Chapter 4. Wave Refraction, Diffraction, and Reflection 4.1 Three-Dimensional Wave Transformation Refraction (due to depth and current) Diffraction (internal or external) High energy region →(Energy transfer)→ Low energy region Internal diffraction External diffraction Refraction, diffraction, shoaling → Change of wave direction and height in nearshore area → Important for longshore sediment transport and wave height distribution → Nearshore bottom change, design of coastal structures 4.2 Wave Refraction Normal incidence on beaches with straight and parallel bottom contours ↓ No refraction, and only shoaling occurs C0 HH= 0 = KHs 0 2Cg Assuming no energy transfer across wave orthogonals, 1 2 1 2 B0 E0 B= EB ⇒ ρgH B= ρ gH B⇒ H = H 0 8 0 0 8 0 B For shoaling and refraction, C0B 0 HH= 0 = HKK0 s r 2C g B 파의 굴절과 천수 현상에 대한 비유적 설명 4.3 Manual Construction of Refraction Diagrams Wave crest method: Crest lines → Orthogonals (⊥ to crest lines) → Calculate K r Orthogonal method (Ray tracing method): • Simpler than wave crests method ( K r is directly calculated from orthogonals) • Based on Snell’s law l CΔ t l CΔ t sinα =1 = 1 ; sinα =2 = 2 1 x x 2 x x sinα C L 1 =1 = 1 (Snell’s law) sinα 2 C2 L2 sinα1 sin α 2 sinα 2 sinα 3 = → α 2 , = → α 3 , L CC1 2 CC2 3 Straight and parallel bottom contours B0 B B0 B cosα 0 = , cosα = ⇒ = x x cosα0 cos α B cosα K =0 = 0 r B cosα where ⎛ C ⎞ −1 ⎜ ⎟ α = sin⎜ sinα 0 ⎟ by Snell’s law ⎝ C0 ⎠ Graphical method is tedious and inaccurate. Moreover, it needs refraction diagrams for different wave periods and directions. Therefore, computational methods have been developed (see Dean, R.G. and Dalrymple, R.A., 1991, Water Wave Mechanics for Engineers and Scientists, World Scientific, pp. 109-112) 4.4 Numerical Refraction Analysis Let t = travel time along a ray of a wave moving with speed C(,) x y . Let the wave is located at [x( t ), y ( t )] at time t . Then dx = C cosθ (1) dt dy = C sinθ (2) dt θ (,)?x y = dCdt 1 ∂C dx= dscosθ = −dnsinθ dθ≅tan d θ = − = − ds dn C ∂n dy= ds sinθ = dn cosθ ds= Cdt ∂θ 1∂C 1 ⎛ ∂C ∂x ∂C ∂y ⎞ 1 ⎛ ∂C ∂C ⎞ = − = − ⎜ + ⎟ = −⎜ − sinθ+ cos θ ⎟ ∂s C ∂n C ⎝ ∂x ∂n ∂y ∂n⎠ C ⎝ ∂x ∂y ⎠ 1 ⎛ ∂C ∂C ⎞ = ⎜sinθ− cos θ ⎟ (4.5) C ⎝ ∂x ∂y ⎠ ∂θ ∂θ ∂s ∂θ ∂C ∂C = = C = sinθ− cos θ (3) ∂t∂ s ∂t ∂s ∂x ∂y Solve equations (1), (2), (3) for x() t , y() t , and θ (,)x y . Problems of ray tracing method: (1) Wave ray crossing B K =0 → ∞ at ray − crossing point r B HHKK= 0 s r → ∞ at ray − crossing point (2) Interpolation is needed in the region of sparse orthogonals To resolve these problems, finite-difference method is used, which solves Eq. (4.6) for θ on (,)x y grid. Wave height is also calculated at each grid point using conservation of energy equations. 4.5 Refraction by Currents (read text) 4.6 Wave Diffraction Based on Penney and Price (1952) solution Use diffraction diagrams in Shore Protection Manual (SPM, Figs. 2.28-2.39), in which θ varies from 15° to 180° with increment of 15° . In the figures, x and y are given in units of wavelength. Therefore, we need to calculate L for given T and d . Waves passing through a gap ( BL≤ 5 ) Normal incidence: BL/= 0.5, 1.0, 1.41, 1.64,L , 5.0 (SPM Figs. 2.43-2.52) Oblique incidence: BL/= 1.0;θ = 0° , 15 ° , 30 ° , 45 ° , 60 ° ,75 ° (SPM Figs. 2.55-2.57) Approximate method for oblique incidence: Use diffraction diagrams for normal incidence by replacing B by B' . See SPM Fig. 2.58 for comparison with exact solutions. If BL> 5 (wide gap), use diffraction diagram for each breakwater separately, by assuming no interaction between breakwaters. Waves behind an offshore breakwater KKKKK2= 2 + 2 + 2 cosθ DDL DR DDLR where cosθ = phase difference between two diffracted waves: ⎧ l ⎛ l ⎞⎫ θ =⎨ − INT⎜ ⎟⎬× 2π ⎩Lx ⎝ Lx ⎠⎭ For normal incidence, θ =0 ° , cosθ = 1.0 4.7 Combined Refraction and Diffraction More general approach: Use mild-slope equation (Berkhoff, 1972, Computation of combined refraction-diffraction, Proceedings of 13th International Conference on Coastal Engineering, ASCE, pp. 471-490). 4.8 Wave Reflection HCHr= R i where CR = reflection coefficient ≤ 1.0 ⎧0.1 ~ 0.2 sand beach ⎪ ⎪0.4 Tetrapod on breakwater CR ≅ ⎨ ⎪0.7 rock beach ⎩⎪0.9 vertical wall .