2.4 Linear Decomposition of Irregular Waves Purposes of Wave Decomposition: 1)Calculating one resultant wave property based on the records of different types of resultant wave properties, statistically or deterministically. such as wave kinematics based on wave elevation, or wave elevation based on pressure; 2) Calculating the same property at locations other than that of the record, it is necessary to solve the inverse problem. Steps how to decompose an irregular wave field deterministically based on the measurements recorded at fixed points in the context of linear wave theory.

2.4.1 Uni-directional or Long-Crested Irregular Waves The amplitudes and initial phases of individual free (or linear) waves are related to the measurement of a resultant wave property through FFT and inverse FFT.

where A elevation amplitude, initial phase, frequency f(m)-----time series of a measured wave property, M total data points of f(m) used in FFT, linear transfer function relating elevation amplitude to the amplitude of the recorded wave property, the corresponding phase delay.

Table 2.1 Transfer Functions & Phase Delay for Various Wave Properties.

2.4.2 Directional or Short-Crested Irregular Waves No sufficient data for 2-D FFT (such as image data process). The wave sensors used in Lab & Field Measurements are Range from 3 – 20. Covariance When, m=f it reduces to auto-covariance function. The Fourier Transform of an auto-covariance/covariance is known as the power/cross spectrum.

where F m (n) the FFT coefficient of f m (n) and ‘*’ the complex conjugate. The integer number ‘n’ is related to frequency. A power spectrum is real. A cross spectrum is in general complex, the real part-- Cospectrum the imaginary part----Quadrature spectrum. A cross spectrum between measurements (m and l) is related to the corresponding wave elevation cross spectrum through,

A cross spectrum is related to a wavenumber-frequency spectrum through, Based on LWT, k is related to. Hence, is reduced to directional wave spectrum,. A cross spectrum is related to the corresponding directional wave spectrum through Given measurements, power and cross spectra among them at each discrete frequency, can be calculated. Various methods developed for resolving directional spreading or deriving a directional wave spectrum are used to solve Equation (2.29).

I. Longuit-Higgins (1963) Method.

II. Data Adaptive Methods Maximum Likelihood Method (MLM), (Isobe et al. 1984) Maximum Entropy Method (MEM), Bayesian Method.

2.4.3 Determining The Initial Phases of Directional Waves Reading Assignment 1.Initial Phases of free (linear) waves cannot be retained in the analysis of using cross spectra. 2.Many free waves are used in MLM. 3.Reducing the number of free waves (2J+1), still resemble the wave spreading, wave energy and main direction. How many directional free waves used at each discrete frequency depends on a spreading factor.

4. Using least square fitting to determining the initial phases of free waves.