Water Wave Mechanics Part 1 PDF

OUTLINE Wave Mechanics PART1 q Wave theory vs wave analysis q Wave characteristics, parameters and wave theories q Linear wave theory: solution q Dispersion relation: solution and physics q Wave particle velocities and accelerations q Particle paths q Pressure field M2198 WATER WAVES AND SEA LEVE...

OUTLINE Wave Mechanics PART1 q Wave theory vs wave analysis q Wave characteristics, parameters and wave theories q Linear wave theory: solution q Dispersion relation: solution and physics q Wave particle velocities and accelerations q Particle paths q Pressure field M2198 WATER WAVES AND SEA LEVEL WATER WAVE MECHANICS Part 1 Iñigo Losada Wave Mechanics Master Coastal Hazards-Risks, Cimate Change Impacts and Adaptation (COASTHazar) M2198 WATER WAVES AND SEA LEVEL 400.00 10.00 800.00 20.00 1200.00 30.00 40.00 Wave analysis vs wave theory 2.00 0.00 0.00 0.00 1.00 0.00 -1.00 Random process Statistical, probabilistic and spectral approach WAVE ANALYSIS Wave: deterministic mathematical model Regular monochromatic waves WAVE THEORY Wave Mechanics Wave Mechanics Wave: periodic and uniform oscillation of the water surface M2198 WATER WAVES AND SEA LEVEL Waves M2198 WATER WAVES AND SEA LEVEL 4 Swell waves M2198 WATER WAVES AND SEA LEVEL Wave Mechanics Wave Mechanics From wave theory to wave analysis M2198 WATER WAVES AND SEA LEVEL 5 Wave characteristics - Wave period T, - Wave length L. - Wave height, H Crest amplitude, AC. - Trough amplitude, AS M2198 WATER WAVES AND SEA LEVEL trough crest Wave characteristics zero up-crossing point H zero down-crossing point Wave mathematical description: - Uniform and periodic oscillation - Progressive and standing (MWL), Mean water level M2198 WATER WAVES AND SEA LEVEL Wave Mechanics (SWL), Still water level Average water surface elevation at any instant, excluding local variation due to waves and wave set-up, but including the effects of tides, storm surges and long period seiches Wave Mechanics 8 7 H y c Wave characteristics z η Long-crested waves M2198 WATER WAVES AND SEA LEVEL Wave characteristics η, free surface h, water depth Wave number k (1/m) Angular frequency ω (1/s) L x L T C= w k Wave celerity C= Wave Mechanics 2p k= L 2p w= T Wave Mechanics Cyclic frequency f (cycles/second=Hz Hertz) M2198 WATER WAVES AND SEA LEVEL f = 1 T 10 9 Ly (a) y x ky kx a k Wave crest 2 Wave crest 1 L cos a = Lx y Progressive waves Wave Mechanics æ ky ö ÷ è kx ø Vector direction provides the direction of wave propagation The scalar wave number becomes a vector Oblique incident waves Ly = 2p Ly L sin a L cos a ky = (b) Progressive vs standing waves Standing waves M2198 WATER WAVES AND SEA LEVEL a L Wave characteristics z Lx x Lx = 2p Lx kx = Wave Mechanics a = tan -1 ç k = k sin a k x = k cos a   y  k = kx i + k y j  k = k x2 + k y2 M2198 WATER WAVES AND SEA LEVEL 12 11 µ = kh o Wave Mechanics h L Deep water depth 10 p < kh<p kh< 10 p kh>p Relative water depth Intermediate water depth Intermediate Wave Mechanics Shallow water depth Shallow H/h or H/L<<1 Nondimensional parameters-wave theories Solitary Wave Cnoidal Waves Stokes Waves Small Amplitude Waves No wave theory represents all wave conditions H/h -> relative wave height H/L -> wave steepness M2198 WATER WAVES AND SEA LEVEL h L Relevant nondimensional parameters h 1 > L 2 1 h 1 < < 20 L 2 ³1 h L h 1 < L 20 Stokes waves << 1 kh o Shallow water waves M2198 WATER WAVES AND SEA LEVEL Deep 13 2 3 3 æ ö æ h ö2 ÷ ç ç ÷ g÷ ÷ gT 2 ç ç tanh ç 2p ÷ L= ç 2p ç T ÷ ÷ ç çç ÷÷ ÷ è ø ÷ ç è ø -1/ b (error < 1.7 %) Wave Mechanics ω 2 = gk tanh ( kh) Full equation can be solved numerically using Newton-Raphson or other approaches (error less than 0.75 %) Dispersion relation-approximations Fenton y McKee, (1990) Guo, (2002) y = x 2 ëé1 - exp( - x b ) ûù y = kh x = hs / gh b = 2.4908 M2198 WATER WAVES AND SEA LEVEL Linear wave theory- 2-D Progressive wave L w = T k H cos ( kx − ω t ) 2 H/h or H/L<<1 2 gT æ 2p h ö L= tanh ç ÷ 2p è L ø gT æ 2p h ö tanh ç ÷ 2p è L ø C= ¶u ¶t ax(x,z,t) and az(x,z,t) ax = ¶F p + + gz = 0 Þ ¶t r ¶w ¶t = - gz + az = r p Wave pressure p(x,z,t) - Wave Mechanics 16 x 15 ¶F ¶t Acceleration of the water particles Velocity of the water particles u(x,z,t) and w(x,z,t) ¶F ¶F u=w=¶x ¶z Solution for a 2-D (x,z) progressive wave travelling in the “x” axis positive direction z at wave celerity C Wave celerity C= 𝑯𝒈 𝐜𝐨𝐬𝐡 𝒌 𝒉 + 𝒛 𝐬𝐢𝐧(𝐤𝐱 − 𝛚𝐭) 𝟐𝝎 𝐜𝐨𝐬𝐡 𝒌𝒉 Velocity potential 𝚽(𝒙, 𝒛, 𝒕) 𝜱 𝒙, 𝒛, 𝒕 = − = Includes periodic Wave free Surface 𝜼(𝒙, 𝒕) changes in space and time ⎡ 1 ∂Φ ⎤ η ( x, t ) = ⎢ ⎥ ⎣ g ∂t ⎦ z=0 Dispersion relationship ω 2 = gk tanh ( kh) M2198 WATER WAVES AND SEA LEVEL Dispersion relationship - physics Different wave fronts with T= 2, 6, 12, 18 s 2 6 113.28 0.055 12 174 0.036 18 Constant water depth h=10 m. Solving T (s) 48 0.131 18 gT 2 = 1.56T 2 ( metros ) 2p In the table d=h, Lo is the wavelength in deep water in (m) and n is a coefficient that relates wave celerity, C, with wave group celerity, Cg 1æ 2kh ö C g = nC ® n = çç 1 + ÷ 2 è sinh ( 2kh) ø÷ L0 = ω 2 = gk tanh ( kh) 9.7 6.24 9.44 1.006 8 L (m) 3.12 k (1/m) C (m/s) Higher wave periods -> longer wave lengths Wave Mechanics Higher wave periods -> higher celerity-> waves travel faster Frequency dispersion -> f(T) M2198 WATER WAVES AND SEA LEVEL Dispersion relationship-Full solution Wave Mechanics For a given depth, d and wave period T: 1.Calculate Lo and d/Lo 2.For d/Lo obtain from the table: d/L->L , kd->k, tankh, etc M2198 WATER WAVES AND SEA LEVEL Deep and shallow water depth limits Wave Mechanics gT 2 = 1.56T 2 ( metros ) 2p e x - e- x e x + e- x sinh x = 2 2 Function cosh x = Deep water depth (kh > π) L0 = gT = 1.56T ( m / s ) 2p ω 2 = gk tanh ( kh ) ≈ gk C0 = M2198 WATER WAVES AND SEA LEVEL Dispersion relationship - physics Single wave front T=10 s h (m) 2 6 99.71 0.063 12 116.77 0.053 18 Varying depths h=2, 6, 12 y 18 m. Solving 73.1 0.085 11.67 0.14 9.97 43.7 7.36 L (m) 4.36 k (1/m) C (m/s) 20 L decreases with the water depth -> linked to shoaling and increasing wave steepness waves propagate slowlier as they move into shallower water C decreases with water depth -> • 19 The part of the front in shallower depths propagates slowlier than the part in deeper depths-> change in wave angle-> linked to refraction Wave Mechanics • M2198 WATER WAVES AND SEA LEVEL x>0 u max, w=0 ( Wave Mechanics u max ) u=0, w max 1 ω2 1 ω 2 = gk tanh ( kh) ⇒ = cosh ( kh) gk sinh ( kh) Hg cosh k ( h + z ) Φ=− sin ( kx − ω t ) 2ω cosh ( kh) Particle velocities and accelerations Progressive waves ¶w az = ¶t components ¶F w=¶z Velocity components f(x,z,t) ¶F u=¶x Acceleration f(x,z,t) ¶u ax = ¶t Velocity ) H cosh ( k ( h + z ) ) u= ω cos ( kx − ω t ) 2 sinh ( kh) ( ) Hgk cosh k ( h + z ) = cos ( kx − ω t ) 2ω cosh ( kh) ( ) Hgk sinh k ( h + z ) sin ( kx − ω t ) 2ω cosh ( kh) ( H sinh k ( h + z ) w= ω sin ( kx − ω t ) 2 sinh ( kh) = M2198 WATER WAVES AND SEA LEVEL Deep and shallow water limits Shallow water depth (kh < π/10) L = gh T ω 2 = gk tanh ( kh) ≈ gk 2 h C = gh Non dispersive waves C is not a function of T Wave Mechanics If in shallow water depths astronomical tide and wind waves propagate at the same celerity for a given h M2198 WATER WAVES AND SEA LEVEL 22 Particle velocities and accelerations kh< ) Hgk cos ( kx − ω t ) 2ω ( = p 10 ( No “z” dependence ) ) velocities (i.e. astronomical tides, storm surge, tsunamis…) ( ) Hgk cosh k ( h + z ) cos ( kx − ω t ) 2ω cosh ( kh) ( ) H sinh k ( h + z ) ω sin ( kx − ω t ) 2 sinh ( kh) ( 23 1 ω2 1 = cosh ( kh) gk sinh ( kh) Hgk sinh k ( h + z ) sin ( kx − ω t ) 2ω cosh ( kh) w= = ω 2 = gk tanh ( kh) ⇒ Wave Mechanics 24 Relevant for long waves h/L<<1 Hgk k ( h + z ) sin ( kx − ω t ) 2ω max x>0 Wave Mechanics umax 1 = w kh w= u= Shallow water Deep and shallow water depth limits ) Hgk cosh k ( h + z ) cos ( kx − ω t ) 2ω cosh ( kh) ( u= ) Hgk sinh k ( h + z ) sin ( kx − ω t ) 2ω cosh ( kh) ( w= M2198 WATER WAVES AND SEA LEVEL ) H cosh k ( h + z ) u= ω cos ( kx − ω t ) 2 sinh ( kh) Particle velocities and accelerations ) Accelerations for progressive waves ( ) ∂u H 2 cosh k ( h + z ) = ω sin ( kx − ω t ) ∂t 2 sinh ( kh) ( ( ) cosh k ( h + z ) H = gk sin ( kx − ω t ) 2 cosh ( kh) ( sinh k ( h + z ) H gk cos ( kx − ω t ) 2 cosh ( kh) sinh k ( h + z ) ∂w H = − ω2 cos ( kx − ω t ) ∂t 2 sinh ( kh) =− M2198 WATER WAVES AND SEA LEVEL Particle trajectories H 1 H , b= 2 kh 2 Shallow water depth En z = 0, a = H cosh ( k ( h + z1 ) ) 2 sinh ( kh ) H 1 , b =0 2 kh a= H sinh ( k ( h + z1 ) ) 2 sinh ( kh) En z = - h, a = b= M2198 WATER WAVES AND SEA LEVEL z 2 x2 + =1 a2 b 2 Water particle trajectories linear wave theory Closed trajectories in g h Intermediate water depth En z = 0, a = En z = - h, a = a= b= 1 , b =0 H cosh ( k ( h + z1 ) ) 2 sinh ( kh ) a= H kz e =a 2 H kz e 2 Circles Deep water depth b= 25 26 H 1 H , b= 2 tanh ( kh ) 2 H 2 sinh ( kh ) H H En z = 0, a = , b = 2 2 Deep water depth Intermediate water depth Asymptotic values in deep water Wave Mechanics H sinh ( k ( h + z1 ) ) 2 sinh ( kh) H En z = - h, a = b = kh 2e Wave Mechanics Ellipse axes α and β. 𝜁 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑤𝑎𝑡𝑒𝑟 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑜𝑣𝑒𝑟 𝑎 𝑤𝑎𝑣𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 H 1 HT a= = 2 kh 4p Shallow water depth 𝜉 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑤𝑎𝑡𝑒𝑟 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑜𝑣𝑒𝑟 𝑎 𝑤𝑎𝑣𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 Asymptotic values in shallow water Hæ zö b = ç1+ ÷ 2è hø en z = - h, b = 0 M2198 WATER WAVES AND SEA LEVEL Pressure field ) p r = - gz + ¶F ¶t Dynamic pressure No information between z=0 and the crest!! (i.e. in the absence of waves) Hydrostatic pressure ¶F p + + gz = 0 Þ ¶t r ( Wave Mechanics 27 28 cosh ( kh) cosh ( k ( h + z ) ) Wave Mechanics con K P = H cosh k ( h + z ) cos ( kx − ω t ) − h ≤ z ≤ 0 2 cosh ( kh) - General expresión based on Bernoulli equation Progressive waves p = − ρ gz + ρ g Kp (z) pressure response function p = − ρ gz + ρ gη K P = ρ g (η K P − z ) In phase with the free surface M2198 WATER WAVES AND SEA LEVEL Wave particle trajectories In this figure d=h M2198 WATER WAVES AND SEA LEVEL ( ) L cos a = Lx a L S = Lkxx x + k y y − ω t = x x + k y − ωt) (a) y k cos (θ ) x + k sin (θ ) y − ω t y Oblique incident waves z L Wave phase S, x Free Surface η(x,y,t) H η ( x, y, t ) = cos ( k 2 ) y kx aθ x ky k Wave Mechanics y k is now a vector ¶F ¶y Velocity-alongshore component v (b) =- ¶v ¶t 29 30 acceleration-alongshore component ay = We add this term between z=0 and the crest!! Wave Mechanics Hg cosh k ( h + z ) sin k x x + k y y − ω t 2ω cosh ( kh) ( Velocity potential Φ (x,y,z,t) Φ=− M2198 WATER WAVES AND SEA LEVEL Pressure field p = r g (h - z ) 0 < z < h M2198 WATER WAVES AND SEA LEVEL WATER WAVE MECHANICS Part 1 Iñigo Losada Wave Mechanics Master Coastal Hazards-Risks, Cimate Change Impacts and Adaptation (COASTHazar) M2198 WATER WAVES AND SEA LEVEL