Chapter 4 Water Waves

2.20 Marine Hdrodnamics, Fall 2018 Lecture 15 Copright c 2018 MIT - Department of Mechanical Engineering, All rights reserved. Chapter 4 Water Waves 2.20 - Marine Hdrodnamics Lecture 15 4.1 Exact (Nonlinear) Governing Equations for Surface Gravit Waves, Assuming Potential Flow Free surface definition = η( x, z, t) or F( x,, z, t) = 0 z x B( x,, z, t) = 0 Unknown variables Velocit field: v (x,, z, t) = φ (x,, z, t) Position of free surface: = η (x, z, t) or F (x,, z, t) = 0 Pressure field: p (x,, z, t) Governing equations Continuit: 2 φ = 0 < η or F < 0 Bernoulli for P-Flow: φ t + 1 2 φ 2 + p pa ρ + g = 0; < η or F < 0 Far wa, no disturbance: φ/ t, φ 0 and p = p a 1 atmospheric ρg hdrostatic

Boundar Conditions 1. On an impervious boundar B (x,, z, t) = 0, we have KBC: v ˆn = φ ˆn = φ n = U ( x, t) ˆn ( x, t) = U n on B = 0 Alternativel: a particle P on B remains on B, i.e., B is a material surface. For example if P is on B at t = t 0, P stas on B for all t. so that, following P B is alwas 0. B( x P, t 0 ) = 0, then B( x P (t), t) = 0 for all t, DB Dt = B + ( φ ) B = 0 on B = 0 t For example, for a flat bottom at = h B = + h = 0 ( ) ( ) DB φ Dt = ( + h) = 0 φ = 0 on B = + h = 0 =1 2. On the free surface, = η or F = η(x, z, t) = 0 we have KBC and DBC. KBC: free surface is a material surface, no normal velocit relative to the free surface. A particle on the free surface remains on the free surface for all times. DF Dt = 0 = D ( η) = φ Dt vertical velocit η t φ x η x slope of f.s. φ z η z slope of f.s. on = η still unknown DBC: p = p a on = η or F = 0. Appl Bernoulli equation at = η: φ t + 1 2 φ 2 non-linear term + g η still unknown = 0 on = η 2

4.2 Linearized (Air) Wave Theor Random seas: see figures in Appendix A below Assume small wave amplitude compared to wavelength, i.e., small free surface slope A λ << 1 Vp Wave height H crest SWL H=2A Wave amplitude A Water depth h trough wavelength Wave period T Consequentl We keep onl linear terms in φ, η. φ λ 2 /T, η λ << 1 For example: () =η = () }{{ =0 + η } () =0 +... Talor series keep discard 3

4.2.1 BVP In this paragraph we state the Boundar Value Problem for linear (Air) waves. = 0 = -h 2 φ φ + g = 0 t 2 2 φ = 0 φ = 0 Finite depth h = const Infinite depth GE: 2 φ = 0, h < < 0 2 φ = 0, < 0 BKBC: FSKBC: FSDBK: φ = 0, = h φ 0, φ = η t, = 0 2 φ + g φ t 2 = 0 φ t + gη = 0, = 0 Introducing the notation {} for infinite depth we can rewrite the BVP: Constant finite depth h {Infinite depth} { 2 φ = 0, h < < 0 2 φ = 0, < 0 } (1) φ = 0, = h { φ 0, } (2) 2 φ t + g φ { 2 2 = 0, = 0 φ t + g φ } 2 = 0, = 0 (3) Given φ calculate: η (x, t) = 1 φ g t p p a = ρ φ }{{ t} dnamic =0 ρg hdrostatic { η (x, t) = 1 φ g t { p p a = ρ φ }{{ t} dnamic =0 } ρg hdrostatic } (4) (5) 4

4.2.2 Solution Solution of 2D periodic plane progressive waves, appling separation of variables. We seek solutions to Equation (1) of the form e iωt with respect to time. Using the KBC (2), after some algebra we find φ. Upon substitution in Equation (4) we can also obtain η. φ = ga ω sin (kx ωt) cosh k ( + h) cosh kh η = using (4) A cos (kx ωt) {φ = gaω sin (kx ωt) ek } η = A cos (kx ωt) using(4) where A is the wave amplitude A = H/2. Exercise Verif that the obtained values for φ and η satisf Equations (1), (2), and (4). 4.2.3 Review on plane progressive waves (a) At t = 0 (sa), η = A cos kx periodic in x with wavelength: λ = 2π/k Units of λ : [L] λ x k K = wavenumber = 2π/λ [L -1 ] (b) At x = 0 (sa), η = A cos ωt periodic in t with period: T = 2π/ω Units of T : [T ] T t ω = frequenc = 2π/T [T -1 ], e.g. rad/sec (c) η = A cos [ k ( x ωt)] k Units of ω [ ] L k : T Following a point with velocit ω ( ω ) k, i.e., x p = t + const, the phase of η does k not change, i.e., ω k = λ T V p phase velocit. 5

4.2.4 Dispersion Relation So far, an ω, k combination is allowed. However, recall that we still have not made use of the FSBC Equation (3). Upon substitution of φ in Equation (3) we find that the following relation between h, k, and ω must hold: 2 φ t + g φ 2 = 0 ω 2 cosh kh + gk = 0 ω 2 = gk tanh kh φ= ga ω sin(kx ωt)f() This is the Dispersion Relation ω 2 = gk tanh kh { ω 2 = gk } (6) Given h, the Dispersion Relation (6) provides a unique relation between ω and k, i.e., ω = ω(k; h) or k = k(ω; h). Proof 1 C kh kh =f(c) tanh kh kh C ω2 h g C kh = from (6) = tanh kh (kh) tanh (kh) obtain unique solution for k Comments - General As ω then k, or equivalentl as T then λ. - Phase speed V p λ T = ω k = g k tanh kh {V p = } g k Therefore as T or as λ, then V p, i.e., longer waves are faster in terms of phase speed. - Water depth effect For waves the same k (or λ), at different water depths, as h then V p, i.e., for fixed k V p is fastest in deep water. - Frequenc dispersion Observe that V p = V p (k) or V p (ω). This means that waves of different frequencies, have different phase speeds, i.e., frequenc dispersion. 6

4.2.5 Solutions to the Dispersion Relation : ω 2 = gk tanh kh Propert of tanh kh: tanh kh = cosh kh = 1 e 2kh 1 + e 2kh = To determine regime: C ω 2 h/g long waves shallow water { {}}{ kh for kh << 1. In practice h < λ/20 1 for kh > 3. In practice h > λ 2 short waves deep water Shallow water waves Intermediate depth Deep water waves or long waves (C < 0.1) or wavelength or short waves (C > π) kh << 1 Need to solve ω 2 = gk tanh kh kh >> 1 h < λ/20 given ω, h for k h > λ/2 (given k, h for ω - eas!) ω 2 = gk kh ω = gh k (a) Use tables or graphs (e.g.jnn fig.6.3) ω 2 = gk λ = gh T ω 2 = gk tanh kh = gk λ = g 2π T 2 k k = λ λ = V p V p = tanh kh (b) Use numerical approximation (hand calculator, about 4 decimals ) i. If C > 2: deeper kh C(1 + 2e 2C 12e 4C +...) If C < 2: shallower kh C(1 + 0.169C + 0.031C 2 +...) ( λ(in ft.) 5.12T 2 (in sec.) ) No frequenc dispersion Frequenc dispersion Frequenc dispersion V p = g g gh V p = k tanh kh V p = 2π λ 7

4.3 Characteristics of a Linear Plane Progressive Wave Wave kinematics: see figures in Appendix B below k = 2π λ ω = 2π T H = 2A MWL A Vp λ h η(x,t) = x Define U ωa Linear Solution: η = A cos (kx ωt) ; 4.3.1 Velocit field φ = Ag ω cosh k ( + h) cosh kh sin (kx ωt), where ω 2 = gk tanh kh 1 u(x, 0, t) U o = Aω cos (kx ωt) tanh kh v(x, Velocit on free surface v(x, = 0, t) Velocit field v(x,, t) 0, t) V o = Aω sin (kx ωt) = η t u = φ x = Agk ω = Aω U cosh k ( + h) cos (kx ωt) cosh kh cosh k ( + h) cos (kx ωt) v = φ = Agk sinh k ( + h) sin (kx ωt) ω cosh kh sinh k ( + h) = Aω sin (kx ωt) U u U o = cosh k ( + h) cosh kh e k 1 deep water shallow water v V o = sinh k ( + h) e k 1 + h deep water shallow water u is in phase with η v is out of phase with η 8

Velocit field v(x, ) Shallow water Intermediate water Deep water 9

4.3.2 Pressure field Total pressure p = p d ρg. Dnamic pressure p d = ρ φ t. Dnamic pressure on free surface p d (x, = 0, t) p do Pressure field Shallow water Intermediate water Deep water cosh k ( + h) p d = ρgη p d = ρga cos (kx ωt) cosh kh cosh k ( + h) = ρg η cosh kh p d = ρge k η p d p do same picture as u U o p d ( h) p do = 1 (no deca) p = ρg(η ) hdrostatic approximation p d ( h) p do = 1 cosh kh p d ( h) p do = e k p = ρg ( ηe k ) x V p = gh x V p = gλ 2π kh << 1 kh >>1 p p( h) d p do p( h) Pressure field in shallow water p p( h) d o p do p( h) Pressure field in deep water 10

4.3.3 Particle Orbits ( Lagrangian concept) Let x p (t), p (t) denote the position of particle P at time t. Let ( x; ȳ) denote the mean position of particle P. The position P can be rewritten as x p (t) = x + x (t), p (t) = ȳ + (t), where (x (t), (t)) denotes the departure of P from the mean position. In the same manner let v v( x, ȳ, t) denote the velocit at the mean position and v p v(x p, p, t) denote the velocit at P. ( x', ') ( x, ) P ( x, P P ) v p = v( x + x, ȳ +, t) = TSE v p = v ( x, ȳ, t) + v x ( x, ȳ, t) x + v ( x, ȳ, t) +... v p = v ignore - linear theor To estimate the position of P, we need to evaluate (x (t), (t)): x = dt u ( x, ȳ, t) = cosh k (ȳ + h) = A = dt v ( x, ȳ, t) = cosh k (ȳ + h) dt ωa cos (k x ωt) sin (k x ωt) sinh k (ȳ + h) = A cos (k x ωt) sinh k (ȳ + h) dt ωa sin (k x ωt) Check: On ȳ = 0, = A cos (k x ωt) = η, i.e., the vertical motion of a free surface particle (in linear theor) coincides with the vertical free surface motion. It can be shown that the particle motion satisfies x 2 a 2 + 2 b 2 = 1 (x p x) 2 a 2 + ( p ȳ) 2 b 2 = 1 cosh k (ȳ + h) sinh k (ȳ + h) where a = A and b = A, i.e., the particle orbits form closed ellipses with horizontal and vertical axes a and b. 11

crest V p (a) deep water kh >> 1: a = b = Ae k circular orbits with radii Ae decreasing exponentiall with depth k A trough A k Ae (b) shallow water kh << 1: A Vp = gh A a = = const. ; b = A(1+ ) kh h decreases linearl with depth A/kh V p (c) Intermediate depth P V p A Q S Q S R R λ 12 R

4.3.4 Summar of Plane Progressive Wave Characteristics f() Deep water/ short waves kh > π (sa) Shallow water/ long waves kh << 1 cosh k(+h) = f cosh kh 1 () e k 1 e.g.p d cosh k(+h) e.g.u, a = f 2 () e k 1 kh sinh k(+h) e.g. v, b = f 3 () e k 1 + h 13

C (x) = cos (kx ωt) S (x) = sin (kx ωt) (in phase with η) (out of phase with η) η A = C (x) u = C (x) f Aω 2 () v = S (x) f Aω 3 () p d ρga = C (x) f 1 () A = C (x) f 3 () x A = S (x) f 2 () a = f A 2 () b = f A 3 () b a 14

Appendix A: Random Seas 15

Appendix B: Wave kinematics 16