Long Wave Runup. Outline. u u. x 26 May 1983 Japan Sea (Shuto, 1985) Some Analytical Solutions in Long Wave Theory: Runup and Traveling Waves
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1 Some Aalytical Solutio i og Wave Theory: uup ad Travelig Wave Ira Didekulova og Wave uup Ititute of Cyberetic at Talli Uiverity of Techology, Etoia Ititute of Applied Phyic Wave iduced by high-peed ferrie Mvi_.avi Outlie uup of ymmetric log wave o a beach: ifluece of the wave hape uup of aymmetric log wave o a beach: ifluece of the wave teepe uup o the beach of pecial profile ( oreflectig beach): travelig wave olutio og wave ruup i Talli Bay Tuami Wave Shape at Japaee Coat Noliear Shallow Water Theory [( αx η ) u] = 6 May 98 Japa Sea (Shuto, 985) u u u g = (Carrier & Greepa, 958) h ( x) = αx
2 Hodograph Traformatio Φ Φ Φ = σ σ σ σ Implicit form Φ η = u g Φ u = σ σ (Carrier & Greepa, 958) = g( h η) Φ x = u αg σ Φ t = αg σ σ Explicit Solutio for Movig Shorelie if Icidet Wave i give Far from Shorelie where it i iear Peliovky & Mazova, 99 u u( = U t r ( = α αg u( dt Noliear Coordiate ad velocity of movig horelie x( r( u( Firt Step Solutio of iear Equatio For Wave Traformatio o a Beach Secod Step Noliear Movig Shorelie η( = Icidet Wave = A i ( ωt ϕ ) ( = π π A i ωt ϕ = d U ( = α dt u u( = U t r ( = α αg u( dt iear Coordiate ad velocity of o-movig horelie x= ( U( iear Coordiate ad velocity of o-movig horelie x=, (, U( Noliear Coordiate ad velocity of movig horelie x(, r(, u( Motivatio Icidet Wave Shape - - Oe-cale wave hape Time are aalyzed i literature Sie wave ad ie pule oretz ad Gau pule Solito ad N-wave Doe uup Height deped o the Icidet Wave Shape where the Icidet Wave i ymmetrical? For periodic wave ye! Phy. Fluid, 988, v., No.
3 For igle wave??? Icidet wave hape ued i literature:. Solitary Wave. Gauia Pule. oretz Pule ad everal other f f ( x) = ech ( x) f ( x) = exp( x x ( x) = ) Wave egth (Duratio) Defiitio for Pule Sie Wave = π H How thi formula ca be modified? H = μ We ugget to ue legth of the wave o / level philoophy ued to defie the igificat wave propertie f Icidet Wave Shape: example ( x) co ( πx) f = f ( x) ech ( x) f = = πh uup Height max = μ H ruup.5 = = ζ.5 = = - - ζ μ olito ie loretz rudow 8 6 Velocity 8 U max = H μu g α Breakig Parameter 6 μ U ie olito loretz rudow ruup 6 μ Br 8 loretz olito ie
4 Parameterized Formula U ruup ruup =.5H H =.5 g α rudow U rudow =.5H H = 7 g α 5 mi Idoeia Tuami, December 6 Shock Wave Applicatio: H Br = α 6 Fat Etimate of uup Characteritic J&J Cale Caadia Couple Icidet wave could be aymmetric! (two cale for face ad back lope) Noliear Deformatio V = x η( x, = η t V ( η) V = g( h η) gh ( g( h gh) u = η) Noliear Deformed Wave a Iput for uup Formula
5 .5.5 =, Br =.5 ( r( uup of teep wave.5.5 =, Br = ( r( uup of teep wave -.5 Movig horelie -.5 Movig horelie - π t π π =, Br =.5 U( u( - π t π π =, Br = U( u( - π t π π - π t π π 5, - uup накат = πa Steepet wave peetrate ilad igificatly udow откат velocity скорость 8 6 U / ~ накат uup Wave move quickly ilad откат udow крутизна teepe / крутизна S teepe Cocluio: Steep Wave Peetrate Ilad over arger Ditace ad with Greater Velocity, tha a ymmetric oe -ad Slowly ito the Sea Formula for Solitary Wave uup ca be Parameterized Thee reult are importat for egieerig etimate og Wave Dyamic above Iclied Bottom of a Special Profile h( x) u = u g = [ ] Wave Equatio iear hallow water theory η c t x ( x) gh( x) c = ( x) = 5
6 η Travelig wave olutio for arbitrary bottom profile gh x ( x) = η( x) = A( x) v( x, x τ ( x) v v = τ trogly ihomogeeou media A ( x) =? τ ( x) =? h( x) =? ~ / A(x) x τ(x)~ x / h(x)~ x / Boudary coditio Firt all, the wave equatio hould be olved o emi-axi (<τ< ) The atural boudary coditio for the reduced wave equatio i thi poit τ= i v( τ =, = It provide the boudede of the water diplacemet o the hore. I thi cae the domai ca be exteded to the whole axi (- <τ< ) ad iitial coditio hould be cotiued for τ< with ig iverio of the water diplacemet ( imagiary mirror reflectio coditio). Water diplacemet ad velocity o a beach of pecial profile η(x).. t= t= t= t= t= t= -. 5 h(x) u(x) The wave amplificatio whe the wave approache the hore ad it differetiatio o the horelie x= x= x= - η( dηi ( t τ ) ( = τ dt 5 x 5 x t Maximal ruup height if a icidet wave i the olutio of the Korteweg de Vrie equatio Ag η( = Aech t h / A A A max = = ~ A h α h / where α i the mea lope of a beach / A A max =.8 ~ A α h 5 / for a beach of pecial profile for a plae beach Syolaki (987) Cocluio The ruup of olitary wave of moderate amplitude o a beach of pecial profile lead to more eergetic amplificatio tha for the beach of cotat lope The hape of the water ocillatio i the horelie i determied by the firt derivative of the icidet wave hape 6
7 Wave iduced by high-peed ferrie Variability i hip chedule (Super)Star 7: SuperSeaCat 7:5 Vikig XPS 8: Nordic/Baltic Jet :5 SuperSeaCat : (Super)Star : Baltic Price : SuperSeaCat : (Super)Star : Nordic/Baltic Jet 5: Superfat 5: SuperSeaCat 6:5 Norladia 7: Vikig XPS 8: (Super)Star 8: SuperSeaCat 9: Nordic/Baltic Jet 9: (Super)Star : SuperSeaCat : 8: : : : 6: 8: : : Time Variability i hip route Ship wave propagatio Experimet i Talli Bay Wave tatitic 8 Water urface elevatio, mm Experimet i Talli Bay (Aega 8, Naiaar 9) Water urface elevatio, mm Time o 8 July 8 The record of water level elevatio o 8 July 8 after adjutmet of the traducer to a higher poitio (.55 m above the mea water level of thi day) :5 : :5 : Time o 8 July 8 Filtered water urface elevatio, mm Group Group Group :5 : :5 : Time o 9 July 8 The wake created by Star/Supertar o 9 July 8: (a) origial recordig (left pael), (b) low-pa filtered water level data with cut-off frequecy.5 (right pael) 7
8 Beach profile chage Wave hape ad wave ruup height, m Aega beach profile 7.6.8, -, -, -,6 -,8 -, -, -, -,6 -,8 -, ditace, m 7: 8: height, m Aega beach profile 8.6.8, -,5 -, -,5 -, -,5 5 5 ditace, m 7: : Water urface elevatio, mm SuperStar, 9 July 8 :7. :8. :8. Time o 9 July 8 height, m Aega beach profile 6.7.8, -,5 -, -,5 -, -, ditace, m 7:5 8: wave height, m umber of wave Star, July 8 wave height (amplitude), m umber of wave SuperSeaCat, 9 Jue 8 max A max H max Numerical imulatio ad compario with meauremet Directio of the future work Aalyi of extreme wave tatitic Statitical characteritic of the ruup of oliear wave o a beach og-wave ruup Numerical imulatio of log-wave propagatio Sedimet traport Never uderetimate the upredictability of rough ea! 8
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