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 = s HK 0 2Cg

Assuming no energy transfer across wave orthogonals,

1 2 1 2 B0 0 BEBE ⇒= ρ = ρ =⇒ HHBgHBgH 0 8 0 0 8 0 B

For shoaling and refraction,

C B00 = HH 0 = 0 KKH rs 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 ΔtC l ΔtC sinα == 11 sin; α == 22 1 x x 2 x x sinα C L 1 1 == 1 (Snell’s law) sinα 2 C2 L2

1 sinsin αα 2 sinα 2 sinα 3 = → α 2 , = → α 3 , L 1 CC 2 2 CC 3

Straight and parallel bottom contours

B0 B B0 B cosα 0 = cos, α ⇒= = x x cos 0 cosαα B cosα K 0 == 0 r B cosα where

⎛ C ⎞ −1 ⎜ ⎟ α = ⎜ sinsin α 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 yxC ),( . Let the wave is located at [tytx )(),( ] at time t . Then

dx = C cosθ (1) dt dy = C sinθ (2) dt θ yx = ?),(

dCdt 1 ∂C dsdx cosθ −== dnsinθ tan dd θθ −=−=≅ ds dn C ∂n dsdy sinθ == dn cosθ = Cdtds

∂θ ∂C 11 ⎛ ∂C ∂x ∂C ∂y ⎞ 1 ⎛ ∂C ∂C ⎞ −= −= ⎜ + ⎟ ⎜−−= sin + cosθθ ⎟ ∂ Cs ∂ Cn ⎝ ∂x ∂n ∂y ∂ ⎠ Cn ⎝ ∂x ∂y ⎠

1 ⎛ ∂C ∂C ⎞ = ⎜sin − cosθθ ⎟ )5.4( C ⎝ ∂x ∂y ⎠ ∂θ ∂θ ∂s ∂θ ∂C ∂C = = C = sin − cosθθ (3) ∂ ∂st ∂t ∂s ∂x ∂y

Solve equations (1), (2), (3) for tx )( , ty )( , and θ yx ),( .

Problems of ray tracing method:

(1) Wave ray crossing

B K 0 rayat −∞→= crossing point r B

= 0 KKHH rs rayat −∞→ 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 yx ),( 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 ( ≤ 5LB )

Normal incidence: LB = L 0.5,,64.1,41.1,0.1,5.0/ (SPM Figs. 2.43-2.52)

Oblique incidence: LB = θ = 75,60,45,30,15,0;0.1/ °°°°°° (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 > 5LB (wide gap), use diffraction diagram for each breakwater separately, by assuming no interaction between breakwaters.

Waves behind an offshore breakwater

222 ++= 2 KKKKK cosθ DD L DR DD RL

where cosθ = phase difference between two diffracted waves:

⎧ l ⎛ l ⎞⎫ θ ⎨ −= INT⎜ ⎟⎬× 2π ⎩Lx ⎝ Lx ⎠⎭

For normal incidence, θ 0°= , θ = 0.1cos

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

= HCH iRr

where CR = reflection coefficient ≤ 0.1

⎧ 2.0~1.0 sand beach ⎪ ⎪ 4.0 Tetrapod on breakwater CR ≅ ⎨ ⎪ 7.0 beachrock ⎩⎪ 9.0 wallvertical