# wave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:

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1 3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition, roup velocity and Superposition of linear plane proressive waves. Oblique Plane Waves: z k = k k k z ( k, ) k z θ (Lookin up the y-ais from below the surface) Consider wave propaation at an anle θ to the -ais k {}}{ η =A cos( k cos θ + kz sin θ ωt) = A cos (k + k z z ωt) φ = A cosh k (y + h) sin (k cos θ + kz sin θ ωt) ω cosh kh ω =k tanh kh; k = k cos θ, k z = k sin θ, k = k + kz

2 . Standin Waves: + Same A, k, ω, no phase shift η =A cos (k ωt) + A cos ( k ωt) = A cos k cos ωt φ = A cosh k (y + h) cos k sin ωt ω cosh kh 90 o at all times y t = 0, T, T, T 3T t =,, antinode node T 3T 5T t =,, η φ nπ = sin k = 0 at = 0, k = nλ Therefore, φ = 0. To obtain a standin wave, it is necessary to have perfect reflection at the =0 wall at = 0. Define the reflection coefficient as R A R A I ( ). y A I = A R R = A R A I =

3 3. Oblique Standin Waves. η I =A cos (k cos θ + kz sin θ ωt) η R =A cos (k cos (π θ) + kz sin (π θ) ωt) z η R θ θ θ R θ I θ θ R = π θ I η I Note: same A, R =. k k z z ωt {}}{{}}{ η T = η I + η R = A cos (k cos θ) cos (kz sin θ ωt) }{{}}{{} standin wave in propaatin wave in z and Check: λ = π k cos θ ; V P z = ω k sin θ ; λ z = π k sin θ φ η sin (k cos θ) = 0 on = 0

4 4. Partial Reflection. ηi + ηr R: Comple reflection coefficient η I =A I cos (k ωt) = A I Re { e i(k ωt)} η R =A R cos (k + ωt + δ) = A I Re { R e i(k+ωt)} R = R e iδ, R = A R A I η T =η I + η R = A I Re { e ( i(k ωt) + Re ik)} η T [ =A I + R + R cos (k + δ) ] η T A I λ + R At node, antinode node At antinode, η T = η T min = A I ( R ) at cos (k + δ) = or k + δ = (n + ) π η T = η T ma = A I ( + R ) at cos (k + δ) = or k + δ = nπ kl = π so L = λ R = η T ma η T min η T ma + η T min = R (k)

5 5. Wave Group: waves, same amplitude A and direction, but ω and k very close to each other. V P η =R ( Ae i(k ω t) ) η =R ( Ae i(k ω t) ) V P ω, =ω, (k, ) and V P V P η T = η + η = R { Ae i(k ω t) [ + e i(δk δωt)]} with δk = k k and δω = ω ω A V λ π = δk V V P P T = π ω π = λ λ k T = π δω η T ma = A when δk δωt = nπ η T min = 0 when δk δωt = (n + ) π = V t, δkv t (δω) t = 0 then V = δω δk

6 In the limit, δk, δω 0, V = dω dk k k k, and since ω = k tanh kh ( ω ) ( V = + kh ) }{{} k sinh kh }{{} n (a) deep water kh >> n = V = (b) shallow water kh << n = V = (no dispersion) (c) intermediate depth < n < V Appear V Disappear VP

7 Wave Enery - Enery associated with wave motion. For a sinle plane proressive wave: Potential enery PE (per unit surface area of wave) 0 PE without wave = ρydy = ρh h η PE with wave = ρydy = ρ (η h ) h PEwave = ρη = ρa cos (k ωt) PEwave = 4 ρa averae over one period or one wavelenth Wave enery: E = PE + KE = ρa [ cos (k ωt) + ] Kinetic enery KE (per unit surface area of wave) KEwave = η h dy ρ (u + v ) = = 4 ρa }{{} for deep water KE A KE const in,t to leadin order KEwave = 4 ρa for any h averae over one period or one wavelenth Averae wave enery E (over period or wavelenth): E = ρa [ P E + KE ] = ρa = E s, which is the Specific Enery: total averae wave enery per unit surface area. Linear waves: PE = KE = E s (equipartition). E E s PE= Es cos ( k ωt) PE = Es cos (k = E PE Nonlinear waves: KE > PE. ½ KE KE= E = E s cos = + cos

8 Enery Propaation - Group velocity S E = E s per area V Consider a fied control volume V to the riht of screen S. Conservation of enery: dw dt }{{} rate of work done on S = de dt }{{} rate of chane of enery in V = J- }{{} enery flu left to riht where e.. A = 3m, T = 0 sec J- = 400KW /m η ( ) dφ J- = pu dy with p = ρ dt + y and u = dφ d h J- = ( ( ω ) ρa) [ ( )] + kh = E (nv sinh kh p ) = EV }{{}}{{} k }{{} E n }{{} V

9 Conservation of enery equation F F ( ), F F( ) E = E = h = h(). E t = 0, V E = constant in for any h(). ( ) J- J- t = E J- = J- + J- +. V = constant (i.e. constant depth, δk << k) i.e. wave packet moves at V. E t + J- = 0, but J- = V E E t + ( V E ) = 0 ( ) t + V E = 0, so E = E ( V t) or A = A ( V t)

10 Steady ship waves, wave resistance D A Vp = E = ρa ( )( A ) F = V E = ρ L E = 0 ahead of ship = 0 C.V. Wave resistance dra on ship D. Rate of work done = rate of enery increase D + J- = d ( ) EL = E dt deep water = {}}{ (E E / ) = E = D force / lenth 4 ρa enery / area Question: Amplitude A =? (depends on, eometry). Let l effective lenth. L L Superimpose a bow wave (η b ) and a stern wave (η s ): η b = a cos (k) and η s = a cos (k ( + l)) η T = η b + η s A = η T ma = a sin ( kl) D = 4 ρa = ρa sin ( kl), V p = = so k = / deep water k D = ρa sin ( l )

11 D ρa where ship lenth F = Steady ship waves (deep water) = ship speed = k = ; so k = and λ = π L =ship lenth, l L D W =ρa sin ( ( ) ( ) ) l = ρa sin = ρa sin FL FL D W ρa 0 Small, lots of wave cancellation D w ~ small π F = F π = F = "hull speed" = hull 0.56 π L hull L

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