Regular Waves using Stokes Linear Theory

8/20/2019 Regular Waves using Stokes Linear Theory

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AN INTRODUCTION

WAVE MECHANICS


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Scope

• Wave generation

• Regular Linear waves

• Wave Charecteristics


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Introduction

• Ocean surface waves cause periodic loads on allman-made structures in the sea

• Responses: accelerations, displacements, internal

loads• ffects of waves ! resulting motions on ships:

" #dded resistance

" Impaired safet$

" #ffect operations of weapons ! e%uipment

" #ffect aircraft& helo operations

" #ffect humans


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Wave generation

• Waves generated '$ a ship or an$ other floating structure which ismoving, either at a constant forward speed or '$ carr$ing out anoscillator$ motion(

• Waves generated '$ the interaction 'etween wind and the sea

surface(• Waves generated '$ astronomical forces: )ides(

• Waves generated '$ earth%ua*es or su'marine landslides: )sunamis(

• Interaction of ocean currents can create ver$ large wave s$stem

• +ree surface waves generated in fluids in partiall$ filled tan*s suchas fuel or cargo tan*s on a ship(

• o single mathematical solution

• #ppro.imations re%uired: 'e aware of simplifications


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)sunami


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Wind generated wave s$stems• )he si/e of the wave s$stem is dependent on the following

factors• Wind Strength :

" )he faster the wind speed, the larger the energ$ transfer to the sea(

" Larger waves are generated '$ strong winds(

• Wind 0uration : " )he longer wind 'lows, the greater the time the sea has to 'ecomefull$ developed at that wind speed(

• Water 0epth : " Wave heights are affected '$ water depth(

" Waves traveling to 'each will turn into 'rea*ing wave '$ a deptheffect(

• +etch " +etch is the area of water that is 'eing influenced '$ the wind(

" )he larger the fetch, the more efficient the energ$ transfer 'etween

wind and sea(


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W i n d E n e r g y

Energy Dissipation

due to viscous friction

Fully Developed Wave

(Wind energy =Dissipation Energy)

Swell (low frequency long wave)

Small Wave or dying out

(Wind energy Dissipation Energy

Wave creation se%uence


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Wind-generated waves

• Sea " )rain of waves driven '$ the prevailing local

wind field

" Short-crested with the lengths of the crests onl$a few 12-34 times the apparent wavelength

" 5er$ irregular

" 6ulti-directional

" Crests are fairl$ sharp

" #pparent wave period ! apparent wave lengthvar$ continuousl$


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Wind-generated waves

• Swell " Waves which have propagated out of the area

and local wind in which the$ were generated

" o longer dependent upon the wind " Individual waves are more regular and the

crests are more rounded

" Lengths of the crests are longer: several 17-84

times the virtual wave length

" Wave height is more predicta'le


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Superposition principle• Wind waves are ver$ irregular

• Can 'e seen as a superposition ofman$ simple, regular harmonic wavecomponents, each with its ownamplitude, length, period or fre%uenc$

and direction of propagation• )o anal$/e complicated wave s$stems,it is necessar$ to *now the propertiesof the simple harmonic components " time and location-dependent pressure in

the fluid " relation 'etween wave length and wave

period

" energ$ transport, etc(


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Regular Waves: 0efinitions

• Origin ! conventions

• Crest, )rough, #mplitude 1ζa 4, 9eight 19 2 ζa 4• Wave length 1λ4, Wave ;eriod 1)4• Wave steepness 9& λ•


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=asic Categories

• 0eep water waves 1short waves4 " )he water is considered to 'e deep if the water depth, h,

is more than half the wavelength, λ " )hus, h& λ > ?&2 or λ &h @ 2

" )hese 1relativel$4 short waves do not AfeelA the seafloor(

• Shallow water waves 1long waves4 " )he water is considered to 'e shallow if the water

depth, h, is less than ?&2B of the wave length, λ

" )hus, h& λ @ ?&2B or λ &h > 2B( " )he sea floor has a ver$ large influence on the

characteristics of these 1relativel$4 long waves(


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Linear Wave theor$

• ;rogressive harmonic wave:ζ ζ

a cos1*.- ωt4

• Linear wave theor$: watersurface slope is ver$ small

• Wave steepness is small• 9armonic displacements,

velocities, accelerations ! pressures have linear relationwith wave surface elevation

• ;rofile of such a wave loo*sli*e sine& cosine

• 6otion of water particle inwave depends on depth 'elow

SWL


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Relations for Linear Waves• Continuit$ 1Laplace e%uation4

• =oundar$ Conditions " Sea 'ed

" +ree surface d$namic

" +ree surface *inematic

• 0ispersion relation: ω 2 = g.k. tanh(kh)- 0eep water: ω 2 = g.k or λ ≈ 1.56 T 2- Shallow water: ω =k.√gh or λ = T.√gh

• ;hase velocit$:

- 0eep water: c = √(g/k) or c ≈1.25√ λ ≈ 1.56 T- Shallow water: c= √gh 1critical velocit$A4


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)raDectories of water

particles


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Wave group


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Eroup 5elocit$

• In deep water,

cg c&2

• In shallow water,

cg c

Regular Waves using Stokes Linear Theory

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