Directional Waves in the Nearshore Coastal Region of Perth ...

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Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanThe following formula shows the general relationship between the cross power spectrum for apair of wave properties and the wavenumber frequency spectrum (Hashimoto 1995):*(#) H ( k ,#) H ( k,#) exp[ " ik( x x )] S( k #)$mn= ! mnn"m,kdkEquation 2-8where " mn(!) is the cross-power spectrum between the m- and n-th wave properties,! the angular frequency,k is the wavenumber vector,m( k,! )H is the transfer function from the water surface elevation to the m-th waveproperty,i is the imaginary constant,xmandproperty,xnthe respective location vector (x,y) of the probe for the m- and n-th waveS ( k,! ) the wavenumber-frequency spectrum* is the conjugate complex.The wavenumber k is related to the frequency f by the dispersion relationship given inEquation 2-1. Through the use of this relationship, the wavenumber-frequency spectrumS ( k,! ) can be expressed as a function of f and ! (Hashimoto 1995):#2$*( f ) H ( f ,") H ( f ,!)[ cos{ k( x cos!+ y sin!)} % i sin{ k( x cos!+ y sin!)}] S( f ! ) d!= &mn mnmnmnmnmn,0Equation 2-9wherex !mn= xnxm, ymn= yn! ymWhen S ( k,! ) and ( f ,! )for " mn(!) and ( f )1995).mnS are non-negative functions and satisfy the fundamental equations! functions, they are termed as the directional spectrum (Hashimoto2.3.3 Methods for directional wave analysisPopular conventional methods for directional wave analysis include the direct Fouriertransformation method (DFTM), parametric method, maximum likelihood method (MLM)and extended maximum likelihood method (EMLM), details of which can be found inLiterature Review 15
Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanHorikawa (1988). However, there are inherent drawbacks to these methods as they sometimesestimate erroneous directional spectra due to inaccuracies (e.g. inaccuracy of the cross-powerspectra). Hence, the maximum entropy principle method (MEP), Bayesian directionalspectrum estimation method (BDM) and extended maximum entropy principle method(EMEP) were developed to mitigate the drawbacks of conventional methods, and have sinceprovided a powerful means for reliably estimating an accurate directional wave spectrum(Hashimoto 1995).The MEP method derives the directional wave spectrum using location-dependent threequantitymeasurements related to random wave motion (eg, pitch-roll-and-heave data). Itprovides the directional estimates with a higher resolution than that obtained by theconventional methods, however it is not a general method because it can only utilise a threequantitymeasurement and does not consider errors in the cross-power spectra (Hashimoto1995). The BDM is a general method because it can utilise multi-quantity measurementsobtained from mixed instrument arrays and provide the highest resolution whilst providing areliable and robust method for estimating the directional spectrum and cross-power spectracontaminated by errors. The main drawback of the BDM is that it utilises a time-consumingiterative computation and cannot always be used for three-quantity measurements (Hashimoto1995). The EMEP method contains the benefits of the BDM and can use three-quantitymeasurements and yield equivalent results in a shorter time (Hashimoto 1995). Moreinformation on the EMEP method is provided in Appendix A.16 Literature Review
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