2.4 Linear Decomposition of Irregular Waves

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2.4 Linear Decomposition of Irregular Waves

• Purposes of Wave Decomposition
• 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.

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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.
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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.
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Table 2.1 Transfer Functions Phase Delay for
Various Wave Properties.
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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 , mf it reduces to auto-covariance
function. The Fourier Transform of an
auto-covariance/covariance is known as the
power/cross spectrum.
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where Fm(n) ----- the FFT coefficient of
fm(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,
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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).
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I. Longuit-Higgins (1963) Method.
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II. Data Adaptive Methods Maximum Likelihood
Method (MLM), (Isobe et al. 1984) Maximum
Entropy Method (MEM), Bayesian Method.
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2.4.3 Determining The Initial Phases of
Directional Waves Reading Assignment

• Initial Phases of free (linear) waves cannot be
  retained
• in the analysis of using cross spectra.
• Many free waves are used in MLM.
• Reducing the number of free waves (2J1), 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.

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