4.4 Superposition of Linear Plane Progressive Waves

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Ἰφιγένεια Αλεξιάδης
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1 .0 Marine Hydrodynamics, Fall 08 Lecture 6 Copyright c 08 MIT - Department of Mechanical Engineering, All rights reserved..0 - Marine Hydrodynamics Lecture Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k = k v k k z ( k, ) k z θ (Looking up the y-ais from below the surface) Consider wave propagation at an angle θ to the -ais k {}}{ η =A cos( k cos θ + kz sin θ ωt) = A cos (k + k z z ωt) φ = ga cosh k (y + h) sin (k cos θ + kz sin θ ωt) ω cosh kh ω =gk tanh kh; k = k cos θ, k z = k sin θ, k = k + kz

2 . Standing Waves Figures: see Appendi A + Same A, k, ω, no phase shift η =A cos (k ωt) + A cos ( k ωt) = A cos k cos ωt φ = ga cosh k (y + h) cos k sin ωt ω cosh kh 90 o at all times y t = 0, T, T, T 3T t =,,L A amplitude antinode node T 3T 5T t =,, L η φ nπ = sin k = 0 at = 0, k = nλ φ Therefore, = 0. To obtain a standing wave, it is necessary to have perfect =0 reflection at the 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 Standing 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 =. and k k zz ωt {}}{{}}{ η T = η I + η R = A cos (k cos θ) cos (kz sin θ ωt) }{{}}{{} standing wave in propagating wave in z Check: λ = π k cos θ ; V P = 0; λ z = π k sin θ ; V P z = ω k sin θ φ η sin (k cos θ) = 0 on = 0 3

4 4. Partial Reflection Figures: see Appendi A 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) ] ηt A I λ free surface wave envelope + R At node, antinode node η T = η T min = A I ( R ) at cos (k + δ) = or k + δ = (n + ) π At antinode, 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) 4

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 g λ g π = δk V V P P T = π ω π = λ λ k T = π g δω η T ma = A when δk δωt = nπ η T min = 0 when δk δωt = (n + ) π g = V g t, δkv g t (δω) t = 0 then V g = δω δk 5

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

7 4.5 Wave Energy - Energy Associated with Wave Motion. For a single plane progressive wave: Potential energy PE PE without wave = PE with wave η h 0 h Energy per unit surface area of wave ρgydy = ρgh KE wave = η h Kinetic energy KE dy ρ (u + v ) ρgydy = ρg (η h ) Deep water = = 4 ρga }{{} PE wave = ρgη = ρga cos (k ωt) Finite depth = KE const in,t Average energy over one period or one wavelength PE wave = 4 ρga KEwave = 4 ρga at any h to leading order Total wave energy in deep water: E = PE + KE = ρga [ cos (k ωt) + Average wave energy E (over period or wavelength) for any water depth: E = ρga [ P E + ] = ρga = E s, KE E s Specific Energy: total average wave energy per unit surface area. ] Linear waves: PE = KE = E s (equipartition). Nonlinear waves: KE > PE. E E s PE= Es cos ( k ωt) = PE PE = E ½ = KE E = KE Recall: cos = + cos 7

8 4.6 Energy Propagation - Group Velocity S E = E s per area V Consider a fied control volume V to the right of screen S. Conservation of energy: dw dt }{{} rate of work done on S = de dt }{{} rate of change of energy in V = J- }{{} energy flu left to right where η ( ) φ J- = pu dy with p = ρ t + gy and u = φ h J- = ( ( ω ) ρga) [ ( )] + kh = E (nv sinh kh p ) = EV g }{{}}{{} k }{{} E n }{{} V g e.g. A = 3m, T = 0 sec J- = 400KW /m 8

9 4.7 Equation of Energy Conservation F F ( ), F F( ) E = E = h = h(). ( ) J- J- t = E J- = J- + J- + E t + J- = 0, but J- = V ge E t + ( Vg E ) = 0 E t = 0, V ge = constant in for any h().. V g = constant (i.e., constant depth, δk << k) ( ) t + V g E = 0, so E = E ( V g t) or A = A ( V g t) i.e., wave packet moves at V g. 9

10 4.8 Steady Ship Waves, Wave Resistance D A Vp = E = ρga ( )( ga ) F = Vg E = ρ L E = 0 ahead of ship = 0 C.V. Wavelength of generated waves To obtain the wave length, observe that the phase speed of the waves must equal in order to have stationary wave system relative to the ship. For deep water, we therefore have Ship wave resistance drag D w = ω k = deep water g =, or λ = π k g Rate of work done + Flu of energy into CV = rate of energy increase D w + J- = d ( ) EL = E dt D w force / length Amplitude of generated waves deep water = {}}{ (E E / ) = E = 4 ρga energy / area D w A The amplitude A depends on and the ship geometry. Let l effective length. L ll To approimate the wave amplitude A 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) envelope amplitude D w = 4 ρga = ρga sin ( kl) D w = ρga sin ( 0 ) gl

11 Summary Steady ship waves in deep water. = ship speed g = k = ; so k = g and λ = π g L =ship length, l L D w =ρga sin ( ( ) ( ) ) gl = ρga sin = ρga sin Fr L Fr L D w ρga ma at: F l = π hull 0.56 gl gl hull L 0 π Increasing Fl =, where l L gl Small speed Short waves Significant wave cancellation D w ~ small

12 Appendi A: Reflected and standing waves

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