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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