Chapter 4. Wave Refraction, Diffraction, and Reflection

Chapter 4. Wave Refraction, Diffraction, and Reflection

4.1 Three-Dimensional Wave Transformation

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

₀ ⁰ ₀

2 K H

C H C

H _s

g

=

=

Assuming no energy transfer across wave orthogonals,

B H B H B

gH B

gH B

E B

E0 _{0 0}² ₀ ² ₀ ⁰

8 1 8

1 = ⇒ =

⇒

= ρ ρ

For shoaling and refraction,

r s g

K K B H

B C H C

H ₀ ^{0 0} ₀

2 =

=

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

x t C x l x

t C x

l Δ

= Δ =

=

= ^{1 1} ₂ ^{2 2}

1 ; sin

sinα α

2 1 2 1 2 1

sin sin

L L CC = α =

α (Snell’s law)

L sin ,

, sin sin

sin

3 3

3 2

2 2

2 2 1

1 α α α α α

α _{= → = →}

C C

C C

Straight and parallel bottom contours

α α α

α , cos cos cos

cos

0 0 0

0

B B x

B x

B = ⇒ =

=

α α cos cos ₀

0 =

= B

K_r B

where

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⁻ ⎛ ₀

0

1 sin

sin α

α C

C by Snell’s law

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 = travel time along a ray of a wave moving with speed . Let the wave is located at at time t . Then

t C(x,y)

[

]

θ cos dt C

dx = (1)

θ sin dt C

dy = (2)

? ) , (x y = θ

θ

θ sin

cos dn

ds

dx= =− ds

n C C dn

d dCdt

d ∂

− ∂

=

−

=

≅ 1

tan θ θ

θ

θ cos

sin dn ds

dy = =

Cdt ds=

) 5 . 4 ( cos

1 sin

cos 1 sin

1 1

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂

= ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂

− ∂

−

⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ +∂

∂

∂

∂

− ∂

∂ =

− ∂

∂ =

∂

y C x

C C

y C x

C C

n y y C n x x C C n C C s

θ θ

θ θ θ

y C x

C C s

t s s

t ∂

− ∂

∂

= ∂

∂

= ∂

∂

∂

∂

= ∂

∂

∂θ θ θ θ θ

cos

sin (3)

Solve equations (1), (2), (3) for x(t), y(t), and θ(x,y). Problems of ray tracing method:

(1) Wave ray crossing

at ray crossingpoint point crossing ray

at

0 0

−

∞

→

=

−

∞

→

=

r s r

K K H H

B K B

(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 grid. Wave height is also calculated at each grid point using conservation of energy equations.

) , (x y

4.5 Refraction by Currents

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 are given in units of wavelength. Therefore, we need to calculate

y L for given T and d .

Waves passing through a gap (B≤5L)

Normal incidence: B/L=0.5,1.0,1.41,1.64,L,5.0 (SPM Figs. 2.43-2.52)

Oblique incidence: B/L=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 (wide gap), use diffraction diagram for each breakwater separately, by assuming no interaction between breakwaters.

L B>5

Waves behind an offshore breakwater

θ cos

2 2

2 2

D D D

D

D K K K K

K = + +

R L R

L

where cos = phase difference between two diffracted waves: θ

π

θ INT ×2

⎭⎬

⎫

⎩⎨

⎧ ⎟⎟

⎠

⎜⎜ ⎞

⎝

− ⎛

=

x

x L

l L

l

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 13^th International Conference on Coastal Engineering, ASCE, pp. 471-490).

4.8 Wave Reflection

i R

r C H

H =

where C_R = reflection coefficient ≤1.0

⎪⎪

⎩

⎪⎪

⎨

⎧

≅

wall vertical 9

. 0

beach rock 7

. 0

breakwater on

Tetrapod 4

. 0

beach sand 2

. 0

~ 1 . 0 CR