widely used in offshore engineering practice): Fast Fourier Transform (FFT) is used under an assumption that a wave field is a linear superposition of free waves, which is known as linear wave theory or linear spectral methods. Owing to the assumption of linear wave theory, BWs are treated or approximated as FWs in using FFT to decompose a wave field. When waves are steep, the approximation may result in large errors. (examples to be studied) Additional difficulty (or errors) in decompose directional irregular waves: Limited number of wave measurements. Data adaptive methods (MLM, MEM etc) are hence used.

8.2 Decomposition Wave measurement records: resultant wave = all FWs + their interactions (BWs). Before decomposing a wave field into its FWs, we need to calculate BWs and then to decouple or subtract them from the measurements. In order to compute BWs, we must know FWs. Hence, the decomposition of a wave field is achieved through iterative processes. (see the flow chart) For convergence of iterations, perturbation methods for computing BWs must yield a unique and convergent solution.

Figure 8.1: Flowchart of decomposition

When ocean waves are steep and of a broad-banded spectrum, significant interactions may occur between FWs of either close frequencies or quite different frequencies. To ensure the solutions for BWs are convergent, two complementary perturbation methods, namely MCM & PMM, are employed. Because the computation of BWs uses two different perturbation methods, the related wave models later are named as Hybrid Wave Models (HWM). The solutions are truncated. The BWs whose magnitudes are of order higher than the truncated order are neglected and thus result in errors of HWM.

A HWM truncated at higher order is more accurate, but its formulation and the following numerical implementation are much more complicated and intensive in computation. Wave records inevitably involve measurement noises or errors and the perturbation methods are based upon potential wave theory that neglects the viscosity & the effects of wind. Hence, whether or not a high-order HWM should be developed and used depends on required accuracy (your mission objective) and quality of the measurements. For most applications, a HWM truncated at 2 nd or 3 rd order seems sufficient. In the following description, HWMs is truncated at 2 nd order and the water depth is assumed to be uniform and intermediate with respect to LWs.

Extending General Solution to A Wave field of Multiple Free Waves MCM solution (Intermediate depth for LW)

PMM solution

Band Division in Wave Frequency Domain MCM solution is used for computing BWs resulting from the interaction between FWs of relatively close wavelengths. PMM solution is used for BWs resulting from the interaction between FWs of quite different wavelengths. To apply the two methods respectively to an irregular wave train of a broad-banded spectrum, FWs of relatively close frequencies are bundled into bands in the frequency domain. Depending on the broadness of a spectrum, it can be divided into six or more bands in the frequency domain. Bands 0 – N. Sketch of band division in Figure 8.2.

Figure 8.2: Band division of a typical wave spectrum

Ocean waves of a single-peak spectrum (6 bands, N=5). In Bands 0 & 5: Energy of FWs is insignificant. FWs in Band 0: their interaction with other FWs is neglected. In Bands 1- 4: Energy of FWs is significant. Hence, we only consider interactions among FWs in Bands 1 to 4. Band 1 is chosen to start at the lowest freq. where a FW has significant amplitude. (percentage of the amplitude at spectral peak) The cut-off freq. is set at the end of Band 5, the cut-off freq. of FWs is at the end of Band 4. Band 4 is chosen to end at the highest freq. where a FW has significant amplitude. Wave energy in Band 5 is assumed due solely to BWs and used as a constrain condition for the solution of free waves in Band 4.

Freq.s of any pair of FWs located in the same or neighboring bands are relatively close and so are their wavelengths. The interaction between them is described by the MCM solution. Freq.s of any pair of FWs located in two different bands separated by at least one band between them are quite different. Their interaction is described by the PMM solution. The bandwidth of an individual band (Band 1 to 4) is limited by a constraint that the equivalent maximum steepness of all FWs in this band is much smaller than unity.

Subtraction of Bound Waves from Resultant Wave Spectra Modulation of a SW by a LW is of 2 nd order while the influence of the SW on the LW is at most of 3 rd order. Hence, the subtraction of BWs from the measured wave properties is conducted in the order from low- to high-freq. bands. 1)BWs resulting from LW bands 1 & 2 (Bands 1 & 2) 2)Interaction between LW band 1 and SW band 1 (Band 3) 3)BWs resulting from LW band 2 and SW band 1. 4)Interaction between LW bands 1&2 and SW band 2. 5)Bws resulting from SW bands 1 and 2.

8.3 Superposition or Prediction 1)Knowing the information of free waves: Uni-directional waves: Amplitude, initial phase as a function of freq. Band Division. (from wave decomposition) Directional waves: Amplitude, initial phase, direction angle as a function of freq. Band Division 2)Computing wave characteristics of free waves and their interaction using MCM or PMM solution depending on band division at a given location and time. 3)Superposition

8.4 Uni-Directional Hybrid Wave Model (HWM) A UHWM assumes: 1)irregular waves are uni-directional or long crested. 2)water depth, is uniform and of intermediate depth to LWs but deep with respect to SWs Band Division in the original UHWM (5 bands) Additional constraint is the wavelength of the first free wave in Band 2 (short- wave band 1)

Flow Chart Resultant Wave Property Free Wave Components Decouple Nonlinear Effects Decomposition Free Wave Components Resultant Wave Property Nonlinear Effects Prediction

Two major computational components: decomposition and superposition. The input to the decomposition is a time series of wave properties at a fixed point. In the original UHWM, the input is limited to wave elevation. The extension of the UHWM to allow for other wave characteristics as the input, say wave induced dynamic pressure, was made later by Meza et al. (1999). The duration of the time series, T, must be limited. 1) Average amplitudes of FWs are assumed to be stationary. In nature, the amplitude of FWs may change due to resonant wave- wave interactions, that is, the energy transfer among FWs. Resonant wave-wave interactions are weak interactions and their effects become significant after duration of hundreds of wave periods.

The longer T is, there are more FWs involved in the decomposition because of a smaller frequency increment, The CPU time will increase significantly with the increase in number of free waves in Bands 1 to 3. Hence, we usually choose T<50 Tp. Discontinuity caused by FFT (see Fig 8.3) Remedy: Using a (trapezoid) window function to avoid discontinuity at the start and end. If we need to analyze a very long wave record. Remedy: 1) Chop it into several segments; 2) Overlap between two neighboring segments; 3) window function applied to each segments

Fig 8.3: Periodicity of Discrete FFT causes discontinuity See Equ. (2.26) & (2.27) of Section 2.4 for why the FFT assumes the period T in a time series

Fig 8.4 a: Apply a trapezoid window function to avoid discontinuity. Fig 8.4 b: A long wave record can be divided into several segments and an overlap between the two neighboring segments is necessary to avoid discontinuity in prediction.

The output of the decomposition is: 1)FWs’ amplitudes and initial phases as a function of frequency; 2)information of band division and water depth. They are used as the input to the prediction part. Input to prediction also includes wave properties to be predicted at a given duration and location. The output of the prediction is resultant wave characteristics of measured irregular wave train, such as wave elevation, kinematics and pressure, as a function of time at any fixed points nearby the measured location. Limitation of HWM: In the vicinity of the measurement locations.

8.5 Directional Hybrid Wave Model (DHWM) Ocean waves are directional or short-crested. Hence, it is desirable to extend a HWM to allow for wave directionality. A DHWM is deterministic and considers both wave directionality and wave non-linearity up to 2 nd order. It is unique in 3 respects. 1) It subtracts BWs from the measurements and then decomposes a directional irregular wave field into free waves without the assumptions of random initial phases and a prior directional spreading function. 2) BWs are calculated using two complementary perturbation methods: MCM and PMM, which a common feature of HWMs.

3) It renders deterministic predictions of directional irregular waves. The predictions can be used to compare with other independent measurements deterministically. The currently DHWM uses a variety of wave characteristics, measured by pressure transducers, surface piercing wave gauges and velocimetry as input for the wave decomposition. Since all wave properties can be computed based on the potential of free waves, it can be in principle extended to allow for other measurements, such as wave slopes and accelerations as well.

The decomposition of a DHWM consists of 3 major steps: 1)wave direction estimation, 2)initial phase estimation, and 3)computation and subtraction of the BWs from measurements. While the third step is similar to a UHWM, the first two steps are unique to a DHWM because of limited number of wave records. 1) To achieve relatively fine resolution in wave direction using as few as 3 wave records, the estimate of wave spreading as a function of direction is based on data-adaptive methods (MLM, MEM etc). For deterministic decomposition, no smoothing or averaging is applied to amplitude-frequency spectra.

Knowing the directional spreading at each freq., a limited number of directional FWs are chosen at each freq. such that their amplitudes and directions conserve the total energy and approximately resemble the energy directional spreading. 2) The initial phases of the FWs are determined by minimizing the square of the differences between the measurements and the resultant of predicted free and BWs. The steps for determining wave directional spreading, the choice of limited number of free waves to represent the directional spreading and the fitting of initial phase of free waves are similar to those described in Section 2.4. We only elaborate the differences between nonlinear and linear decomposition of a directional wave field.

3) Once the initial phases, amplitudes and directions of FWs are computed, the nonlinear interactions between them can be calculated and then subtracted from the corresponding measurements. Similar to the decomposition of long-crested irregular waves, the decomposition of a measured directional irregular field is accomplished through iterative processes.

8.6 Numerical verification of HWMs Consistent Test: Using the prediction part of a HWM to generate a numerical wave train (given the information of free waves) The time series of the numerical wave train is used as an input to the decomposition part of HWM. The results will be the free waves. Comparing the results with the input to the prediction to see if they are consistent. Limitation Test: Testing the extreme cases.