Chapter 6 - Water Waves

2.20 Marine Hdrodnamics, Fall 2011 Lecture 14 Copright c 2011 MIT - Department of Mechanical Engineering, All rights reserved. Chapter 6 - Water Waves 2.20 - Marine Hdrodnamics Lecture 14 6.1 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: Governing equations p(x,,z,t) Continuit: 2 φ = 0 < η or F < 0 Bernoulli for P-Flow: t + 1 2 φ 2 + p pa ρ +g = 0; < η or F < 0 Far wa, no disturbance: / t, φ 0 and p = p a }{{} atmospheric ρg }{{} hdrostatic 1

Boundar Conditions 1. On an impervious boundar B(x,,z,t) = 0, we have KBC: v ˆn = φ ˆn = 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 t +( φ )B = 0 on B = 0 For example, for a flat bottom at = h B = +h = 0 ( )( ) DB Dt = ( +h) = 0 = 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 Dt ( η) = }{{} vertical velocit η t x η x }{{} slope of f.s. 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: t + 1 2 φ 2 } {{ } non-linear term + g η }{{} still unknown = 0 on = η 2

6.2 Linearized (Air) Wave Theor 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

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

6.2.2 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) coshk( +h) cosh kh η = }{{} using (4) Acos(kx ωt) where A is the wave amplitude A = H/2. {φ = gaω sin(kx ωt)ek } η }{{} = Acos(kx ωt) using(4) Exercise Verif that the obtained values for φ and η satisf Equations (1), (2), and (4). 6.2.3 Review on plane progressive waves (a) At t = 0 (sa), η = Acoskx periodic in x with wavelength: λ = 2π/k Unitsofλ : [L] λ x k K = wavenumber = 2π/λ [L -1 ] (b) At x = 0 (sa), η = Acosωt periodic in t with period: T = 2π/ω UnitsofT : [T] T t ω = frequenc = 2π/T [T -1 ], e.g. rad/sec (c) η = Acos [ k ( x ωt)] k Unitsof 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

6.2.4 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 2 = 0 ω 2 coshkh+gksinhkh = 0 ω 2 = gktanhkh φ= ga ω sin(kx ωt)f() This is the Dispersion Relation ω 2 = gktanhkh { 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 = }{{} from (6) (kh) tanh(kh) C kh = tanhkh 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 tanhkh {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

6.2.5 Solutions to the Dispersion Relation : ω 2 = gktanhkh Propert of tanh kh: tanhkh = sinhkh coshkh = 1 e 2kh 1+e 2kh = 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 or wavelength or short waves kh << 1 Need to solve ω 2 = gktanhkh 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 = gktanhkh = gk λ = g 2π T2 k k = λ = V p ( = tanhkh λ(in ft.) 5.12T λ V 2 (in sec.) ) p (b) Use numerical approximation (hand calculator, about 4 decimals ) i. Calculate C = ω 2 h/g ii. If C > 2: deeper kh C(1+2e 2C 12e 4C +...) If C < 2: shallower kh C(1+0.169C +0.031C 2 +...) No frequenc dispersion Frequenc dispersion Frequenc dispersion V p = g g gh V p = k tanhkh V p = 2π λ 7

6.3 Characteristics of a Linear Plane Progressive Wave k = 2π λ ω = 2π T H = 2A MWL A Vp λ h η(x,t) = x Define U ωa Linear Solution: η = Acos(kx ωt); φ = Ag ω 6.3.1 Velocit field coshk( +h) cosh kh 1 u(x,0,t) U o = Aω tanhkh cos(kx ωt) sin(kx ωt),where ω 2 = gktanhkh Velocit on free surface v(x, = 0,t) Velocit field v(x,, t) v(x,0,t) V o = Aωsin(kx ωt) = η t u = x = Agk ω = Aω }{{} U coshk( +h) cos(kx ωt) cosh kh coshk( +h) cos(kx ωt) v = = Agk sinhk( +h) sin(kx ωt) ω cosh kh sinhk( +h) = }{{} Aω sin(kx ωt) U u coshk( +h) = U o cosh kh e k 1 deep water shallow water v V o = sinhk( +h) e k deep water 1+ h 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

6.3.2 Pressure field Total pressure p = p d ρg. Dnamic pressure p d = ρ t. Dnamic pressure on free surface p d (x, = 0,t) p do Pressure field Shallow water Intermediate water Deep water p d ( h) p do p = p d = ρgη = 1 (no deca) ρg(η ) } {{ } hdrostatic approximation coshk( +h) p d = ρga cos(kx ωt) cosh kh coshk( +h) = ρg η cosh kh p d p do same picture as u U o p d ( h) p do = 1 cosh kh p d = ρge k η 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

6.3.3 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) = coshk(ȳ +h) = A = dt v( x,ȳ,t) = coshk(ȳ +h) dt ωa cos(k x ωt) sin(k x ωt) sinhk(ȳ +h) = A cos(k x ωt) sinhk(ȳ +h) dt ωa sin(k x ωt) Check: On ȳ = 0, = Acos(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 coshk(ȳ +h) sinhk(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

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