6.4 Superposition of Linear Plane Progressive Waves

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1 .0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( 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) ga cosh k (y + h) φ = 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 + Same A, k, ω, no phase shift η =A cos (k ωt)+ A cos ( k ωt) =A cos k cos ωt ga cosh k (y + h) φ = cos k sin ωt ω cosh kh 90 o at all times y t = 0, T, T, A amplitude node 3 t = T, T, L T 3 T 5T antinode t =,, L η φ nπ nλ = sin k = 0 at = 0, = k Therefore, φ =0. To obtain a standing wave, it is necessary to have perfect reflection at the wall at = 0. A R Define the reflection coefficient as R ( ). A I y A I = A R A R A I R = =

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 η θ I θ θ R θ I θ R = π θ I Note: same A, R =. and k k z z 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: π π ω λ = ; V P =0; λ z = ; V Pz = k cos θ k sin θ k sin θ φ η sin (k cos θ) =0 on = 0 3

4 4. Partial Reflection η I + ηr { } η I =A I cos (k ωt) = A I Re e ik ωt { } ik ωt η R =A R cos (k + ωt + δ) = A I Re R e R: Comple reflection coefficient A R R = R e iδ, R = AI { ( )} η +Re ik T =η I + η R = A I Re e ik ωt [ ] η T =A I + R + R cos (k + δ) free surface η T wave envelope A I λ + R At node, antinode node η T = η T = A I ( R ) at cos (k + δ) = ork + δ = (n +) π At antinode, η T = η T = A I ( + R ) atcos (k + δ) = or k + δ = nπ λ kl = π so L = = η T η T R η T + η T = 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 ik ω t ( ) η =R Ae ik ω t V P ω, =ω, (k, )and V P V P { [ ]} η + e iδk δωt T = η + η = R Ae ik ω t with δk = k k and δω = ω ω A V g λ g π = δ k V V P P T = π ω π =λ λ k π T = g δω η T = A when δk δωt = nπ g = V g t, δkv g t (δω) t = 0 then V g = η T = 0 when δk δωt = (n +) π δω δk 5

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

7 6.5 Wave Energy - Energy Associated with Wave Motion. For a single plane progressive wave: Potential energy PE PE without wave = ρgydy = ρgh PE with wave η h h Energy per unit surface area of wave ρgydy = ρg (η h ) PE wave = ρgη = ρga cos (k ωt) KE wave = η h Deep water = = Finite depth = Kinetic energy KE dy ρ (u + v ) ρga }{{} KE const in,t Average energy over one period or one wavelength PE wave = ρga KEwave = 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 [ + ] = ρga = E s, PE 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 ( ω t) PE = E ½ Recall: cos = + cos KE = E = 7

8 6.6 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 de = = J- }{{} dt }{{} dt }{{} rate of work done on S rate of change of energy in V energy flu left to right where η ( ) dφ φ J- = pu dy with p = ρ + gy and u = dt h ( ) ( ω ) [ ( )] kh J- = ρga + = E (n )= EV kh g }{{}}{{} k }{{} E n }{{} e.g. A = 3m, T = 0 sec J- = 400KW /m V g 8

9 6.7 Equation of Energy Conservation E = E(, ) = ( ) h = h() ( ) J- J- t = E J- J- = J- + + E J- + = 0, but J- = V g E t E ( ) + V g E =0 t E. = 0,V g E = constant in for any h(). t. V g = constant (i.e., constant depth, δk << k) ( ) + V g E = 0, so E = E ( V g t)or A = A ( V g t) t i.e., wave packet moves at V g. 9

10 6.8 Steady Ship Waves, Wave Resistance D A Vp = U U E = ρga g = V E = ( U )( ρga ) L E = 0 ahead of ship = 0 C.V. Ship wave resistance drag D w Rate of work done = rate of energy increase d ( ) D w U + J- = EL = EU dt deep water {}}{ / D w = (EU EU ) = E = ρga D w A force / length U energy / area Amplitude of generated waves The amplitude A depends on U and the ship geometry. Let l effective length. To approimate the wave amplitude A superimpose a bow wave (η b ) and a stern wave (η s ). η b L + - l = a cos (k) and η s = a cos (k ( + l)) η T = η b + η s ( ) A = η T =a sin kl envelope amplitude ( ) ( ) gl D w = ρga = ρga sin kl D w = ρga sin U Wavelength of generated waves To obtain the wave length, observe that the phase speed of the waves must equal U. For deep water, we therefore have ω deep g U = U = U = U, or λ = π k water k g - 0

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

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