Chapter 2. Two-Dimensional Wave Equations and Wave Characteristics

Chapter 2. Two-Dimensional Wave Equations and Wave Characteristics

2.1 Surface Gravity Waves

− Classification of waves (See handout, Shore Protection Manual, 1984, Fig. 2-1):

• capillary waves: T ≤0.1s; surface tension is important

• gravity waves (or wind waves): 1≤ T ≤30s; mostly generated by wind, and gravity is primary restoring force; the most important in coastal engineering

- seas (풍파): waves generated by local wind - swell (너울): waves propagating out of storms

• infragravity waves: 30s≤ T ≤5min; important for surf beat and harbor oscillation

• long-period waves: T ≥5min; storm surge, tsunamis, tides

− Regular versus irregular waves:

• regular ( or sinusoidal, monochromatic) waves: uniform period and amplitude

• irregular ( or random) waves: superposition of a number of sinusoidal waves with different periods, amplitudes, directions, and phases (See handout, Goda, 2000, Fig. 2.8):

2.2 Small-Amplitude (or Linear) Wave Theory

d = water depth; L= wavelength; T= wave period; C= wave celerity (speed);

H= wave height; a=H/2= wave amplitude; η = surface displacement;

δ = boundary layer thickness; ν_t= eddy viscosity

Assuming inviscid and incompressible fluid and irrotational flow motion, velocity potential φ(x,z,t) exists, which satisfies the Laplace equation (cf. Elementary Fluid Mechanics, Street et al.):

2 0

2

2 2

∂ = +∂

∂

∂

z x

φ φ

↑

=0

∂ +∂

∂

∂ z w x

u (continuity equation)

u x

∂

= ∂φ

, w z

∂

= ∂φ ← definition of velocity potential

Boundary value problem

Assume the free surface varies sinusoidally in both space and time so that )

2 cos(

) ,

( H kx t

t

x σ

η = −

where

k = 2Lπ (wave number) T

σ = ²π (wave angular frequency) Ú

f =T1 (wave frequency)

At t =0, H kx

x cos

) 2 0 ,

( =

η

At x=0, H t

H t

t σ σ

η cos

) 2 2 cos(

) , 0

( = − =

Small-amplitude wave theory assumes L∼O(d)

H /L = wave steepness << 1 η ∼O(H)

t∼O(T) x∼O(L)

z∼O(d)∼O(L) u x

∂

= ∂φ ∼ _⎟

⎠

⎜ ⎞

⎝

⎛ T

O H

φ ∼ ⎟

⎠

⎜ ⎞

⎝

⎛ T O HL

DFSBC

( )

∂ +∂ + +

+ z

t gz p

w

u 0 at

2

1 2 2

( )

∂ +∂ +

+ z

g t w

u 0 at

2

1 2 2

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

2 2

T

H (gH) ⎟

⎠

⎜ ⎞

⎝

⎛ T2

HL

<<1

⎟⎠

⎜ ⎞

⎝

⎛ L

H ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ L

gT²

( )

φ η

η = =

∂

+∂ z

g t 0 at

Taylor series expansion about z =0 gives

0

0 0

=

⎟ +

⎠

⎜ ⎞

⎝

⎛

∂ +∂

∂ + ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂ +∂

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂ +∂

=

=

=

L

z z

z g t

z g t

gη φt η φ η η φ

η

(gH) ⎟

⎠

⎜ ⎞

⎝

⎛ T2

HL ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ L g H

2

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

2 2

T H

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ L gT²

(

)

⎠

⎜⎜ ⎞

⎝

⎛

L H L gT²

_⎟

⎠

⎜ ⎞

⎝

⎛ L H

Now

0 at

0 =

∂ =

+∂ z

gη φt (LDFSBC)

Assume

) sin(

) ( ) , ,

(x z t Z z kx σt

φ = −

Substitution into the Laplace equation and applying BBC and LDFSBC give

) cosh sin(

) ( cosh ) 2

, ,

( kx t

kd z d k g

t H z

x σ

φ = σ + −

KFSBC

η η ₌η

∂ + ∂

∂

= ∂ z

u x

w t at

Order of magnitude analysis and Taylor series expansion about z =0 gives

at =0

∂

= ∂ z

w ηt

(LKFSBC)

Using LDFSBC,

1 at 0

2

2 =

∂

− ∂

∂ =

= ∂ z

g t

w ηt φ

Using w=∂φ/∂z,

0 at

2 0

2

=

∂ = + ∂

∂

∂ z

g z t

φ

φ (LCFSBC)

Dispersion Relationship

Substitution of φ into LKFSBC gives kd

gk tanh

2 =

σ (dispersion relationship)

L d kd gL

k g k

kd gk

k T

C L π

π

σ 2

2 tanh tanh tanh

=

=

=

=

=

L d g L

T

π π

π 2

2 tanh )

2 (

2 2

=

L d g

T

L π

π tanh2

2 = 2

L d

C gT π

π tanh2

= 2

L d

L gT π

π tanh2 2

2

= ← proves

L gT²

∼O(1) in previous order of magnitude analysis

Waves disperse due to difference of T(=2π /σ) ⇒ frequency dispersion

The wave period is constant as a wave propagates from deep to shallow water (i.e., varies). But

) (T

d L and C vary as d varies.

The dispersion relationship should be solved for k (or L) for given d and σ (or ) by iteration using Newton-Raphson method. Approximate formulas are available.

T

Eckart (1951):

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛ d

gk g

2

2 tanh σ

σ

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

g d T

L gT ₂

2

2 4

2 tanh

π π

Hunt (1979):

∑

+ +

= ₆

1 2

2

1 )

(

n n ny c y y

kd

where

g d

y=σ² / and c₁ =0.6666666666,c₂ =0.3555555555,c₃ =0.1608465608, 0065407983

. 0 , 0217540484 .

0 , 0632098765 .

0 _{5 6}

4 = c = c =

c .

2.3 Wave Classification

Relative depth = L

d or _⎟

⎠

⎜ ⎞

⎝⎛=

L kd 2πd

In deep water, tanhkd ≅1, so that

π

π π

2

2 2

2

0

0 0

L gT

gL gT C

=

=

=

No dependence on d

In shallow water, tanhkd ≅kd, so that

gd

C = → No dependence on T (Shallow water wave is non-dispersive) T

gd L=

In intermediate depth water,

↓

↓

⇒

=

= C L d

L d L

L C

C 2 , as

tanh

0 0

π

2.4 Wave Kinematics and Pressure

) sinh cos(

) ( cosh

) cosh cos(

) ( cosh 2

t kd kx

z d k T

H

t kd kx

z d k kg

H u x

π σ σ σ φ

+ −

=

+ −

∂ =

= ∂

) sinh sin(

) (

sinh kx t

kd z d k T

H

w φz =π + ₋σ

∂

= ∂

Acceleration

) sinh sin(

) ( cosh 2

) sinh sin(

) ( cosh

2 2

t kd kx

z d k T

H

t kd kx

z d k T

H t

u z w u x u u t u dt du

π σ σπ σ

+ −

=

+ −

∂ =

≅ ∂

∂ + ∂

∂ + ∂

∂

= ∂

) sinh cos(

) ( sinh 2

) sinh cos(

) ( sinh

2 2

t kd kx

z d k T

H

t kd kx

z d k T

H t

w dt dw

π σ σπ σ

+ −

−

=

+ −

−

∂ =

≅ ∂

Acceleration is 90° out of phase with velocity so that

max max

at w dt

du⎟

⎠

⎜ ⎞

⎝

⎛

max max

at u dt

dw⎟

⎠

⎜ ⎞

⎝

−⎛

Particle orbit

) sin(

) sinh sin(

) ( cosh 2

) sinh sin(

) ( cosh 1

t kx A t kd kx

z d k H

t kd kx

z d k T

udt H

σ σ

π σ ς σ

−

−

= + −

−

=

+ −

−

=

=

∫

) cos(

) sinh cos(

) ( sinh

2 kx t B kx t

kd z d k

wdt H σ σ

ε =

∫

1 ) (

cos ) (

sin^{2 2}

2 2

=

− +

−

⎟ =

⎠

⎜ ⎞

⎝ +⎛

⎟⎠

⎜ ⎞

⎝

⎛ kx t kx t

B

A ε σ σ

ς ∴ Ellipse

In deep water,

kz kd

z d k

H e e

e A H

2 2

) (

=

= ⁺

A H e

e e

B H _kd ^kz

z d

k = =

= ⁺

2 2

) (

∴ Circle

Diameter of the circle decreases exponentially with depth

→ No wave effect on sea bed (See Figure 2.2 in textbook)

In shallow water,

) 1 (

2 f z

kd

A= H ≠ ← A is constant over depth

⎪⎩

⎪⎨

⎧

−

=

= =

⎟⎠

⎜ ⎞

⎝⎛ + + =

=

d z H z d

z H kd

z d k B H

at 0

0 2 at

2 1 ) ( 2

The ellipse becomes flatter and flatter with depth (see Figure 2.2 in textbook)

Water particle motions in deep, shallow, and intermediate-depth waters

Wave pressure

The unsteady Bernoulli equation for wave motion is

( )

2 1

inear) small(nonl

2

2 =

∂ +∂ + +

+ p gz t

w

u φ

43 ρ 42 1

Using

) cosh sin(

) ( cosh

2 kx t

kd z d k g

H σ

φ = σ + −

we have

) cosh cos(

) ( cosh

2 kx t

kd z d k g H

gz gz t p

σ ρ

ρ ρ φ ρ

+ − +

−

=

∂

− ∂

−

=

The first term is the hydrostatic pressure and the second term is the dynamic pressure due to wave motion. The dynamic pressure is positive under the wave crest but it is negative under the trough. See Figure 2.3 of textbook. The above equation is valid below the still water level. Above the still water level, the pressure is assumed to be hydrostatic so that

η η

ρ − < <

= g z z

p ( ) for 0

Since

) 2 cos(kx t

H σ

η = −

the dynamic pressure is related to the free surface elevation by

η ρ

σ ρ

kd z d g k

t kd kx

z d k g H

p_d

cosh ) ( cosh

) cosh cos(

) ( cosh 2

= +

+ −

=

kd z d g k

p_d cosh

) (

cosh +

=

∴ ρ η

Therefore the surface waves can be measured using a pressure transducer.

2.5 Energy, Power, and Group Velocity

Total energy (E) = potential energy (E_p) due to displacement of free surface + kinetic energy (E_k) due to water particle movement Potential energy

( )

L g H

L gd dx g d

gdL d dE

E

L L

p p

2

2 0

2 0

16

2 2

h 2 wavelengt one

per

ρ

η ρ ρ

ρ

=

− +

=

⎟⎠

⎜ ⎞

⎝

− ⎛

=

∫

∫

Potential energy per unit surface area is

²

16gH L

Ep E^p ρ

=

=

Another view

L g H H

g HL

E_p ²

2 16 16 2

ρ π

ρ π × × =

=

Kinetic energy

(

)

E ^L

k d

2 0

0 2 2

16 2

ρ

ρ _{+ =}

≅

∫ ∫

2

16gH L

Ek E^k ρ

=

=

k

p E

E =

∴

Now the total energy per unit surface area is

⎟⎠

⎜ ⎞

⎝

= ⎛ ⋅

= +

= ^{2 2} _{2 2}

m Joule m or

m N 2

8 g a

g H E

E

E p k ρ ρ

Wave power

Wave power (P) = rate of work done = energy flux (F) =

s or J s

m N

0 ⋅

∫

Cn E

kd kd gH k

udzdt T p

P ^T

d d

=

⎥⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝⎛ +

=

=

∫ ∫

2 sinh 1 2 2 1 8

1 1

2 0

0

ρ σ

where

⎟⎠

⎜ ⎞

⎝⎛ +

= kd

n kd

2 sinh 1 2 2 1

Defining the group velocity as

nC C_g =

we have

Cg

E F P= =

Now, the group velocity is the velocity at which wave energy is transferred. When we generate waves using a wavemaker in a wave flume, the leading edge of the generated waves propagates at the speed of group velocity.

In deep water,

, 2 2

1 ₀

0

C C

n→ ∴ _g =

In shallow water, n → ,1 ∴ C_g = C

Wave shoaling

In steady state, energy conservation equation is constant

2

1 = P =

P

( ) ( )

2 2 2 2 1

1 2

1 8

1 8

1ρgH C n = ρgH C n

2 2

1 1

2 2

1 1

1 2

n L

n L n

C n C H

H = =

If the location 1 is in deep water,

2 2

2

0

2 2

0

2 2

0 0

0 2

Cg

C n

C C n

C n C H

H = = =

s g

K C H

H C

H_{2 0} ⁰ ₀

2 2 =

=

where K_s is the shoaling coefficient given by

0 0

2 H

H C

K C

g

s = =

The shoaling coefficient is shown in Figure 2.5 of textbook as a function of the relative depth, d /L.

2.6 Radiation Stress and Wave Setup

Radiation stress = excess momentum flux due to waves = pressure force (due to dynamic pressure)

+ momemtum transfer (analogous to Reynolds stress of turbulent flow)

[ ]

plane subscript

1st

=

⎥ =

⎦

⎢ ⎤

⎣

=⎡

nn ns

sn ss

S S

S S S

( )

⎠

⎜ ⎞

⎝

⎛ −

⎟=

⎠

⎜ ⎞

⎝⎛ +

=

−

− +

=

∫

∫

2 1 2

sinh 2 2 1 8

) 1

( ²

2 0

n kd E

gH kd dz

gz dz

u p

Sss ^ηd ρ s d ρ ρ

( )

( )

0 = ⊥

⎪⎭

⎪⎬

⎫

=

=

=

=

∫

∫

−

−

n

d s n

ns

d n s

sn u

dz u u S

dz u u S

η Q

η

ρ ρ

( )

⎠

⎜ ⎞

⎝⎛ −

=

=

−

− +

=

∫

∫

1 2

sinh 8

) 1

( ²

2 0

n kd E

gH kd dz

gz dz

u p

Snn ^ηd ρ n d ρ ρ

When a wave is propagating with an angle θ with respect to x -axis,

l S

l S

s S

n S

lS_xx = _ss Δ + _nn Δ = _ss Δ + _nn Δ

Δ cosθ sinθ cos²θ sin²θ

( )

( )

⎢⎣⎡ + −

=

⎥⎦

⎢ ⎤

⎣

⎡ ⎟ −

⎠

⎜ ⎞

⎝⎛ −

⎟ +

⎠

⎜ ⎞

⎝

⎛ −

=

⎟⎠

⎜ ⎞

⎝⎛ −

⎟ +

⎠

⎜ ⎞

⎝

⎛ −

=

+

=

2 1 1 cos

cos 2 1

cos 1 2 2 1

2 sin cos 1

2 2 1

sin cos

2

2 2

2 2

2 2

θ

θ θ

θ θ

θ θ

n E

n n

E

n E n

E

S S

S_{xx ss nn}

Likewise,

( )

⎢⎣⎡ + −

= 2

1 1 sin²θ n E S_yy

θ θcos nsin

E S_xy =

Wave setup and setdown

Outside surf zone,

↑

E , n↑ as d ↓

↑

∴ S_xx as d ↓ → setdown

Inside surf zone,

↓

E , n↑ as d ↓ but E↓>> n↑

↓

∴ S_xx as d ↓ → setup

Force-momentum flux balance gives

( )

( )

xx g d d S g d d R

S + + = + + 2 +

2 1 2

2

1 '

2 ' 1

2

1ρ ρ

Taylor series expansion about center line gives

( ) ( )

( ) ( ) ( )

x d d d dx g

d d x g

d d dx g

x S S

d dx d x g

d d dx g

x S S

xx xx

xx xx

∂ + ∂

⎥⎦ −

⎢⎣ ⎤

⎡ +

∂ + ∂ +

∂ + +∂

=

⎥⎦⎤

⎢⎣⎡ +

∂

− ∂ +

∂ +

−∂

2 ' 2 '

' 1 2

1 2

' 2 2

' 1 2

1 2

2 2

2 2

ρ ρ

ρ

ρ ρ

(

) (

)

2

1 2 =

∂ + ∂

⎥⎦ −

⎢⎣ ⎤

⎡ +

∂ + ∂

∂

∂ dx

x d d d g dx d d x g

x dx

S_xx ρ ρ

( ) (

) (

)

' 2 2

1 =

∂ + ∂

∂ − + + ∂

∂ +

∂

x d d d x g

d d d

d x g

S_xx ρ ρ

( )

' =

∂ + ∂

∂ +

∂

x d d d x g

S_xx ρ ← Eq. (2.52) except d →d+d'

Wave setdown outside surf zone is given by

kd k d H

2 sinh 8 ' 1

− 2

=

Inside surf zone, wave height is assumed to be proportional to water depth, that is

(

)

H =γ +

where γ is a constant of about 0.9. Also, inside surf zone where the shallow water condition is satisfied, n=1 so that

( )

2

2 '

16 3 8

1 2 3 2 3 2

2n 1 E gH g d d

E

S_xx ⎟≅ = = +

⎠

⎜ ⎞

⎝

⎛ −

= ρ ρ γ

Substituting in (2.52),

( ) ( ) ( )

' ' 8 '

3 2 =

∂ + ∂

∂ + + + ∂

x d d d x g

d d d

d

gγ ρ

ρ

x d x

d

∂

− ∂

∂ =

⎟∂

⎠

⎜ ⎞

⎝⎛ + ^{2 2}

8 3 ' 8

1 3γ γ

x d x

d x

d

∂

⎟⎟ ∂

⎠

⎜⎜ ⎞

⎝

⎛ +

−

∂ =

∂ +

−

∂ =

∂ ⁻¹

2 2

2

3 1 8 8

1 3 8 3 '

γ γ

γ ← Eq. (2.54)

Integration gives

1 1

3 2

1 8

' d C

d ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ +

−

=

−

γ

− ₁

1

3 2

1 8 ) (

' d d C

d _b ⎟⎟ _b +

⎠

⎜⎜ ⎞

⎝

⎛ +

−

=

−

γ

) 3 (

1 8 ) ( ' '

1

2 d d

d d

d _b ⎟⎟⎠ _b −

⎜⎜ ⎞

⎝

⎛ + +

=

−

γ

where is the wave setdown at breaking point, given by Eq. (2.53). Now, the wave setup at the still water line contour is given by

) ( ' d_b d

b

b d

d d d

1

3 2

1 8 ) ( ' ) 0 ( '

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

= γ

2.7 Standing Waves, Wave Reflection

perfect reflection (H_r =H_i) partial reflection (H_r <H_i)

⎩⎨

⎧

+

−

direction -

negative in

g propagatin )

sin(

direction -

positive in

g propagatin )

wave sin(

e progressiv

x σt

kx

x σt

kx

t H kx

t kx H

t kx t

kx t

kx t

H kx

t H kx

t H kx

s p p

p p

s

σ σ

σ σ

σ σ

σ σ

η

cos 2 cos

cos cos

) sin sin cos

cos sin

sin cos

2 (cos

) 2 cos(

) 2 cos(

=

=

− +

+

=

+ +

−

=

t kd kx

z d k H g

t kd kx

z d k H g

t kx t

kx t

kx t

kd kx z d k H g

t kd kx

z d k H g

t kd kx

z d k H g

s p p

p p

s

σ σ σ σ

σ σ

σ σ σ

σ σ σ σ

φ

sin cosh cos

) ( cosh 2

sin cosh cos

) ( cosh

) sin cos cos

sin sin

cos cos

cosh (sin ) ( cosh 2

) cosh sin(

) ( cosh ) 2

cosh sin(

) ( cosh 2

− +

=

− +

=

−

− + −

=

+ +

− + −

=

t kd kx

z d H k

t kd kx

z d k H gk

u x

s s s s

σ σ

σ σ φ

sin sinh sin

) ( cosh 2

sin cosh sin

) ( cosh 2

= +

= +

∂

=∂

t kd kx

z d H k

t kd kx

z d k H gk

w z

s s s

s

σ σ

σ σ φ

sin sinh cos

) ( sinh 2

sin cosh cos

) ( sinh 2

− +

=

− +

∂ =

= ∂

t H_s kx

s σ

η cos cos

= 2

Under the pure standing wave, at t =0orT/2, the water particle motion stops for an instant so that only potential energy exists, while at t =T/4or3T/4, the water surface becomes flat so that only kinetic energy exists. Also, w=0 at nodes, while at antinodes.

=0 u

( )

( )

Hi = η + η

( )

( )

Hr = η − η

Reflection coefficient,

i r

r H

C = H .

For pure standing wave,

( )

( )

0 .

=1 Cr

Transmission coefficient,

i t

t H

C = H .

If there is no energy dissipation through the structure, or (energy is conserved)

2 1

2 + _t =

r C

C H_r² +H_t² = H_i²

2.8 Wave Profile Asymmetry and Breaking

Wave breaking In deep water,

H

u_c ∝ , C ≠ f(H): H ↑ → u_c ↑ → wave breaking if u_c >C

In shallow water,

c ↑

u and C↓ as d ↓ → wave breaking

Miche (1944) formula ← for any constant depth

L d L

H 2π

7tanh 1

max

⎟ =

⎠

⎜ ⎞

⎝

⎛

In deep water,

(

)

(

)

(

)

On sloping beach,

Spilling breaker: small H₀/ L₀ and m (beach slope) Plunging breaker ↓

Surging breaker: large H₀/ L₀ and m

Given , , → (Fig. 2.11) → (Fig. 2.12), where is the unrefracted deepwater wave height, which is the deepwater wave height of the waves propagating normal to the shore with straight and parallel bottom contours where no wave refraction but only wave shoaling occurs. If wave refraction occurs as well, then

should be

0'

H T m H_b d_b H₀'=H/K_s

0' H

r s

r s s

K K H

K K H K

H₀'= H = ⁰ = ₀

It is the most dangerous for coastal structures when plunging breaker hits the structures.

The most dangerous point is about from the breaker initiation point, where is the distance from breaker initiation to plunging point. Therefore, it is needed to design the structure for the waves breaking at seaward of the structure. For plane slope,

Xp

5 .

0 X_p

Xp

5 . 0

25 . 0 0 0/ 95

.

3 ⎟⎟

⎠

⎞

⎜⎜

⎝

= ⎛

m L H H

X

b p

For slope with submerged bar,

81 . / 1

63 .

0 ^{0 0} ⎟⎟+

⎠

⎞

⎜⎜

⎝

= ⎛

m L H H

X

b p

2.9 Wave Runup

Smooth, impermeable slope

Saville (1957): Laboratory experiments with tanβ =0.1

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

, ' , '

' cot ² ₀

0

0 H

d gT fn H

H

R α _s (Figure 2.15 of textbook)

'↑ H0

R as ⁰₂' ↓ gT

H (milder waves) and cotα ↓ (steeper structure slope)

If cotα ≤1.0 (or steeper than 1:1 slope), slope effect is negligible.

Rough, permeable slope

si

rp rR

R =

where is runup on rough, permeable slope, is runup on smooth, impermeable slope, and is the coefficient representing the effect of roughness and permeability of the slope (See Table 2.1 of textbook).

Rrp R_si

0 .

<1 r