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

Directional Waves in the NearshoreCoastal Region of Perth, WesternAustraliaHuey Jean TanSupervisor: Professor Charitha PattiaratchiThis thesis is submitted in partial fulfilment of the requirements for the degreeof Bachelor of Engineering at the University of Western Australia

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanABSTRACTThe conventional approach for wave analysis is the relatively simple monochromatic waveapproach, which is still widely used in coastal and offshore applications. This traditionalmethod does not consider the direction of wave propagation, which is an important factor toconsider in order to realistically characterise the natural surface waves of the ocean.Directional wave methods are more complex, time consuming and expensive than theconventional monochromatic wave approach. However, it has been proven that directionalwave analysis offers a more accurate representation of natural ocean waves, and that there aresignificant differences between the transformations of monochromatic and directional waves.Directional waves in the nearshore coastal region influence the direction of littoral drift on thebeach, which, in turn, affects the morphology on the beach. The beaches along the coastlineof Perth, Western Australia, exhibit seasonal variations in beach morphology. The main aimof the present study was to carry out an analysis of the directional waves in the nearshoreregion of the coastal waters of Perth. This was achieved by analysing the directional wavedata collected at Cables Artificial Surfing Reef (ASR) in August 1999 (i.e. winter) and at CityBeach during January and February 2001 (i.e. summer). More specifically, due to the fact thatwave growth and the direction of wave propagation are directly affected by wind, this studyinvestigated the wave response to the changing wind climate. The summer data (i.e. at CityBeach) indicated that the propagation of nearshore waves were predominantly in the northeastdirection, whereas the winter data (i.e. at Cables ASR) displayed directional waves in apredominant east-southeast direction. This is in agreement with the observations by Masselink& Pattiaratchi (2001) of the prevailing northward sediment transport during the summermonths and southward longshore transport during the winter months. Swell waves duringsummer is predominantly in the east-northeast direction, whereas the dominant swell directionin winter is east-southeast. The daily sea breeze events during summer generate sea waves inthe northeasterly direction. Sea waves generated by storm events change from southeasterly tonortheasterly with the passage of mid-latitude depressions, however the direction of swellwaves remain constant in the east-southeasterly direction. During the summer samplingperiod, the mean significant wave height (H s ) was 0.77 m, and the maximum H s achievedduring a sea breeze event was 1.33 m. For the winter sampling period, the mean H s was 1.17Abstracti

m, and a maximum H s of 2.75 m was achieved during a storm event. The results from thepresent study indicated that the wave response time to changes in wind is 3.5 to 4 hours.

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanACKNOWLEDGMENTSThere are many who I would like to acknowledge for their contribution in helping mecomplete this thesis project:To my supervisor, Professor Charitha Pattiaratchi, for your knowledge, advices, patience andencouragement throughout the duration of this project. Without your invaluable guidance, thisproject would not be possible.To my family, for your help, prayers and encouragements. Thanks especially to my parentsfor understanding my huge absence from the family home, and for consciously preparingnutritious food when I have meals at home.To my fiance, Robert Marlow, for your tremendous support and help, especially withcomputer stuff. Thank you, Rob, for your listening ear, for sticking by me even when I was atmy worst, and for all those times that you told me to "Go get 'em, tiger (*roar*)!" Youprovided me with an outlet from all this ‘thesis stuff’. I don't think I could have sanely made itto the end without your constant moral support, humour and cooking and cleaning.To my fellow final year friends in the Class of 2003: you have my gratitude for your words ofencouragement. Special thanks to Christina Young for your advice and for helping me outwith missed lectures.Thanks to my church Endeavour Christian Gathering and all my Christian buddies whoremembered and kept me in prayer.And last, but certainly not the least (not by a long shot), I thank You and praise You, my Godin heaven, for giving me life, blessings and your gift of eternal salvation.Acknowledgmentsiii

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanTABLE OF CONTENTSABSTRACTiACKNOWLEDGMENTSiii1. INTRODUCTION 12 LITERATURE REVIEW 32.1 BACKGROUND INFORMATION ON OCEAN WAVES 32.1.1 CLASSIFICATION OF PROGRESSIVE OCEAN SURFACE WAVES............................................. 42.1.2 FACTORS INFLUENCING NEARSHORE WAVES.................................................................... 52.2 THEORIES AND METHODS OF WAVE ANALYSIS 62.2.1 REGULAR WAVES ............................................................................................................ 62.2.2 IRREGULAR WAVES ......................................................................................................... 72.2.3 SIGNIFICANT WAVE HEIGHT AND PEAK PERIOD.............................................................. 112.2.4 SPECTRAL WAVE POWER ............................................................................................... 112.3 DIRECTIONAL WAVE SPECTRUM 132.3.1 MEASUREMENT METHODS .............................................................................................. 132.3.2 FUNDAMENTAL EQUATIONS FOR DIRECTIONAL SPECTRUM ESTIMATION............................ 142.3.3 METHODS FOR DIRECTIONAL WAVE ANALYSIS ................................................................ 152.4 ENVIRONMENTAL SETTING 172.4.1 SITE CHARACTERISTICS.................................................................................................. 172.4.2 CLIMATE ....................................................................................................................... 182.4.3 REGIMES OF SEA BREEZE AND STORM EVENTS IN PERTH ................................................. 202.4.4 OFFSHORE WAVE CONDITIONS....................................................................................... 233 APPROACH 253.1 DATA COLLECTION 253.1.1 MEASURING DEVICE ...................................................................................................... 253.1.2 DATA COLLECTION TECHNIQUE ..................................................................................... 253.2 DATA ANALYSIS 27ivTable of Contents

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan3.2.1 DIRECTIONAL ANALYSIS................................................................................................. 293.2.2 NON-DIRECTIONAL ANALYSIS......................................................................................... 293.2.3 WIND ANALYSIS............................................................................................................. 294 RESULTS AND DISCUSSION 304.1 INTRODUCTION 304.2 SEA BREEZE CYCLES 324.2.1 OVERALL PATTERN ........................................................................................................ 324.2.2 HOURLY TRENDS ........................................................................................................... 354.3 STORM EVENTS 414.3.1 OVERALL PATTERN ........................................................................................................ 414.3.2 HOURLY TRENDS............................................................................................................ 444.4 SWELL AND SEA COMPONENTS 504.4.1 CITY BEACH .................................................................................................................. 504.4.2 CABLES ASR ................................................................................................................. 524.5 WAVE RESPONSE TIME 544.6 WAVE REFLECTION 584.6.1 CITY BEACH .................................................................................................................. 584.6.2 CABLES ASR ................................................................................................................. 664.6.3 DISCUSSION OF THE REFLECTION COEFFICIENTS AT CITY BEACH AND CABLES ASR......... 735 CONCLUSIONS 746 REFERENCES 75APPENDICES 78APPENDIX A: “EMEP” FORMULATION 78APPENDIX B: MATLAB ® SCRIPTS 82Table of Contentsv

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanLIST OF FIGURESFigure 2.1: Distribution of the ocean surface wave energy (after Kinsman 1965) ................................... 3Figure 2.2: The wave parameters of a two-dimensional simple progressive wave propagating in thepositive x-direction (after USACE 2002) ........................................................................................... 7Figure 2.3: Wave parameters for an irregular and random sea state (USACE 2002) ............................... 8Figure 2.4: The surface elevation time series and spectrum of a regular wave (USACE 2002)............ 10Figure 2.5: The surface elevation time series and spectrum of an irregular wave (USACE 2002) ....... 11Figure 2.6: An example of a directional spectrum (USACE 2002) ......................................................... 14Figure 2.7: Locality map of study area (adapted from Masselink & Pattiaratchi 2001)......................... 17Figure 2.8: A typical summer sequence for Perth (from BOM 1993) ..................................................... 22Figure 2.9: Typical weather pattern for winter for Perth (from BOM 1993) .......................................... 23Figure 3.1: InterOcean S4DW Current Meter (from ISI 2004a) .............................................................. 25Figure 3.2: A wave spectral energy plot indicating the frequency cut-offs for the swell and seacomponents of the wave field............................................................................................................ 28Figure 4.1: Wind and wave conditions measured in the nearshore region of City Beach from 21January to 09 February 2001: ............................................................................................................ 34Figure 4.2: Typical wave sequence induced by the sea breeze cycle measured in the nearshore regionof City Beach on the 25 – 26 January 2001...................................................................................... 40Figure 4.3: Wind and wave conditions measured in the nearshore region of Cables Artificial SurfingReef from 11 August to 27 August 1999:......................................................................................... 43Figure 4.4: Typical wave pattern induced by a storm associated with the mid-latitude depression,measured in the nearshore region of Cables ASR on the 23 – 24 August 1999............................. 49Figure 4.5: Nearshore significant wave height and peak wave period and their respective swell and seacomponents at City Beach for the period 21 January to 09 February 2001 ................................... 51Figure 4.6: Nearshore significant wave height and peak wave period and their respective swell and seacomponents at Cables Artificial Surfing Reef for the period 11 August to 27 August 1999........ 53Figure 4.7: Hourly sequence of polar plots of wave energy in terms of wave frequency and direction ofwave travel measured in the nearshore region of City Beach on 31 January 2001........................ 57Figure 4.8: Wave information at 06:00 on 21 January 2001 at City Beach. ........................................... 61Figure 4.9: Wave information at 17:00 on 26 January 2001 at City Beach. ........................................... 63Figure 4.10: Wave information at 10:00 on 31 January 2001 at City Beach. ......................................... 65Figure 4.11: Wave information at 09:00 on 17 August 1999 at Cables ASR. ........................................ 68Figure 4.12: Wave information at 09:00 on 18 August 1999 at Cables ASR. ........................................ 70Figure 4.13: Wave information at 18:00 on 18 August 1999 at Cables ASR. ........................................ 72viList of Figures

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanLIST OF TABLESTable 3.1: Details of the S4DW current meter deployments.................................................................... 26Table 3.2: Wave parameters extracted from the data in the frequency domain ...................................... 28Table 4.1: Bi-hourly wind data for 31 January 2001 collected at Swanbourne. ..................................... 57List of Tablesvii

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan1. INTRODUCTIONOcean surface waves are the major factor in determining the geometry and composition ofbeaches and significantly influence the planning and design of harbours, waterways, shoreprotection measures, coastal structures and other coastal works. Surface waves derive theirenergy from wind, and a significant amount of this energy is dissipated in the nearshoreregion. The wave energy shapes the coastline, transports and sorts bottom sediments, andexerts forces on coastal structures; hence the information obtained from wave climate data isimportant for planning and design in coastal and offshore engineering. The local wave climateis also important in other fields, such as shipping, commercial fishing, coastal research andrecreation (Lemm 1996).The conventional method to analyse wave records is through the relatively easymonochromatic wave approach, however this simple method does not consider the directionof wave propagation. The analysis of directional wave is accomplished via the spectralmethod. The main drawback of utilising spectral analysis is the relatively complex and timeintensivecomputations involved, which is the main reason why engineers are reluctant toabandon the monochromatic wave approach. However, the uses of computer models andmultidirectional wave basins have indicated that there are significant differences between thetransformations of monochromatic and directional waves (Goda 1995), and that directionalwave data offer a more accurate representation of natural ocean waves.Directional data is often needed for the design of harbour and coastal structures, and for theanalysis of sediment transport and pollution control. The loading and performance of manymarine structures are directly affected by wave directions, and the consideration of directionaldistribution of wave energy can assist in obtaining an efficient structural design. Takingdirectional wave data into consideration usually results in reduced design wave loads andpossibly a lighter structure, which would reduce construction costs for the structure (Wiegel1981). According to Hashimoto (1995), it is apparent that “with the ever-increasing demandfor the utilisation of coastal areas… there is a need for directional wave data to ensureadequate though cost-effective safety margins be implemented during the design process ofcoastal structures.” (p. 140). The directional wave spectrum is now a standard tool ofIntroduction 1

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanengineering practice in the planning and designing of many harbour and coastal projects(Goda 1995).The aim of the present study is to carry out an analysis of the nearshore directional waves inthe Perth coastal region. Perth is the capital city of Western Australia, and is located on thesouth-west coast of the Australian continent. Beaches along the coastline of Perth exhibit anapparent seasonality in beach morphology, however not all beaches are characterised by thesame seasonal cycle (Masselink & Pattiaratchi 2001). Some of the Perth beaches becomewider during summer (i.e. accretion) and narrower during winter (i.e. erosion) whereas othersexperience summer erosion and winter accretion (Masselink & Pattiaratchi 2001). It isgenerally accepted that seasonal beach cycles are due to seasonal variations in the incidentwave energy conditions; however the study by Masselink & Pattiaratchi (2001) demonstratedthat the seasonal changes on most beaches along the Perth coasts are better explained by theseasonal reversal in the littoral drift direction. The direction of littoral drift depends largelyupon the direction of the waves incident on the beaches (i.e. nearshore waves), which is, inturn, affected by the direction of wind. During summer, northward sediment transport prevailsdue to sea breeze activity blowing in a predominantly south-southwest direction; howeverduring winter, the longshore sediment transport is towards the south due to southwardflowingcurrents generated by northwesterly storms (Masselink & Pattiaratchi 2001). Due to alack of nearshore directional wave data at the time of the study by Masselink & Pattiaratchi(2001), conclusions could only be drawn on the basis of the wind speed and direction. Themain objective of the analysis is to quantify the directional wave response to changes inmeteorological conditions (specifically storm and sea breeze events). The information fromthis study can then be applied to the findings by Masselink & Pattiaratchi (2001). This is thefirst time a study of nearshore directional waves is done for the Perth coastal region..2 Introduction

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2 LITERATURE REVIEW2.1 BACKGROUND INFORMATION ON OCEAN WAVESWind-generated waves are important as energy-transfer agents: the waves first obtain energyfrom wind and then transfer the energy across the expanse of the ocean to the coastal zone. Aspectrum of ocean surface waves proposed by Kinsman (1965) is depicted in Figure 2.1, andit suggests that wind-driven waves contribute the largest amount of the energy from the oceanto the beach and nearshore physical system. In particular, surface gravity ocean waves withperiods between 3 and 25 s are the major influence on the geometry of beaches, planning anddesign of coastal works (USACE 2002). Therefore, there is an importance to gain anunderstanding of these waves and the associated generated forces for the planning and designof coastal projects.This section provides a brief description of the theories and methods of surface ocean waveanalysis, and a general overview of the factors affecting waves in the nearshore region.Figure 2.1: Distribution of the ocean surface wave energy (after Kinsman 1965)Literature Review 3

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2.1.1 Classification of Progressive Ocean Surface WavesThe classification of progressive waves propagating on the ocean surface is usually based onwave period T or frequency f. The spectrum of ocean waves in Figure 2.1 shows the differenttypes of surface waves on the ocean and the importance of gravity wave energy. Gravitywaves can be classified into two types of surface waves: swell and sea.Swell WavesSwells are long-period waves that have propagated far from the region of generation and areno longer under the influence of significant wind action (Kinsman 1965). Swell waves aremore uniform in height, period and direction (i.e. monochromatic) with well-defined longcrests (i.e. individual wave crests are several wavelengths in extent) (USACE 2002). Swellwaves that have originated from a storm area appear to propagate in a predominant direction(i.e. unidirectional), and although swells are milder in nature than sea waves, it is the swellwaves that transfer energy across the ocean to the coastal zone (Komar 1998). Swells aremore persistent than sea waves and therefore are important for predicting sediment transport.The direction of these predominant swells can be related to the geometry of the shorelines(Silverster & Hsu 1993).Sea WavesThe term sea waves is used to describe waves that are still within the area of generation(USACE 2002). Sea waves are short crested (i.e. the length of individual wave crests mayonly be a wavelength or two in extent) and are highly complex and irregular due to the windsgenerating a whole spectrum of waves with a wide range of periods and heights (USACE2002, Komar 1998). The ocean surface under sea conditions appears “steeper, more ruggedand confused” than swell (Kinsman 1965 p. 22). As sea waves propagate away from the areaof generation, they develop into swell waves. Based on previous work on offshore wavestudies in the Perth area, Lemm (1996) arbitrarily defined sea waves as waves with periodsbelow 8 s and swell waves defined as waves with periods above 8 s.Wave characteristicsWaves of different periods can originate in the region of generation but in time, the variouswave components will separate from each other. The longer period waves will propagate4 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanfaster than shorter period waves, and will reach distant destinations first. The shorter periodcomponents may arrive at the same destination several days later (USACE 2002).In the area of wave generation, energy is transferred from the shorter period waves to thelonger period waves (USACE 2002). Waves can travel large distances without losing muchenergy, but energy can be dissipated internally within the fluid, by interaction with the air,and by friction with the bottom sediment. Due to the fact that short-period wave componentslose energy more easily than long-period components, the periods of swell waves tend to belonger than those of sea waves.2.1.2 Factors Influencing Nearshore WavesOcean surface waves undergo certain transformations as they approach the coastline. Thespeed of wave propagation decreases as water depth decreases, with those of longerwavelengths (such as swell waves) slowing down first. The water depth may vary at differentpoints along the length of the wave, hence different sections of the wave may travel atdifferent speeds (BOM 1993). Waves travelling through shallow water are strongly affectedby seabed bathymetry and currents, and a sloping or undulating sea bed can cause largechanges in wave height and direction of propagation. For instance, shoals can focus wavesand sometimes waves can double in height behind a shoal (USACE 2002). The magnitude ofwave changes depends on the wave period and direction and the distribution of wave energyin frequency and direction.The following points outline the factors affecting wave propagation from deep into shallowwater (USACE 2002):• Propagation effectsThese result from the convergence or divergence of waves caused by shape of bottomtopography, causing wave energy to focus or spread out. Propagation effects includerefraction, shoaling and diffraction. Diffraction is also caused by presence of coastalstructures that interrupt wave travel.• Sink mechanismWave energy is dissipated due to friction, percolation and breaking• Source mechanismThe addition of wave energy due to windLiterature Review 5

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan• Wave-current and wave-wave interactionsWave transformation analyses for nearshore waves are discussed in detail in USACE (2002)2.2 THEORIES AND METHODS OF WAVE ANALYSIS2.2.1 Regular WavesIn nature, ocean surface waves are three-dimensional, irregular and random in amplitude,period and direction (Horikawa 1988). The ocean surface typically varies in time and isunsteady, and therefore the ocean surface cannot be adequately described in its fullcomplexity. The simplest method to deal with irregular waves is to reduce them to arepresentative two-dimensional monochromatic wave of infinite wave crest, to enable theapplication of the vast knowledge available on periodic waves (Horikawa 1988). Wavetheories have been developed only as approximations to real waves based on variousassumptions. The most elementary wave theory is the linear first-order wave theory (alsocalled small-amplitude or Airy wave theory), and this theory is widely used in coastalengineering and design because it can be applied with ease whilst giving a reasonableapproximation of wave characteristics for a wide range of wave parameters (USACE 2002).The nonlinear, finite amplitude periodic wave theories include Stokes, Cnoidal and solitarywave theory. The details of these theories can be found in several texts such as Horikawa(1988), Kinsman (1965) and USACE (2002).A simple, sinusoidal progressive wave passing a fixed point in the ocean (as depicted inFigure 2.2) can be represented by the horizontal spatial coordinates x and time t. Thefollowing wave parameters are used to describe a simple sinusoidal oscillatory wave:x – horizontal spatial coordinatest – timeL – wavelength: horizontal distance between corresponding points on two successive wavesT – wave period: time interval between two successive crests at a given pointC – wave celerityω – angular or radian frequency," = 2!Tk – wave number,k = 2!Lθ – phase," = kx # ! tH – wave height: vertical distance to its crest from the preceding trough6 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanη – elevation of the water surface, " = cos( kx # ! t)h – water depthH2Figure 2.2: The wave parameters of a two-dimensional simple progressive wave propagating in thepositive x-direction (after USACE 2002)2.2.2 Irregular WavesThe ocean surface is often a combination of several wave components that were individuallygenerated by wind in different regions and have travelled to the observation point. Hence realwave systems would be irregular and random, and successive waves may have differingperiods and heights (Figure 2.3). Therefore, statistical and probabilistic methods have to beutilised to describe the natural time-dependent three-dimensional characteristics of real wavesystems (USACE 2002). There are two methods for treating irregular waves: wave-by-waveanalysis (time domain) and spectral analysis (frequency domain)Literature Review 7

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 2.3: Wave parameters for an irregular and random sea state (USACE 2002)Wave-by-wave AnalysisThe wave-by-wave (or wave train) analysis determines wave properties by obtaining theaverage statistical quantities of heights and periods of individual wave components (USACE2002). The undulations of the water surface from the mean water level are manually identifiedas individual wave using the zero-downcrossing method to obtain representative waveparameters, and using this information, statistical characteristics of the wave record can beestimated and statistics of the record are compiled (USACE 2002). Wave height is the verticaldistance between the highest and lowest points between two successive zero-downcrossing(i.e. crossing the zero line on a negative gradient) points; wave period is horizontal distance(see Figure 2.3). All the local maxima and minima not crossing the zero-line is discarded.This time-domain method requires direct measurements of sea surface, and the wave recordsmust be sufficiently large to obtain reliable statistics. The primary disadvantage to the waveby-waveanalysis is that it does not give an indication about the direction of the waves;therefore a single wave at a point may actually be the local superposition of two smallerintersecting waves from different directions (USACE 2002).Spectral AnalysisSpectral analysis utilises Fourier theory to transform the irregular ocean surface into asummation of simple sine waves, and wave characteristics are defined in terms of itsspectrum. The wave spectrum denotes which frequencies have significant energy content. Thesurface elevation time series in both Figure 2.4 and Figure 2.5compares the spectra of regularand irregular waves. In contrast to the wave-by-wave analysis which attempts to defineindividual waves, the spectral method attempts to describe the distribution of the surfacevariation from the mean level with respect to the frequency of the signal (USACE 2002).8 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanSubsequently, information on individual waves is lost and because spectral analysis is acomplex, linear approach, the representation of nonlinear waves may be distorted (USACE2002). The distribution of the variance is defined with frequency as S(f), assuming that thefunction is continuous in frequency space (i.e. statistically smooth the discrete frequencies toestimate continuum). The function E(f) is often called the frequency energy spectrum, and isdefined as! "0( f )E = # g S df Equation 2-1where ρ is the seawater density and g is the gravitational acceleration (Tucker and Pitt 2001).The spectral approach also allows for the analysis of the variability of waves with respect toperiod and the direction of travel. The directional energy spectrum ( f ,! )E describes howvariance is distributed in frequency f and direction θ. More information on directional wavespectrum is provided in Section 2.3. Spectral wave analysis using Fourier transform isdiscussed in greater detail in Tucker & Pitt (2001), Kinsman (1965) and Horikawa (1988).The following points outline some of the benefits of applying the spectral analysis method tosurface wave records:• The approach is easily implemented on microchip and packaged with gauginginstrument• The spectral theory is the basis for principal successful theories for describing wavegeneration by wind and modelling evolution of natural sea states in coastal regions• It is currently the only widely used approach for measuring wave direction• Fourier/spectral analysis of waves has large technical literature and statistical basisthat can be readily drawn upon.Literature Review 9

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 2.4: The surface elevation time series and spectrum of a regular wave (USACE 2002)10 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 2.5: The surface elevation time series and spectrum of an irregular wave (USACE 2002)2.2.3 Significant Wave Height and Peak PeriodThe concept of significant wave heightH or H1 3is a very useful and important index toscharacterise the heights of the surface waves on the sea. It is defined as the mean of the onethirdhighest waves in a sample record (Kinsman 1965), or in terms of the total energy of thewave system (Tucker and Pitt 2001):H= Hm0 4 m 0Equation 2-2s=where m 0 is the spectral moment or variance (‘energy’) of sea surface elevation given bym ! " 0= S( f )df Equation 2-30It is conventional to denote significant wave height H s as H 1/3 when it is measured directly viathe wave-by-wave analysis and H m0 when it has been estimated from the variance of therecord or the integral of the variance in the spectrum. Both are good estimates of H s and aregenerally very close in value, and it is typical forH to exceed H1 3by 5-10% (Lemm 1996).The mean period of these ‘significant’ waves is termed as the significant wave periodAccording to Kinsman (1965), visual observations of wave heights and periods are goodapproximations ofHsand Ts.sTs.Peak period T p is the reciprocal of the frequency at which the peak of the spectrum occurs(Tucker & Pitt 2001). T p is associated with the largest wave energy and is only obtainablethrough spectral analysis. An example of peak period for natural ocean waves is shown inFigure 2.5b. The spectral representation of swell waves will display a single value of peakperiod and wave energy decays at frequencies on either side. The spectra of storm waves ismulti-peaked, with one peak (although not always the highest) corresponding to swelloccurring at lower frequencies. One or more peaks are associated with storm waves occurringat higher frequencies.2.2.4 Spectral Wave PowerWave power (or wave energy flux) is defined as the rate of energy transmission in thedirection of wave travel across a vertical plane that is perpendicular to the direction of waveLiterature Review 11

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanpropagation and extending down the entire depth (USACE 2002). For a unidirectional wavespectrum, the power transported per metre of the crest length can be derived using spectralenergy defined in Equation 2-5 (Tucker and Pitt 2001):!( f ) S( f )P = ECn = " g C df Equation 2-4gwhere C g is the group velocity, which is defined as the rate of transmission of waveenergy:C g= CnE is termed the specific energy or energy density and is the total average wave energyper unit surface area:E =! gH82shallow water.n = 0.5 in deep water, increases in value in the transition zone to become n = 1 in12 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2.3 DIRECTIONAL WAVE SPECTRUMThe conventional wave analyses based on the monochromatic (i.e. regular) wave approachtend to yield significant error in estimates of wave transformations and structural responses(Goda 1995). The directional spectrum is a more realistic representation of sea waves, and canproduce a more accurate solution than that obtained via conventional methods. Thedirectional wave spectrum provides a probabilistic view of sea waves, and it expresses thedistribution of wave energy as a function of wave frequency and wave propagation direction(see Figure 2.6). The directional spectral characteristics of ocean surface waves stronglyinfluence the wave transformations due to diffraction, refraction and reflection, therefore theprecise estimation of the directional spectral characteristics is an important step in the initialplanning and design stages for the construction of coastal and offshore structures (Hashimoto1995).2.3.1 Measurement methodsThe measurement of a directional spectrum usually involves obtaining the measurement ofeither the same hydrodynamic parameter (e.g. surface elevation or pressure) at a series ofnearby locations, or different parameters (e.g. pressure and two components of horizontalvelocity) at the same point. These records are then cross-correlated through a cross-spectralanalysis, and a directional spectrum is obtained (USACE 2002). There are two maincategories of measurement techniques for estimating directional spectra (Horikawa 1988):• Instrument-based direct measurement methods: these include wave gauge arrays(pressure or velocity gauges arranged in a variety of shapes), directional buoy system(pitch-roll-and-heave or heave-and-tilt methods) and multi-component current meter• Remote sensing methods: examples include microwave and optical techniquesA more detailed discussion on the methods of directional measurement can be found inTucker & Pitt (2001).Literature Review 13

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 2.6: An example of a directional spectrum (USACE 2002)2.3.2 Fundamental equations for directional spectrum estimationUsing the wave parameters amplitude A, frequency f and wave propagation direction θ , thedirectional wave spectrum is defined asf + df"+ d"! !f"( f ,! )1 A 2 i= S( f ," ) dfd"2Equation 2-5S is the directional wave spectral density function, which is often expressed as theproduct of the frequency spectrum S ( f ) and the directional spreading function G (!| f )(Goda 1995):( f , ) S( f ) G( ! f )S ! = | Equation 2-6The function G ( | f )2#0("| f ) d = 1! has no dimension and is normalised such that! G "Equation 2-714 Literature Review

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

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2.4 ENVIRONMENTAL SETTING2.4.1 Site CharacteristicsFigure 2.7: Locality map of study area (adapted from Masselink & Pattiaratchi 2001).The locations indicated by the letter and number ‘S4’ are the deployment sites of the InterOcean S4current meters utilised in this investigation. More information in Section 3.1.Literature Review 17

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanThe waves travelling into the shallower waters of the Perth metropolitan area (Figure 2.7) aremodified by a number of processes, including refraction due to changes in bathymetry anddiffraction due to exposed reefs and islands. The continental shelf along the Perth coastalregion generally has a mild slope, but the actual bathymetry may be rather complex due to thepresence of submerged limestone ridges (Masselink & Pattiaratchi 2001). The offshore ridges,shallow reef chains and the islands (Rottnest and Garden Islands) have significant shelteringeffects on the inshore wave field from the offshore wave conditions. In areas of themetropolitan coastal waters, a northwesterly wind will generate higher seas than asouthwesterly of similar strength because of the sheltering effect of the islands in thesouthwest direction (BOM 1993).2.4.2 ClimateThe ‘winter-wet south-west’ of Western Australia experiences a Mediterranean-type climatewhich is characterised by hot, dry summers and cool, wet winters (Gentilli 1972). The majorinfluence on the weather pattern along the south-west coast is the position of the ‘subtropicalridge axis’, which is a belt of semi-permanent anticyclonic high pressure systems encirclingthe southern hemisphere between latitudes of 25 and 40 degrees south (BOM 1993).According to Linacre (1977), Perth is Australia’s windiest city, being exposed to westerlygales in winter and to strong sea breezes in summer. The average annual wind speed for Perthis 4.3 ms -1 , which is approximately 30% above those of other capitals. Perth also experiencesa maximum gust of 43ms -1 , far exceeding those of most other capital cities of Australia(Linacre 1977). There are distinct variations in both the seasonal and diurnal wind patterns.The summer wind pattern is characterised by the daily sea breeze cycle, and the winterweather is dominated by the passage of low pressure systems crossing the south-westernregion of Australia every 7 to 10 days.Subtropical Ridge Axis and the Anticyclonic High Pressure SystemDuring summer, the subtropical belt of high pressure is over most of the south-west region,reaching near 40 o S and preventing most cold fronts from approaching the south coast withany strength (BOM 1993, Gentilli 1972). During winter, the belt is displaced to the north ataround 25 o S to 30 o S, allowing cold fronts and strong westerly winds to frequently penetratethe southern half of WA in some strength (BOM 1993).18 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanDue to the normal breakdown of the high pressure belt into anticyclonic cells, the prevailingair circulation over the southwest region is anticlockwise. Along the north side of theanticyclones, the winds gradually become easterlies (or trade winds) and along the southedge, variable winds come in from the west (Gentilli 1971). Therefore, the seasonal shift ofthe high pressure belt accounts for a dominant onshore (i.e. westerly) component in winterand an offshore (easterly) component in summer along the south-west coast of Australia(Gentilli 1972). The anticyclonic high pressure cells progress eastward with a periodicity offive to six days (Gentilli 1971).Mid-latitude depressionsThe anticyclonic system is periodically disrupted by storm events associated with mid-latitudedepressions. During summer, mid-latitude depressions are usually located far too south ofAustralia to have any direct influence on the local climate. However, during the wintermonths, when the high pressure belt is displaced towards the north, the low pressure cellsassociated with the mid-latitude depressions travel northward. According to Gentilli (1971),the storms associated with mid-latitude depressions occur most frequently in July with anaverage of three cyclonic centres skirting the south-western coast of WA.Tropical cyclonesTropical cyclones are intense low pressure systems that form over the tropical waters ofnorthwest of WA, and the paths of these storms greatly influence the coastal wind conditions.The tropical cyclone season occur between November and May with most cyclonic activityoccurring in January or February (Rose 2001). In most cases, if the cyclone travels to thesouth of Australia, it loses most of its characteristics and becomes indistinguishable from anyfrontal cyclonic depression. However, a few cyclones have reached the southern shore ofAustralia without any apparent loss of intensity (Gentilli 1971), and it has been estimated thattropical cyclones directly affect the Perth metropolitan coastline once in every ten years(Lemm 1996).Literature Review 19

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2.4.3 Regimes of Sea breeze and Storm events in PerthSea breezesThe sea breeze system is a diurnal atmospheric circulation due to the differential heating ofthe land and sea surfaces causing a pressure gradient, which leads to streams of air flowingonshore or offshore until balance is restored (BOM 1993). Sea breezes are a distinctive andwell established feature of the local climate in the south-west of Australia, and are given localnames such as the Fremantle Doctor, Albany Doctor, and Esperance Doctor “from the nameof the coastal locality near which they cross the shore on their welcome way inland.” (Gentilli1971 p.111). A typical sea breeze cycle usually consists of offshore winds in the generaleasterly direction in the morning, followed by the sudden arrival of south to southwesterly(alongshore to cross-shore) winds towards mid-afternoon. The sea breeze is present more than60% of the time during the summer months, and the mean sea breeze velocity at the coastlineis about 8 ms -1 (Masselink & Pattiaratchi 2001). Wind speeds during the summer sea breezeoften exceed 15ms -1 with maximum values of 20 ms -1 (Pattiaratchi et al 1997).In contrast to the ‘typical’ sea breeze system which blows perpendicular to the shoreline, thesea breeze along the west coast of Australia blows parallel to the shoreline in a southerlydirection (Pattiaratchi et al 1997). The reason for the shore-parallel sea breeze system can beattributed to the interaction between the sea breeze system and the synoptic weather patterns.During summer, the Australian continent is generally under the influence of easterly air flow,and due to the heating of the air flow across the continent, a low pressure trough is formedparallel to the coast of Western Australia (i.e. in a north-south direction) (Pattiaratchi et al1997). The location of the trough affects the intensity and direction of the sea breeze. Thesynoptic pressure gradient acts in a northeasterly direction when the low pressure trough islocated inland from the west coast (Pattiaratchi et al 1997). If we consider the sea breezesystem on the West Australian coast as simply due to the differential heating of land and seasurfaces and the Coriolis force, the sea breeze would flow in a southwesterly direction. Hencethe combination of the southwesterly air flow of the sea breeze pressure system and thesoutheasterly air flow of the synoptic pressure results in a southerly sea breeze. When thelocation of the trough is such that the synoptic pressure gradient induces southerly winds, thesea breeze enhances the southerly winds (Pattiaratchi et al 1997). Therefore the summersynoptic patterns in this region heighten the sea breeze system, and very strong sea breezesoccur during the summer months. The deepest sea breezes at Perth occur in November to20 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanDecember, while the strongest ones come later in January or February (Gentilli 1971). Atypical summer weather pattern is depicted in Figure 2.8 (a-d).During the winter months, the general isobaric pattern favours offshore flow and thus landbreezes are strengthened while sea breezes often fail to occur (Gentilli 1971). The strengthand onset of sea breeze are also affected by other meteorological factors. For example, theamount of cloud cover can affect the temperature and the prevailing synoptic situation whichmay either enhance or retard its development (BOM 1993). According to Pattiaratchi et al1997), wind waves that are locally generated by the sea breeze activity are the majormechanism controlling littoral sand transport and nearshore morphology of sandy beaches.When the nearshore wave field is influenced by a sea breeze local wind waves undergodiurnal changes in height, period and incidence angles (Pattiaratchi et al 1997).Storm eventsThe low pressure systems associated with the mid-latitude depressions in winter bringwesterly cold fronts, resulting in strong sustained winds with speeds up to 14ms -1 (Rose2001). With the approach of a depression, the winds are initially from the north and shifts tothe northwest whilst increasing in strength. As the depression crosses the coast, there is arapid shift in the wind direction to become southwesterly, and the mean wind speed decreasesslightly after the change. Storm conditions may be maintained for up to 36 hours, with windsreaching up to 20 ms -1 and instantaneous gusts to 35 ms -1 and will gradually moderate(Masselink & Pattiaratchi 2001, Rose 2001). A typical winter sequence for Perth is shown inFigure 2.9 (a-d).Literature Review 21

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan(a) Day 1: A low pressure trough extends in a generalnorth-south orientation inland of the west coast and ahigh pressure system is centred over the Indian Ocean.The typical wind profile shows a moderatesoutheasterly wind in the early morning, veering to thesouthwest and strengthening by late morning. Windwill gradually tend southeast and moderate overnight.The strongest sea breezes in Perth result from thispattern.(b) Day 2: The low pressure trough has progressedfurther inland and the high over the Indian Ocean iscloser. Wind pattern is similar to Day 1, although seabreeze may arrive later and a little weaker. The windmay turn to a gusty easterly overnight.(c) Day 3: The high pressure cell has progressed furthereast and is now located off the south coast andproducing a hot easterly flow over WA. A fresh tostrong easterly wind early in the morning will moderateduring the day as the temperature rises. A weak seabreeze may arrive late in the afternoon, but the wind islikely to return to the east and freshen during evening.On some days there will not be a sea breeze and the(d) Day 4: The formation of a low pressure trough alongthe west coast is due to the temperature differencebetween the land and sea surfaces. Another high islocated over the Indian Ocean. Winds may remain fromthe northeast until mid-afternoon when a moderate seabreeze is likely to develop.Figure 2.8: A typical summer sequence for Perth (from BOM 1993)22 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan(a) Day 1: The wind early in the morning is a northerly,shifting northwesterly before mid-day andstrengthening by late afternoon(b) Day 2: Northwesterly winds rapidly shift southwestwith the passage of the front and mean wind speeddecreases slightly. Westerly winds soon strengthenahead of a second cold front, associated with a lowpressure cell that developed rapidly in the cold airbehind the first front.(c) Day 3: Cold front progresses on eastward andsqually west to southwesterly winds gradually decreasein strength.(d) Day 4: Conditions have moderated and asouthwesterly wind persists for mosst of the day,moderating slowly by the evening.Figure 2.9: Typical weather pattern for winter for Perth (from BOM 1993)2.4.4 Offshore Wave ConditionsThe offshore wave climate is dominated by moderate energy swell from the south tosouthwest, and a variable wind wave climate is superimposed on the background swell(Masselink & Pattiaratchi 2001). Sea breezes have a strong influence on the offshore waveconditions during summer, therefore the prevailing wave direction is south to southwest.Offshore waves during summer have predominantly low period (less than 8 s) in the range of1-2 m (Lemm 1996). Northwesterly to southwesterly storm waves occur during the winterLiterature Review 23

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanmonths, and the offshore wave climate is characterised by high period (more than 8 s) swelland storm waves of 1.5-2.5 m (Lemm 1996). Hence, for the study area, there is a distinctiveoffshore wave climatic shift from moderate, locally generated seas in summer to higher,distantly generated swell in winter (Lemm 1996). A background swell above 0.5 m was foundto be present all year round (Lemm 1996). The submerged ridges and islands mentioned inSection 2.4.1 offer extensive sheltering of the coastline, particularly from southwesterlywaves (i.e. the prevailing wave conditions), therefore the nearshore wave conditions are lessenergetic than offshore conditions.24 Literature Review

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan3 APPROACH3.1 DATA COLLECTION3.1.1 Measuring DeviceThe wave data in this study was obtained through the use of the InterOcean TM S4DWelectromagnetic current meter (Figure 3.1). The S4DW current meter is designed byInterOcean TM to measure and record the true magnitude and direction of horizontal motionwithout the effect of vertical movement due to wave action (ISI 2004b). The equipmentmeasures velocities using the principles of Faraday's Law of electromagnetic induction. Waterflows through an electromagnetic field generated by the current meter, thus producing avoltage proportional to the flow velocity of the water past the sensor (ISI 2004c). The voltageis sensed by two orthogonal pairs of electrodes located symmetrically on the equator of thesensor and current direction is measured by an internal flux-gate compass (ISI 2004b). TheS4DW instrument also records pressure, which was converted into depth data for the purposesof this study. The equipment samples at a rate of 2 Hz (i.e. once every 0.5 s), and has 20megabytes of memory available for data storage to allow deployments for extended periods(ISI 2004d).Figure 3.1: InterOcean S4DW Current Meter (from ISI 2004a)3.1.2 Data Collection TechniqueThe S4DW current meter was moored at 0.5 m above the sea bed at two inshore locations asshown in blue in Figure 2.7: City Beach and the Cables Artificial Surfing Reef (ASR). Thedetails of each deployment are shown in Table 3.1. The data obtained at City Beach was at ashallower depth than the data obtained at Cables ASR, hence the deployment at City Beachwas closer to shore than the deployment at Cables ASR. The current meter recorded data at 18Approach 25

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanminute bursts for the City Beach deployment, and 20 minute bursts at Cables ASR. Each databurst was continuously recorded beginning on the hour at both locations. Therefore, therewere 18 min! 60sec!2 Hz = 2160 data points recorded for each burst at the City Beachdeployment, and 20 min! 60sec!2 Hz= 2400 data points per burst at the Cables ASRdeployment.Table 3.1: Details of the S4DW current meter deploymentsLOCATIONCity BeachCables ASRDate (inclusive) 21-Jan-2001 to 09-Feb-2001 11-Aug-1999 to 27-Aug-1999Depth ~ 5 m ~ 8 mData points/burst 2160 2400Wind speed and direction data at 30 minute intervals for Swanbourne, which is approximately10 kilometres south-west from the city of Perth (Figure 2.7), was obtained from the Bureau ofMeteorology for the period of both deployments.26 Approach

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan3.2 DATA ANALYSISMajority of the data analysis in this study was carried out using MATLAB ® , a high level,vector-orientated, mathematical programming language developed by The MathWorks Inc.(Appendix B).A sequence of observations made through time may be described as a time series. The datacollected by the S4DW current meter in this study are therefore time series of currentvelocities measured at a fixed location. These time series are stochastic, discrete, uniformlysampled and of finite length. In this study, the time series data were processed using spectralanalysis. Spectral analysis can be used to decompose a time series into its constituentfrequency components, and different frequency bands can be isolated. In particular, thismethod allows the separation of the sea and swell components of the wave field. In this study,sea waves are defined as waves with periods between 2 s and 6.65 s (i.e. 0.15 < f < 0.5) andswell waves are defined as waves with periods between 6.65 s and 20 s (i.e. 0.05 < f < 0.15).The wave spectrum in Figure 3.2 demonstrates the frequency cut-offs for swell and seawaves; this particular wave spectrum shows that the wave field is dominated by sea waveenergy.In this study, the data records were transformed from the time domain to the frequencydomain via the Fast Fourier Transform (FFT). The application of the FFT requires that thenumber of data points in the series should be a power of 2. In addition to this, it is desirable touse a considerable length of data as this increases the accuracy of the conversion performedby the FFT. As mentioned in Section 3.1.2, there were 2160 data points recorded for every 18minute burst at City Beach and 2400 data recorded for every 20 minute burst at Cables ASR.The number of data analysed in the present study is 2048, which is the maximum power of 2value within the data record (N = 2048 = 2 11 ). This ensures optimum spectral resolution andminimum occurrence of error for the given data record. A set of wave parameters useful tothis study was extracted from each data burst (Table 3.2).Approach 27

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 3.2: A wave spectral energy plot indicating the frequency cut-offs for the swell and seacomponents of the wave field.Table 3.2: Wave parameters extracted from the data in the frequency domainSymbol Description Method of Calculation Unitsm 0 Total variance (zeroth moment of spectral( f )dfdensity)! " mS20m 0,sea Variance of sea componentmS( f )df2m 0,swellH sH s,seaVariance of swell componentSignificant wave height estimateSignificant wave height of sea component! 0.50. 15! 0.15 S0. 05( f )dfm 24 m0m4 m 0,seamH s,swell Significant wave height of swell 4 m m0,swellcomponentT p Peak period. Wave period corresponding 1/f p sto the maximum spectral energy.T p,sea Peak period of the sea component 1/f p,sea sT p,swell Peak period of the swell component 1/f p,swell sNote: f p is the spectral peak frequency, the frequency at which the maximum spectral density occurs28 Approach

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan3.2.1 Directional analysisThe directional wave spectra were obtained using the DIWASP (DIrectional WAve Spectra)Toolbox (Version 1.1) for MATLAB ® . DIWASP is a toolbox of MATLAB ® functions for theestimation of directional wave spectra, and there are five different estimation methodsavailable: Direct Fourier Transform Method (DFTM), Extended Maximum LikelihoodMethod (EMLM), Iterated Maximum Likelihood Method (IMLM), Extended MaximumEntropy Method (EMEP) and Bayesian Direct Method (BDM) (Johnson no date). The EMEPestimation method was chosen for use in this study because of its advantages over the othermethods, as briefly discussed in Section 2.3.3. Furthermore, the other methods in DIWASPwere tested on the data and the EMEP method provided the best overall results. The speedand direction data recorded by the current meter were resolved into east-west and north-southcomponents for use with the DIWASP toolbox.3.2.2 Non-directional analysisNon-directional wave parameters (significant wave height and peak period) and the swell andsea components for each burst were obtained using the spectral method (Appendix B). Nondirectionalwave spectra were obtained for each wave burst using FFT techniques and theWelch method (Appendix B). A Hanning window was utilised for frequency band averagingand two segments of 1024 points (i.e. 8.53 minutes) were used for segment averaging,yielding four degrees of freedom in the final spectral estimates.3.2.3 Wind analysisVelocity vectors of the wind speed and direction were plotted via the in-built MATLAB ®function, ‘feather’. However, the plots of bi-hourly data (i.e. the complete wind data set)produced a congested plot which cannot be observed in detail. Therefore, wind data that wererecorded on the hour were extracted to match the hourly wave data for the purpose ofobserving general trends. The complete wind data were used otherwise for detailed analysis.Approach 29

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4 RESULTS AND DISCUSSION4.1 INTRODUCTIONThe results obtained from the methods described in Section 3 are presented along withdetailed discussion in this section:• The wave response to changes in wind speed and wind direction during sea breezes andstorms was investigated in terms of changes in wave height, wave frequency, direction ofwave propagation and spectral density of the waves. These are presented in Sections 4.2.1and 4.3.1 for sea breeze and storm events respectively. The hourly wave patterns inresponse to changes in wind conditions are investigated in Sections 4.2.2 and 4.3.2.• Comparisons between the swell and sea components of the significant wave height andpeak period are presented in Section 4.4.• The wave response time to changes in wind conditions are investigated in Section 4.5.• The reflection of wave energy at City Beach and Cables ASR is considered in Section 4.6.Wave energy in terms of wave frequency and direction of propagation are presented as hourlypolar plots in Sections 4.2.2, 4.3.2, 4.5 and 4.6, which were obtained through the use ofDIWASP. It is worth noting that a significant number of these plots were not suitable foranalysis. These unusable plots showed waves coming from all directions (i.e. the whole 360 operipheral), and this could be due to the EMEP estimation method being relatively sensitive tonoise in the data, hence amplifying erroneous data. Consequently, there were slightdifficulties in obtaining suitable consecutive hourly plots for analysis, and selection of theseplots was subject to the availability of overall good hourly plots.The numerical directions of waves in the discussion of the results are “compass” directions indegrees relative to the north and increases in a clockiwise fashion (i.e. 0 o /360 o is due north,90 o is east, 180 o is south and 270 o is west). Note that this is different to the direction notationon the polar plots presented in Sections 4.2.2, 4.3.2, 4.5 and 4.6.30 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanResults and Discussion 31

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.2 SEA BREEZE CYCLESThe data recorded at City Beach in January/February 2001 were utilised to demonstrate thewave response to the sea breeze cycles experienced in Perth during the summer months.4.2.1 Overall patternThe time series of wind and wave conditions over the sampling period for the City Beachdeployment are depicted in Figure 4.1.The daily south to southwesterly sea breezes are clearly shown in the wind pattern (Figure4.1a). For most of the sampling period, the typical sea breeze cycle prevailed (Section 2.4.3).For each cycle, offshore (easterly) winds blow early in the day, followed by an abruptincrease in wind speed and shift in wind direction to onshore southwesterly winds towardsmiddle of the day. The southwesterly winds persist for most of the evening, and windconditions gradually return to pre-breeze conditions usually in the early hours of the morningwhen the offshore land breeze starts blowing.There was little to no sea breeze activity on days 28, 30 – 31 (i.e. 28, 30 – 31 Jan) and 35 – 37(i.e. 4 – 6 Feb) inclusive, which could be due to various synoptic influences acting to suppresssea breeze events. The events on days 31 – 32 are of interest. Variable northerly windsdominated for the most of day 31 and part of day 32, then rapidly shifting to the southwestlater on day 32. These patterns indicate the possible passage of a storm (Section 2.4.3). Thesouthwesterly wind shift may have strengthened the sea breeze on the latter part of day 32.The maximum and mean wind speeds for the sampling period at the City Beach deploymentwere 23 knots and 11.75 knots respectively.The sea breeze events cause instantaneous changes to the nearshore wave conditions. Therewere distinct peaks of significant wave height H s corresponding to each sea breeze event(Figure 4.1b). During each of the sea breezes, H s usually doubled in height with a generalincrease of approximately 0.6 m in response to the onset of the sea breeze. H s then decreasedas the effects of the sea breeze subside towards the end of the day and the beginning of thenext day. The decrease in H s experienced after each sea breeze was not usually equal to theincrease in H s experienced at the onset of the sea breeze, and vice versa. This is because the32 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanwave field was greatly influenced by the energy of swell waves which are not affected bylocal wind conditions such as the sea breeze events. The effects of the swell and seacomponents of H s for the data collected at City Beach are discussed in Section 4.4.1. Therewere no distinct daily peaks of H s observed on days 28, 30 – 31 and 35 – 37 (inclusive) due toweak or no sea breezes occurring on those days. Furthermore, H s appear less variable duringthe periods of little to no sea breezes. The mean H s for the summer sampling period was 0.77m. A maximum H s of 1.33 m was achieved during a sea breeze event within the samplingperiod.The wave frequency spectra (Figure 4.1c) clearly indicate that during the sea breezes, shortperiod sea waves became superimposed on the background swell. The generation of seawaves begin in response to the start of sea breezes at around 0.3 Hz, decreasing down toaround 0.15 Hz when the sea breezes subsided (i.e. wave period increased from 3 s to 6 s).Swell waves of approximately 0.1 Hz (i.e. wave period of 10 s) were almost always present.The frequency spectra verify the lack of sea breeze generated sea waves on days 28 – 32 and35 – 37, when relatively high energy swell waves dominated instead.The directional wave spectra (Figure 4.1d) show that the dominant wave direction for theduration of the sampling period was in the northeast quadrant (i.e. 0 o to 90 o ), corresponding tothe prevailing southwesterly winds during the summer months.Results and Discussion 33

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.1: Wind and wave conditions measured in the nearshore region of City Beach from 21January to 09 February 2001:(A) wind speed and direction measured at Swanbourne – note that wind speed is given in kilometresper hour (kph); (B) significant wave height; (C) wave frequency in terms of wave spectral energy; (D)wave direction relative to the north in terms of wave spectral energy. The solid line in (A) representsthe wind speed and the arrows are hourly wind vectors with their length proportional to the wind speed(up = southerly; right = westerly; down = northerly; left = easterly). The shading in (C) and (D)reflects increasing levels of spectral energy as defined in the adjacent colour bars (in units of m 2 /Hz)34 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.2.2 Hourly TrendsThe hourly wave patterns induced by the sea breeze cycle are investigated in this section. Thesequence presented in Figure 4.2 illustrates the typical wave patterns that occur during a seabreeze event. This particular sea breeze cycle occurred on 25 – 26 January 2001 at CityBeach. Data analysis for these dates via DIWASP produced generally excellent hourly plots.The sequence begins a few hours after the onset of the wave response to the sea breeze effectoccurring on 25 January at 17:00, and ends on 26 January at 17:00 to encompass a wholeday’s cycle. However, the description presented here shall begin at 06:00 of 26 January inorder to illustrate the wave response to the arrival of the sea breeze.At 06:00 on 26 January, the prevailing southwesterly summer winds have subsided against thestrengthening offshore land breeze. The wave field was dominated by swell waves ofapproximately 0.1 Hz (i.e. wave period of 10 s) flowing in the east-northeasterly direction(60 o – 90 o ), and there were very weak sea waves flowing in the northeasterly direction(approximately 45 o ). By 08:00, the energy of the northeasterly sea waves had significantlyreduced, and the swell waves continued propagating to the east-northeast. Swell conditionsremained the same for the next six hours, as verified by the 13:00 wave pattern. At 14:00, seawaves of approximately 0.32 Hz began propagating to the northeast (approximately 40 o ) inresponse to the onset of the south-southwesterly sea breeze. By 15:00, the sea breezegeneratedwaves of 0.2 – 0.3 Hz have picked up significant energy. The east-northeast(approximately 80 o ) swell component was still present in the background, giving a maximumdifference of 40 o in the directions of wave propagation between the swell and sea componentsduring the sea breeze event. At 16:00 and 17:00, two and three hours after the onset of thewave response to the sea breeze, the propagation of the sea breeze-generated sea wavespersisted in the northeasterly direction against a background easterly swell component.Reflected waves of the same frequency as that of the sea waves were detected almost 180 o inthe opposite direction.In fact, the cessation of the sea breeze effect on the waves was less defined than for the onsetof the sea breeze effect because there was a gradual return to pre-breeze conditions during theevening. This is illustrated in the hourly wave patterns of 17:00 and 21:00 on 25 January and01:00 and 04:00 on 26 January. The pre-breeze swell wave conditions did not return until theearly hours of the next morning.Results and Discussion 35

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanHence, the summer wave patterns on the nearshore coastal zone of Perth waters during thesampling period at City Beach were dominated by east-northeasterly (approximately 80 o )swell waves and northeasterly (approximately 40 o ) sea waves generated by sea breezes. Amaximum difference of approximately 40 o in the directions between the propagation of swelland sea waves was achieved during the sea breeze event. The persistence of the eastnortheasterlyswell waves is due to the fact that swells are not affected by local windconditions (Section 2.1.1). Note that nearshore waves do not respond to the overnight/earlymorning offshore (easterly) winds (i.e. by propagating to the west) due to the insufficientfetch area at the sampling location for sea waves to develop appreciably in response.36 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan1700 on 25 Jan2100 on 25 Jan0100 on 26 Jan0400 on 26 JanResults and Discussion 37

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan0600 on 26 Jan0800 on 26 Jan1300 on 26 Jan38 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan1400 on 26 Jan1500 on 26 Jan1600 on 26 Jan1700 on 26 JanResults and Discussion 39

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.2: Typical wave sequence induced by the sea breeze cycle measured in the nearshore regionof City Beach on the 25 – 26 January 2001The left side of the figure shows hourly polar plots indicating the wave energy in terms of wavefrequency and direction of wave travel. The shading in the polar plots reflects increasing levels ofspectral energy as defined in the adjacent colour bars (in units of m 2 /Hz) The corresponding hourlynon-directional frequency spectrum is shown on the right40 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.3 STORM EVENTSThe data recorded at Cables ASR during August 1999 (i.e. winter) were used to demonstratethe wave response to storm events in the Perth coastal area.4.3.1 Overall patternThe time series of wind and wave conditions over the sampling period for the Cables ASRdeployment are shown in Figure 4.3.The wind pattern (Figure 4.3a) clearly indicate the passage of storms associated with midlatitudedepressions (Section 2.4.3). One storm event occurred on days 229 – 231 (i.e. 17 – 20Aug) and two closely-spaced storms occurred on days 236 – 237 (i.e. 24 – 25 Aug) inclusive.Offshore variable winds initially flow from the north, shifting to the northwest and increasingin speed as the depression approaches the coast. Winds then abruptly shift to southwesterly,indicating that the depression crossed the coast, and the winds decrease slightly in strength.The present wind conditions persist for a few hours before gradually moderating. The meanand maximum wind speeds for the winter sampling period at the Cables ASR deploymentwere 12.05 knots and 34.05 knots respectively. The maximum wind speed was achievedduring the storm event on days 229 – 231.The response of H s to storm events is straightforward and apparent, as shown in Figure 4.3b.During the storm event on days 229 – 231, H s increased more than 1.5 m. At the height of thestorm on day 230, the peak H s of 2.75 m was achieved in response to the highest wind energyfor the sampling period. The mean H s for the winter sampling period was 1.17 m.The wave frequency spectra (Figure 4.3c) clearly show the wave response to storm winds.With the onset of northwesterly winds, short period sea waves became superimposed on thebackground swell. The generation of sea waves began in response to storm winds at 0.3 Hz,decreasing down to around 0.15 Hz for the whole duration of the storm event. The constantbackground swell waves of less than 0.1 Hz (i.e. wave period of more than 10 s) were almostalways present. The highest energy waves were generated in response to the strongest windspeeds, especially for the storm event on days 229 – 231.Results and Discussion 41

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanThe directional wave spectra (Figure 4.3d) indicate that the dominant wave direction for theduration of the sampling period was in the southeast quadrant (i.e. 90 o – 180 o ), correspondingto the prevailing northerly winds in winter. An interesting point to note is that during the latterpart of the storm events, the predominant direction of wave propagation remained to thegeneral southeasterly direction although the wind was blowing from the southwest. This isfurther discussed in Section 4.3.2.42 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.3: Wind and wave conditions measured in the nearshore region of Cables Artificial SurfingReef from 11 August to 27 August 1999:(A) wind speed and direction measured at Swanbourne – note that wind speed is given in kilometresper hour (kph); (B) significant wave height; (C) wave frequency in terms of wave spectral energy; (D)wave direction relative to the north in terms of wave spectral energy. The solid line in (A) representsthe wind speed and the arrows are hourly wind vectors with their length proportional to the wind speed(up = southerly; right = westerly; down = northerly; left = easterly). The shading in (C) and (D)reflects increasing levels of spectral energy as defined in the adjacent colour bars (in units of m 2 /Hz)Results and Discussion 43

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.3.2 Hourly trendsThe hourly wave patterns induced by storm events associated with mid-latitude depressionsare investigated in this section. The sequence presented in Figure 4.4 illustrates the typicalwave patterns that occur during a storm. This particular storm began on 24 August 1999 atCables ASR. This storm event was chosen over the storm that began on 17 August becausethe data analysis using DIWASP produced overall good hourly polar plots for the latter stormevent but not for the former.At 23:00 on 23 August, relatively high energy swell waves were prevailing in the eastsoutheasterlydirection (approximately 110 o ). The swell conditions persisted for two morehours, and at 01:00 on 24 August, waves began responding to the northwesterly winds thatmarked the approach of a mid-latitude depression. East-southeasterly (110 o – 130 o ) sea wavesbegan developing at a frequency of approximately 0.2 Hz, however the wave field remainedprevalently swell conditions. At 02:00 and 03:00, the sea waves shifted to southeasterly (120 o– 150 o ) whilst obtaining more energy from the strengthening northwesterly winds. Note thatthe background east-southeasterly swell was still present, and the energy of reflected seawaves can be detected at approximately 180 o in the opposite direction. At 04:00 and 05:00,the sea waves began to change the direction of propagation to the east (approximately 90 o ),corresponding to the abrupt change in wind direction from northwest to southwest associatedwith the passage of the depression. At 07:00 to 08:00, the change in the direction of stormgeneratedsea waves was completed, with waves of 0.25 Hz propagating to the northeast(approximately 50 o ). Note that the direction of wave propagation of the swell componentremained to the east-southeast (approximately 110 o ) during the change in direction of the seawaves, achieving a maximum difference of approximately 60 o in the directions of propagationbetween the swell and sea waves during the storm. The northeasterly sea wave conditionscontinued for a few more hours before moderating back to almost swell conditions, as verifiedby the wave patterns at 09:00, 10:00 and 13:00. However, the wave field for that particularstorm event did not return to pre-storm swell conditions as at 23:00 on 23 August because theend of this particular storm event was not defined, and another storm followed closelyovernight (Section 4.3.1).Therefore, the winter nearshore wave field for the sampling period at Cables ASR wasdominated by swell waves propagating to the east-southeast (approximately 110 o ). Sea waves44 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanwere generated in variable onshore directions in response to storm events, superimposed onthe east-southeasterly background swells, achieving a maximum deviation to the northeast(approximately 50 o ). Hence a difference of approximately 60 o was obtained between thedirections of wave propagation of the swell and sea components during the storm event.For the duration of the storm event analysed in this section, the energy of swell waves waspredominantly higher than that of sea waves (as observed in the non-directional spectra ofFigure 4.4), implying that swell waves were more important in determining the direction oflittoral transport on the beach. This justifies the observation noted at the conclusion of Section4.3.1 that the dominant wave direction for the duration of the sampling period was in thesoutheast quadrant despite the variable wind conditions experienced during storms.Results and Discussion 45

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2300 on 23 Aug0100 on 24 Aug0200 on 24 Aug0300 on 24 Aug46 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan0400 on 24 Aug0500 on 24 Aug0700 on 24 AugResults and Discussion 47

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan0800 on 24 Aug0900 on 24 Aug1000 on 24 Aug1300 on 24 Aug48 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.4: Typical wave pattern induced by a storm associated with the mid-latitude depression,measured in the nearshore region of Cables ASR on the 23 – 24 August 1999.The left side of the figure shows hourly polar plots indicating the wave energy in terms of wavefrequency and direction of wave travel. The shading in the polar plots reflects increasing levels ofspectral energy as defined in the adjacent colour bars (in units of m 2 /Hz). The corresponding hourlynon-directional frequency spectrum is shown on the rightResults and Discussion 49

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.4 SWELL AND SEA COMPONENTSThe time series of the total, swell and sea components of the nearshore significant waveheight H s and peak wave period T p for the City Beach and Cables ASR deployments areshown in Figure 4.5 and Figure 4.6 respectively. H s is composed of different componentswhich are categorised according to the frequency value. The swell and sea components of H sshown in the second subplot of Figure 4.5 and Figure 4.6 were obtained using the frequencycut off values for swell and sea waves defined in Section 3.2. As previously discussed inSections 2.2.3 and 3.2, T p is the largest wave energy occurring for a certain interval offrequency. Therefore, total T p , as shown in the third subplot of Figure 4.5 and Figure 4.6,represents the largest wave energy spanning both the frequency intervals for the swell and seawaves. The swell and sea components of T p correspond to the peak wave energy in theirrespective frequency cut-off values. Hence, the value of the total T p for any given time canindicate swell or sea conditions.4.4.1 City BeachThe sea component of significant wave height (H s,sea ) shown in the second subplot of Figure4.5 clearly indicates sea breeze events. The peaks of H s,sea correspond to the wave response tothe onset of sea breezes, which is clearly reflected in the total H s . The swell component of thesignificant wave height (H s,swell ) dominated on days 28 – 32 (i.e. 28 Jan – 1 Feb) and 35 – 37(i.e. 4 – 6 Feb) due to a lack of strong sea breezes to generate sea waves, as verified by thewind plot on Figure 4.1a.The total T p plot in the third subplot of Figure 4.5 indicates that the largest wave energy in thewave field for the whole sampling period was dominated by high T p waves (i.e. swells) ofgenerally more than 10 s punctuated with low T p waves (i.e. seas) that correspond to the waveresponse to the onset of sea breezes. The mean and maximum T p for the sampling period was12.15 s and 18.45 s. For most of the sea breeze events that occurred during the samplingperiod, the peak wave period of sea waves, T p,sea , progressively increased from 3 – 4 s to 5 – 6s. There were days when the swell waves had higher energy than that of the sea breezegenerated sea waves, such as days 23 – 25 (i.e. 23 – 25 Jan), and subsequently, the total T p donot indicate low period waves.50 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanIt was previously discussed in Section 4.2.1 that the wind and wave patterns on days 31-32(i.e. 31 Jan – 1 Feb) indicated the possible passage of a storm. H s,sea and H s,swell for that periodshowed the generation of sea waves at the onset of relatively strong northerly winds, and anapparent lag in the response of swell waves of a few hours. This lag is discussed in detail inSection 4.4.2. Total T p for days 31-32 verified that waves with relatively low period (i.e. seawaves) dominated the wave field in response to the onset of the northerly winds on days 31-32. Relatively high period waves dominated the wave field about midday of day 32,corresponding to the lagged response of H s,swell .Figure 4.5: Nearshore significant wave height and peak wave period and their respective swell andsea components at City Beach for the period 21 January to 09 February 2001Results and Discussion 51

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.4.2 Cables ASRThe shape of the total H s in the first subplot of Figure 4.6 is most similar to that of H s,swell inthe second subplot, indicating that the swell component of H s dominated for majority of thesampling period, even during the storm events. The maxima of H s,sea correspond to the waveresponse to storm winds, and contribute to the maxima of the total H s .A lag of a few hours between the response of H s,sea and H s,swell to strong northerly winds at theonset of a storm was previously noted in Section 4.4.1 for the data collected at the City Beachdeployment. The same lagged response is observed in the winter sampling data, and isparticularly obvious in the initial storm stages on days 229 (i.e. 17 Aug) and 236 (i.e. 24 Aug)(second subplot of Figure 4.6). As discussed in Section 2.1.1, swell waves at a location weregenerated in a region far from the sampling location and are no longer under the influence oflocally-generated wind. The eastward passage of a mid-latitude depression would havegenerated waves of different periods to the west of a sampling location. The differentcomponents of the waves will separate from each other, with the longer period waves (i.e.swells) travelling faster than shorter period waves (i.e. seas) (Section 2.1.1). Therefore, thelagged response of H s,swell in a local storm are in fact due to long period components of wavesthat were generated in response to the same storm event at a location to the west of thesampling region.The total T p plot in the third subplot of Figure 4.6 indicates that the largest wave energy in thewave field for the winter sampling period at Cables ASR was dominantly influenced by highperiod swell waves of generally more than 10 s, with an average T p of 14.21 s and a maximumof 21.56 s. The only exception that occurred during the sampling period was on day 229 whenrelatively low period sea waves were initially generated in response to the approach of themid-latitude depression. The minimum T p of 6.67 s occurred during the initial stages of thestorm event on day 229, which is the cut off value between the swell and sea components.This observation implies that the swell component of the peak period dominated the peakenergy wave field for majority of the storm duration. The relatively low T p waves were notsustained for the whole duration of the storm, but initially fluctuated through a large range,and as the response of H s reached its peak in response to the height of the storm event in theearly hours of day 230, high T p,swell of more than 10 s dominated the peak energy wave fieldonce again. The sea component of T p as shown in the final subplot of Figure 4.6 was generally52 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanmaintained at approximately 6 s, decreasing near 4 second during the initial stages of a stormevent.Figure 4.6: Nearshore significant wave height and peak wave period and their respective swell andsea components at Cables Artificial Surfing Reef for the period 11 August to 27 August 1999Results and Discussion 53

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.5 WAVE RESPONSE TIMETo evaluate the wave response time to wind speed and direction, the wind and wave details ofday 31 (i.e. 31 January 2001) from the City Beach deployment data were analysed in detail.This day was chosen because there were good hourly polar plots for each burst for 15consecutive hours coinciding with a distinct change in wave energy and direction. Thesequence of polar plots for each hourly wave burst is from 0600 to 2100 (Figure 4.7). Bihourlywind speed and direction are also tabulated for that day (Table 4.1). As noted inSection 4.2.1, no sea breeze was observed on this day.Two instances shall be used to verify the response time of the waves to the wind. The polargraphs indicate that swell waves were propagating in a westerly direction from 06:00 to09:00. Sea waves began propagating to the south east quadrant at 10:00. Wind data shows thatfrom 00:00 to 06:00, wind was flowing in an easterly to northeasterly direction. At 06:30, thewind changed direction to flow in a northwesterly fashion. This corresponds with the changein wave direction at 10:00. From 10:00 to 14:30, the wind was blowing in a northwesterlydirection. There was a change in wind direction at 15:00 to the west-southwest, and thecorresponding change in wave direction occurred at 19:00.Therefore the time taken for the waves to respond to wind is approximately 3.5 to 4 hours,with an uncertainty of about 0.5 hours due to the fact that bi-hourly wind data were used withhourly wave bursts.54 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan06:00 07:0008:00 09:0010:00 11:0012:00 13:00Results and Discussion 55

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan14:00 15:0016:00 17:0018:00 19:0020:00 21:0056 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.7: Hourly sequence of polar plots of wave energy in terms of wave frequency and directionof wave travel measured in the nearshore region of City Beach on 31 January 2001Table 4.1: Bi-hourly wind data for 31 January 2001 collected at Swanbourne.Date Time (hour) Wind Direction (deg North) Wind Speed (m/s)20010131 0 70 6.6920010131 30 60 8.2320010131 100 50 6.6920010131 130 50 4.6320010131 200 30 3.0920010131 230 20 2.5720010131 300 10 1.5420010131 330 10 2.0620010131 400 20 3.0920010131 430 40 1.5420010131 500 40 2.0620010131 530 70 3.0920010131 600 30 3.6020010131 630 340 4.1220010131 700 360 4.6320010131 730 360 6.1720010131 800 350 4.1220010131 830 30 6.1720010131 900 360 6.6920010131 930 340 7.2020010131 1000 350 8.7520010131 1028 350 8.7520010131 1058 340 10.2920010131 1130 340 11.3220010131 1131 340 11.3220010131 1200 350 11.3220010131 1230 350 9.7720010131 1300 340 9.2620010131 1330 330 9.7720010131 1400 320 8.7520010131 1430 300 6.6920010131 1500 260 8.2320010131 1530 260 8.2320010131 1600 260 7.7220010131 1630 260 8.2320010131 1700 250 7.20Results and Discussion 57

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan20010131 1730 250 6.6920010131 1800 250 7.7220010131 1830 240 6.6920010131 1900 230 7.2020010131 1930 230 7.7220010131 2000 220 8.7520010131 2030 220 9.2620010131 2100 220 8.2320010131 2130 220 8.2320010131 2200 230 7.7220010131 2230 220 7.2020010131 2300 220 6.6920010131 2330 220 6.69Note: the wind direction notation begins at 0 degrees north and increasing in a clockwise fashion,indicating the direction the wind is coming from (e.g. a wind direction of 360 o is a northerly windblowing in from the north). That is, North: 0/360 o , East: 90 o , South: 180 o , West: 270 o4.6 WAVE REFLECTIONThis section compares the energy of reflected waves with that of incident waves via thereflection coefficient, which is defined asX =EEreflectedincidentwhere E reflected and E incident are the respective energy of reflected and incident waves (CharithaPattiaratchi pers. comm. 2004). The reflection coefficient can provide an indication of theamount of wave energy dissipation on the beach, which is an important factor in beachsediment transport. The reflection coefficient can only be accurately obtained with the use ofdirectional wave data. For each of the sampling locations (i.e. City Beach and Cables ASR),three hourly polar plots that best displayed reflected waves were selected (though notnecessarily consecutive hours), and the wave spectrum for the directions of the incident andreflected waves were plotted. For each hour at each sampling location, the ratio of thereflected and incident wave spectra was calculated, and the maximum wave reflectioncoefficients were obtained.4.6.1 City BeachThe following hourly polar plots were selected for wave reflection analysis for City Beach:• 06:00 on 21 January 2001• 17:00 on 26 January 2001• 10:00 on 31 January 200158 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanAt 06:00 on 21 January, waves at approximately 0.08 Hz were propagating to the eastnortheastat 74 o with a corresponding reflected wave component travelling in the direction of292 o at the same frequency (Figure 4.8). The maximum value of E reflected : E incident for this hourwas 0.112, implying that up to 11.2% of the incident wave energy was reflected.At 17:00 on 26 January, waves of approximately 0.25 Hz were propagating in the direction of53 o with a corresponding reflected wave component travelling in the direction of 254 o (Figure4.9). The maximum value of E reflected : E incident obtained for this hour was 0.153, which meansthat up to 15.3% of the incident wave energy was reflected.At 10:00 on 31 January, 0.09 Hz waves were propagating to the east at 86 o with acorresponding reflected wave component propagating in the direction of 274 o (Figure 4.10).The maximum value of E reflected : E incident for this hour was 0.522, implying that as high as52.2% of the incident wave energy was reflected.For the three hours that were analysed for City Beach, the maximum E reflected : E incident was0.522.Results and Discussion 59

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan60 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.8: Wave information at 06:00 on 21 January 2001 at City Beach.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 74 o (previous page); wave spectrum at 292 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.112Results and Discussion 61

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan62 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.9: Wave information at 17:00 on 26 January 2001 at City Beach.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 53 o (previous page); wave spectrum at 254 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.153Results and Discussion 63

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan64 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.10: Wave information at 10:00 on 31 January 2001 at City Beach.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 86 o (previous page); wave spectrum at 274 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.522Results and Discussion 65

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.6.2 Cables ASRThe following hourly polar plots were selected for wave reflection analysis for Cables ASR:• 09:00 on 17 August 1999• 09:00 on 18 August 1999• 18:00 on 18 August 1999At 09:00 on 17 August, waves of approximately 0.22 Hz were propagating in a direction of128 o with a corresponding reflected wave component travelling in the direction of 287 o(Figure 4.11). The maximum value of E reflected : E incident for this hour was 0.127 (i.e. up to12.7% of the incident wave energy was reflected).At 09:00 on 18 August, waves of approximately 0.27 Hz were propagating to the eastnortheastat 79 o with a corresponding reflected wave component propagating in the directionof 274 o (Figure 4.12). The maximum value of E reflected : E incident for this hour was 0.239,implying that as much as 23.9% of the incident wave energy was reflected.At 18:00 on 18 August, waves of approximately 0.29 Hz were propagating to the northeast at55 o with a corresponding reflected wave component travelling in the direction of 228 o (Figure4.13). The maximum value of E reflected : E incident for this hour was 0.981, which implies that upto 98.1% of the incident wave energy was reflected.For the three hours that were analysed for Cables ASR, a maximum E reflected : E incident of 0.981was obtained.66 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanResults and Discussion 67

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.11: Wave information at 09:00 on 17 August 1999 at Cables ASR.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 128 o (previous page); wave spectrum at 287 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.12768 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanResults and Discussion 69

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.12: Wave information at 09:00 on 18 August 1999 at Cables ASR.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 79 o (previous page); wave spectrum at 274 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.23970 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanResults and Discussion 71

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanFigure 4.13: Wave information at 18:00 on 18 August 1999 at Cables ASR.Top to bottom: wave energy in terms of frequency and direction of propagation (previous page); wavespectrum at 55 o (previous page); wave spectrum at 228 o ; and the ratio of the reflected and incidentwave components, yielding a maximum value of 0.98172 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan4.6.3 Discussion of the reflection coefficients at City Beach and Cables ASRThe major factor influencing the wave reflection coefficient at a location is the distance fromthe shore. The energy of reflected waves is higher at a location close to the shore than at a sitelocated further offshore. Hence the wave reflection coefficient should be higher at CityBeach. However, the maximum E reflected : E incident is significantly higher for Cables ASR(0.981) than for City Beach (0.522), implying that more wave energy is reflected at CablesASR. There is insufficient information at this stage to draw reliable conclusions to address theapparent contradiction. Furthermore, there were substantial differences between the threehourly ratio values for both locations, therefore more analysis should be carried out to obtainmore reliable statistics and to determine any outlier values which may indicate erroneousresults.Results and Discussion 73

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan5 CONCLUSIONSThe directional wave data collected in the nearshore coastal waters of Perth, WesternAustralia, during summer and winter clearly show the wave response to the changing windclimate experienced in each season. The summer wave patterns have a predominant eastnortheasterlyswell component, with northeasterly sea waves generated by the sea breezessuperimposed on the background swell. The winter nearshore wave field have relatively highenergy swell waves propagating to the east-southeast, with sea waves generated in variableonshore directions by storm wind conditions associated with the passage of mid-latitudedepressions. The propagation direction of storm-generated sea waves change fromsoutheasterly to northeasterly in response to the varying wind conditions during a storm event,however the direction of swell waves remains constant at east-southeast. The wave responsetime to changes in wind conditions is 3.5 to 4 hours.The constant direction of swell waves in both the summer and winter nearshore wave patternsare in agreement with the observations by Masselink & Pattiaratchi (2001) of the prevailingnorthward sediment transport during the summer months and the southward longshoretransport during the winter months. The findings of this study imply that the energy anddirection of swell waves are more important than those of the sea component in determiningthe directions of sediment transport on the beaches. A recommendation for further workfollowing this study is to conduct detailed investigations on the seasonal sediment transporton beaches, concurrently with the analysis of nearshore directional wave data at the samelocations. This will allow firm conclusions to be drawn on the effect of swell waves on beachsediment transport for the beaches on the coastline of Perth.74 Results and Discussion

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan6 REFERENCESBureau of Meteorology (BOM). 1993, Wind, waves, weather: Perth waters (Jurien Bay toBunbury), Department of the Environment, Sport and Territories.Coastal Engineering Research Center (CERC). 1984, Shore Protection Manual, Department ofthe Army, Waterways Experiment Station, Corps of Engineers, Vicksburg, Mississippi.Gentilli, J. 1971, Australian Climate Patterns, Thomas Nelson Australia.Gentilli, J. 1971, Climates of Australia and New Zealand (World Survey of Climatology Vol 13).Elsevier Publishing Co., Amsterdam.Goda, Y. 1995, ‘Directional wave spectrum and its engineering applications’ in Advances inCoastal and ocean engineering – Vol. 3, ed. P. Liu, World Scientific Publishing Co, Singapore,pp. 67-102.Hashimoto, N. 1995, ‘Analysis of the directional wave spectrum from field data’ in Advances inCoastal and ocean engineering – Vol. 3, ed. P. Liu, World Scientific Publishing Co, Singapore,pp. 103-143.Horikawa, K. 1988, Nearshore Dynamics and Coastal Processes: Theory, Measurement andPredictive Models, University of Tokyo Press, Tokyo.Hsu, S.A. 1988, Coastal Meteorology, Academic Press, San Diego.InterOcean systems, inc. (ISI). 2004a, S4 Current Meter Family, [Online], Available from: [11 May 2004].InterOcean systems, inc. (ISI). 2004b, S4 Current Meter Family – General Description, [Online],Available from: [11 May 2004].InterOcean systems, inc. (ISI). 2004c, S4 Current Meter Family - Theory, [Online], Availablefrom: [11 May 2004].References 75

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanInterOcean systems, inc. (ISI). 2004d, S4 Current Meter Family – Engineering Data &Specifications, [Online], Available from: [11May 2004].Johnson, D. no date, DIWASP, a directional wave spectra toolbox for MATLAB ® : User Manual.Research Report WP-1601-DJ (V1.1), Centre for Water Research, University ofWestern Australia.Komar, P.D. 1998, Beach processes and Sedimentation – 2 nd Ed, Prentice Hall, New Jersey.Lemm, A. 1996, Offshore Wave Climate Perth Western Australia, Department of EnvironmentalEngineering Honours Thesis, University of Western Australia.Linacre, E. & Hobbs, J. 1977, The Australian Climatic Environment, Wiley, Milton.Masselink, G. & Pattiaratchi, C.B. 2001, ‘Seasonal changes in beach morphology along thesheltered coastline of Perth, Western Australia’, Marine Geology, vol. 172, pp. 243-263.Pattiaratchi, C., Hegge, B., Gould, J. & Elliot, I. 1997, ‘Impact of Sea Breeze Activity onNearshore and Foreshore Processes in Southwestern Australia’, Continental Shelf Research, vol.17, no. 13, pp. 1539-1560.Rose, E. 2001, The Dynamics of Flow Between Cockburn Sound and Sepia Depression,Department of Environmental Engineering Honours Thesis, University of Western Australia.Silvester, R. & Hsu, J.R.C. 1993, Coastal Stabilization: innovative concepts, Prentice Hall, NewJersey.Tucker, M.J. & Pitt, E.G. 2001, Waves in Ocean Engineering, Elsevier Ocean Engineering BookSeries Volume 5, Elsevier Science Ltd, Oxford.U.S. Army Corps of Engineers (USACE). 2002, Coastal Engineering Manual, Engineer Manual1110-2-1100, U.S. Army Corps of Engineers, Washington, D.C. (in 6 volumes).76 References

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanWiegel, R.L. 1981, Proceedings of the conference on Directional Wave Spectra Applications,American Society of Civil Engineers, California.References 77

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanAPPENDICESAPPENDIX A: “EMEP” FORMULATIONThe EMEP formulation is characterised by an exponential function with a power that ensure anon-negative value and allow the directional spectrum to fit to a wide range of energydistributions. It is the extension to the MEP formula, therefore a brief explanation for MEPwill be given. Hashimoto (1995).The formula presented in Equation 2-8 in Section 2.3.2 can be rewritten using the uppertriangular component of the Hermitian matrix !( f ) (Hashimoto 1995):2"( f ) = H ( f # ) G( # | f )!$i i, d#Equation A-1 for i = 1,…,N0where N is the number of equations,*Hi( f ,! ) = Hm( f ,!) Hn( f ,!)[ cos{ k( xmncos!+ ymnsin!)} " i sin{ k( xmncos!+ ymnsin!)}]/Wmn( f )!mn( f )and "i( f ) =S( f ) W ( f )mn[ W mn( f ) is a weighting function to normalise and non-dimensionalise the errors in the crosspower spectra].The transfer function ( k,! )Hm# m " m( f ! ) h ( f ) cos ! sin !mH is usually expressed asm, = Equation A-2where h m( f ) and parameters !mand !mare derived from linear wave theory.Rewriting Equation 2-8 with Equation A-2 for a three-quantity measurement taken at thesame point:hm&mn( f )( f ) h ( f ) S( f )2"!(%f )= Gsin*n0$ m + $ n # m + # n| cos % % d%Equation A-3If the measurement provide the water surface elevation ! and the orthogonal components ofthe slope at the surface !xand !y, Equation A-3 becomes:78 Appendices

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan2"(#f ) A (#)! G |id#= BiEquation A-4 for i = 0,…,40where the following conditions apply:A (!) 1, A (! ) cos!, A (! ) sin!, A (! ) 2!, A (! ) 2!0=0 = 11=2=3= cosB , = Q ( f ) { kC ( f )}, Q ( f ) { kC ( f )}B 1 12/11{ }B2 13/1122= { C ( f )!C ( f )} k C ( f ) , = 2C( f ) k C ( f )B3 2222/114= sin= , Equation A-5{ }B4 22/11The directional spreading function G ( | f )! in Equation A-4 can be regarded as a probabilitydensity function defined between 0 to 2π, which can be derived using a method called themaximum entropy approach also known as Jaynes’ principle.Entropy H is defined as2"$#(!| f ) ln G( ! f ) d!H = G|Equation A-60and there exist a function G ( | f )ˆ ! to maximise H under the conditions defined by EquationA-5. This function is called the maximum entropy estimate and uses Lagrange’s unknownmultiplier method in which an auxiliary function with unknown parameters !iis defined as:2* 2*( %42*(L = ) ! G( + | f ) ln G( + | f ) d++ (,0 ) 1) ' 1 ) ! G( + | f ) d+$ + ",i ' Bi) ! G( + | f ) Ai( + ) d+&# = &# $%00i 10Equation A-7! LThe maximum entropy estimate is thus calculated by maximising L ie = 0 :! G4&( | f ) exp ' * ' * ( )" #!G ˆ ) = % 0 ( iA i) Equation A-8$ i=1The following general expression of G ( | f )! is an extension of equation Equation A-8:( | f )G * =2)(exp&'N("{ a ( f ) cos n*+ b sin n*}%#$%d*$! exp&"{ an( f ) cos n*+ bn( f ) sin n*}#'0 n=1Nn=1nnEquation A-9Appendices 79

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanwhere a n( f ) and ( f )model.b nfor n = 1,…,N are unknown parameters and N is the order of theThe application of equation Equation A-1 to the field data need to account for the errors in thecross-power spectra, and the following formula defines the errors:2*N(%!{,i) Hi( + )} exp'"( ancos n+) bnsin n+) $ d+0 & n=1#i=Equation A-10 for i = 1,…, M* N(%! exp'"( ancos n++ bnsin n+) $ d+& n=#-20 1with M being the number of remaining independent equations following the elimination ofmeaningless equations (eg. from the zero co-spectrum). If the values of !iare assumed to beindependent of each other, and the probability of their occurrence is expressed by a normaldistribution with zero mean and varianceˆ ! can be derived by minimizing ! " 2 .2! , G ( | f )Due to the nonlinearity of Equation 8-10 with respect toan, bn, the equation is difficult tosolve, therefore Newton’s technique of linearisation and iteration was utilised to obtain asolution. The solutiona ~ n, b ~ n:an, bncan be expressed in terms of an assumed approximate solutioniabnn= a~~= bnn+ a'+ b'nnEquation A-11wherea ~ n, b ~ n.a'nandb'nare the residuals between the solution an, bnand the approximate solutionSubstituting Equation A-11 into Equation A-10 produces the following linearised equationwith respect toa'n,b'n:N!( a'nXi+ bnYi)n ,n ,# = ZEquation A-12 for i = 1,…,MiN , i"'n=180 Appendices

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanwhere2#{%H ( $ )} F ( $ )! i"i Nd$0ZN , i= Equation A-132#F d$!0N( $ )2)2)&#$ ( FN(*) cos n*d*({ +i' Hi( * )} FN(*) cos n*d*!= $ 00X'!n , iZN , i$2)2)!$ ( FN(*) d*({ +i' Hi( * )} FN(*) d*!$ % 00!"Equation A-142)2)&#$ ( FN(*) sin n*d*({ +i' Hi( * )} FN(*) sin n*d*!= $ 00Y'!n , iZN , i$2)2)!$ ( FN(*) d*({ +i' Hi( * )} FN(*) d*!$ % 00!"Equation A-15FN~( = %'nnEquation A-16$ n=1N&( ) exp ( a~cos n(+ b sin n()" #!Therefore, if an approximate solution ( a' ,nb'n) is initially assumed, then the iterative use ofEquations A-12 to A-16 with Equation A-11 will yield the solution to Equation A-12. Thefollowing Akaike’s Information Criterion (AIC) is incorporated into the iterative calculationsˆ ! .to allow the minimisation to yield a reasonable and smooth result of G ( | f )2( ln 2 ˆ + 1) + 2( 2 + 1)AIC = M "!N Equation A-17where2! ˆ is the estimated variance of !i.Appendices 81

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanAPPENDIX B: MATLAB ® SCRIPTSfunction [return] = interanal(dataset, start, finish, EPMethod, GraphFileName, SavePolarGraphsp, …SaveSpectralPlot, winddat, spare)% Interval Analysis. This function processes the raw data to obtain both directional (via DIWASP) and non-% directional information. This function also allows the user to define the time interval within the data set for the% analysis to be carried out. Required data: suitable wave data file for use under DIWASP, wind data with wind% speed and direction information%% Inputs: dataset: the data file that has been loaded into MATLAB% start: specify the start time of iterations. A value of zero ('0')% will start iteration from the beginning of data% finish: specify the end time of iterations. A value of zero will end% iteration at the end of data file% EPMethod: specify the method to compute directional wave spectra% GraphfileName: the name of the final plot (wind speed, wave height,% wave frequency and wave direction)% SavePolarGraphs: 0 to not run the burst polar plots under DIWASP% >= 1 to run plots for every burst% SaveSpectralPlot: 0 to not run the spectral energy plots for each burst% >= 1 to run plots for every burst% winddat: wind data file% spare: The extra at the end of each burst (i.e. after 2048 data points have been taken out% from each hourly burst. It is 352 for the data collected at Cables ASR and 112 for% City Beachclf% number of points in a burstnumpoints = 2048;%sample ratesrate = 2;% number of points for fftnpoints = 1024;fnd= srate*(1:npoints/2)/npoints;% Find the start point by binary searchingspan = size(dataset, 1);spoint = 1;fpoint = span;if (start == 0 & finish == 0)fpoint = span;elsewhile (span > 1)span = floor(span / 2);mid = spoint + span;if (dataset(mid, 1) < start)spoint = mid;endend% Find the final point by binary searchingspan = size(dataset, 1) - spoint;fpoint = spoint;while (span > 1)span = floor(span / 2);mid = fpoint + span;if (dataset(mid, 1) < finish)fpoint = mid;endendendspointfpointdataset(spoint,1)dataset(fpoint,1)% Find the number of times to iterate (number of bursts)82 Appendices

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanbursts = floor((fpoint - spoint) / (numpoints + spare));% We use this lateralpha = ceil(spoint / (numpoints + spare));% arrays to store wave parameterswstat=ones(bursts,5);wdir=ones(bursts,361);wspect=ones(bursts,50);wavestata=ones(bursts,25);% Parameters for non-directional spectradof=4;% required degrees of freedom for the spectratcut=0.005;% cutoff frequency for long term trendicut=0.05;% cutoff frequency for infragravity wavesscut=0.15;% cutoff frequency for swell waveswcut=0.5;% cutoff frequency for wind waves% Parameters for directional spectra in DIWASPID.fs = 2;ID.datatypes={'pres' 'velx' 'vely'};%ID.depth = mean(data(:,1))+ 0.2 ;ID.layout = [ 0 0 00 0 00.5 0.5 0.5] ;SM.freqs = [0.01:0.01:0.5]; % (these are the frequency bins for the output)SM.dirs = [-180:1:180]; %(these are the directional bins for the output)M.xaxisdir = 90 ;EP.dres = 180; % (this is the number of directions in the calculation)EP.nfft = 1024; %(this is the number of DFTs in the spectral estimation)EP.iter = 100; %(this is the number of algorithm iterations)EP.smooth = 'YES' ;EP.method = EPMethod;for i=1:bursts% define section of record to analysedata=zeros(numpoints,3);epoint = spoint + numpoints - 1;disp(' ');disp(['Analyzing ', num2str(numpoints), ' points from ', num2str(spoint), ' to ', num2str(epoint), ' burst ', num2str(i),' of ', num2str(bursts)])j=spoint;% calculate velocity componentsfor k=1:numpointsdata(k,1)=dataset(j,4);dir=dataset(j,3)*pi/180.;data(k,2)=0.01*dataset(j,2)*sin(dir);data(k,3)=0.01*dataset(j,2)*cos(dir);j=j+1;end% calculate non-directional wave parameters:wavestata(i,2:10)=wavepar(data(:,1),srate,icut);wavestata(i,1)=mean(dataset(j-numpoints:j,1)); % wavepar=[c Tz Hrms Hmax Tmax Hs Ts H10 T10];[f,Pn(:,1),Pnstat]= ...autospec(data(:,1),srate,dof,tcut,icut,scut,wcut);wavestata(i,11:24)=Pnstat; % Pnstat=[Tpeak Tpswell Tpsea ...% Trend_Per Inf_Per Swell_Per Wind_Per Noise_Per ...% Hrms Trend_ht Inf_ht Swell_ht Wind_ht Noise_ht];%plotting wave spectra for each burstif (SaveSpectralPlot > 0)Pnn = Pn(1:256,1);plot(f(1:256), Pnn);xlabel('Frequency (Hz)');ylabel('Spectral Density (m^2/Hz)');% axis([0, 0.5, 0, 5]);savefileb=['spectralburst',num2str(i)];coms=['print -dbitmap ',savefileb];eval([coms]);end% calculate directional wave parametersID.depth = mean(data(:,1))+ 0.5 ;ID.data = data ;Appendices 83

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tan[SMout,EPout,H,Tp,DTp,Dp]=dirspec(ID,SM,EP,{'PLOTTYPE',2,'FILEOUT','sampleout.spec', 'MESSAGE', 0});wstat(i,1)=mean(dataset(j-numpoints:j,1));wstat(i,2)=H;wstat(i,3)=Tp;wstat(i,4)=DTp;wstat(i,5)=Dp;wspec=real(SMout.S);wdir(i,:)=mean(wspec);wspect(i,:)=mean(wspec');% This bit is for plotting that polar graphif (SavePolarGraphsp > 0)savefilea=['burst',num2str(i), sprintf('graph%d.bmp', i)];% coms=['print -djpeg ',savefilea];coms=['print -dbitmap ',savefilea];eval([coms])endspoint = spoint+numpoints+spare;enddirs1=SMout.dirs;ffreqs=SMout.freqs;xaxisdir=SMout.xaxisdir;dirs=xaxisdir*ones(size(dirs1))-dirs1;dirs=dirs+360*(dirs360);[dirs,order]=sort(dirs);clf%%%% Plotting wind data: %%%%H = subplot(411);winddir=(winddat(alpha:alpha + bursts - 1,1)+180)*pi/180;x=winddat(alpha:alpha + bursts - 1,2).*sin(winddir);y=winddat(alpha:alpha + bursts - 1,2).*cos(winddir);z = winddat(alpha:alpha + bursts - 1,2);% There’s a -1 in here for working with wind for every hour (for plotting purposes). An example shows the need% easier. Say we have an array [1 2 3 4 5 6]. The way we're calculating the end index is by saying "we start at 1% and we do 6 bursts *for 1 % inclusive*". If we simply did 1 (the start) + 6 (number of bursts) we'd end up with 7% out of bounds. So we take 1 off because the bursts count the start point.plot(z);axis manualholdfeather(x,y); % wind plothold off;axis tightaxis([1, bursts, -1.2 * max(z), 1.2 * max(z)])span = bursts - 1;set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');% Try calculating xtick dynamically. Make it tick on every day.xtick = ones(1, finish - start + 1);for i = 1:1:finish - start + 1xtick(1, i+1) = 1 + span * i / (finish - start);endset(H, 'XTick', xtick);ylabel('Wind Speed (kph)');colorbar('vert');%%%%Plotting spectral significant wave height %%%%H = subplot(412);plot(wstat(1:bursts,1),wstat(1:bursts,2))axis tight;span = wstat(bursts,1) - wstat(1,1);set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');% This makes the graph choose ugly grid lines and tick points but it's the only way to line up the grid lines% with the wind data which uses a different scale on X. When we're not using wind data we let matlab choose% the tick points.% Calculate the divisionsxtick(1, 1) = wstat(1, 1);84 Appendices

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean Tanfor i = 1:1:finish - start + 1xtick(1, i+1) = round(wstat(1,1) + span * i / (finish - start));endset(H, 'XTick', xtick);ylabel('Wave Height (m)');colorbar('vert');%%%% Plotting wave spectra in a time series %%%%H = subplot(413);pcolor(wstat(1:bursts,1),ffreqs,wspect(1:bursts,:)')set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');set(H, 'XTick', xtick);shading('interp')ylabel('Frequency (Hz)');colorbar('vert');%%%% Plotting directional wave spectra %%%%H = subplot(414);pcolor(wstat(1:bursts,1),dirs,wdir(1:bursts,order)')set(H, 'XGrid', 'on');set(H, 'XTick', xtick);shading('interp')xlabel('Time (day)');ylabel('Direction (degrees)');colorbar('vert');%%%% Saving current image %%%%savefileb = [GraphFileName, '.bmp'];coms=['print -dbitmap ', savefileb];eval([coms]);%%%% Plotting the swell and sea components of significant wave height and peak period %%%%%Significant wave heightH = subplot(411);plot(wstat(1:bursts,1),wstat(1:bursts,2),'k-');set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');ylabel('Significant Height, Hs (m)');axis tight%Swell and sea significant wave heightH = subplot(412);plot(wstat(1:bursts,1),wavestata(1:bursts,22), 'b-', wstat(1:bursts,1),wavestata(1:bursts,23), 'r-');legend('Swell Hs', 'Sea Hs');set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');ylabel('Swell & Sea Hs (m)');axis tight%Peak PeriodH = subplot(413);plot(wstat(1:bursts,1),wavestata(1:bursts,11),'k-');set(H, 'XGrid', 'on');set(H, 'XTickLabel', '');ylabel('Peak Period, Tp (s)');axis tight%Swell and sea peak periodH = subplot(414);plot(wstat(1:bursts,1),wavestata(1:bursts,12), 'b-', wstat(1:bursts,1),wavestata(1:bursts,13), 'r-');legend('Swell Tp', 'Sea Tp');set(H, 'XGrid', 'on');xlabel('Time (day)');ylabel('Swell & Sea Tp (s)');axis tight%Saves plotsavefilec=[GraphFileName, '_swellsea.bmp'];coms=['print -dbitmap ',savefilec];eval([coms]);Appendices 85

Directional waves in the nearshore coastal region of Perth, Western AustraliaHuey Jean TanTrend_Per Inf_Per Swell_Per Wind_Per Noise_Per ...Hrms Trend_ht Inf_ht Swell_ht Wind_ht Noise_ht];function P = periodo(x, srate)% P=periodo(x, srate)% Calculate one-sided periodogram% INPUT: one column time series (x)% sample frequency in hertz (srate)%% The periodogram (P) is normalized so that (length of time series, n):% variance of the (detrended) time series = sum of the spectral estimates.% The binwidth of the last (Nyquist) frequency bin is half that of the others.%% ********************Parseval's Theorum******************************% Variance == sum(x.^2)/n = sum(P(:,2))*(srate/n)%% OUTPUT P=[Freq Density Phase]% Freq - Hz (midpoint of frequency bin}% Density - units^2/Hz% Phase - radians% Authors: Bruce Hegge / Gerd Masselink (17/01/95)% Department of Geography / Center for Water Research% University of Western Australia% Nedlands, 6009% bruce@gis.uwa.edu.au / masselin@cwr.uwa.edu.au% TWO-SIDED PERIODOGRAMn=length(x);% Number of data points in time seriesXx = zeros(n,1);% Pre-allocate output vectorXx = abs(fft(x)).^2; % Power estimates of two-sided periodogramPhaXx = atan(imag(fft(x))./... % Phase estimates of two-sided periodogramreal(fft(x)));% ONE-SIDED PERIODOGRAMselect = [1;ones((n/2)-1,1).*2;1]; % Define first half of periodogramXx = Xx(1:(n/2)+1).*select; % Power estimates of one-sided periodogramXx = Xx/(srate*n);% Normalizing for Parseval's theorumfreq=(((0:n/2)'*srate)/n); % Frequency axis of one-sided periodogramP=[freq Xx PhaXx(1:(n/2)+1)];% One-sided periodogram% P=[freq, density, phase]P(1,:)=[];% Remove the DC valueAppendices 87