Linear Wave Theory fundamental description:

L - wave length

H - wave height

T - period

d - water depth

Shore Protection Manual, 1984

Overview of Waves and Sediment Transport

Most energy on continental shelf - gravity waves consisting of sea and swell

(Jeff Parsons’ web site)

Wave theory characteristics that affect what we see in the bottom boundary layer:

When wave is in “deep” water - d/L > 1/2

orbits circular

waves don’t feel the bottom (and the seabed doesn’t feel the waves)

When wave is in “shallow” water - d/L < 1/25

orbits flatten, become elliptical

wave speed is dependent upon depth, c =(gd)1/2

wave-orbital velocities are felt at the seabed

Linear Wave Theory:

Shore Protection Manual, 1984

Key Linear Wave equations for sediment transport:

(At bottom, z = -d )

Wavelength:

Maximum wave-orbital velocity, cos(Θ) = 1 :

Orbital Excursion:

L

dgTL

2tanh

2

2

LdL

gTHub /2cosh

1

2max

Twhere

ua b

b

2~max

Example: On the Washington shelf, a winter storm could produce waves of 7 m in height with period of 15 seconds. At what depths are these waves felt on the shelf?

Shallow water waves

Speed is dependent on water depth

wave speed, c=(gd)1/2

Leads to wave refraction as shoreline is approached.

Wave boundary layer

Linear wave theory assumed inviscid flow (no friction at bed). We can use linear wave theory above the BBL and develop a viscous boundary layer at seabed.

Because waves oscillate, there is limited time for viscous effects to build. Therefore, the wave boundary layer is thin relative to the current boundary layer.

Results in high shear in u

high u*w

high b

Wave boundary layer thickness is seldom > 10 cm

How do we determine shear stress due to waves?

1. Eddy viscosity concept

Az = u*w z

(time invariant)

2. Wave friction factor (analogous to a drag coefficient)

Time averaged over a wave cycle

2

max

*~

b

ww u

uf

2max

~3

2bwbw uf

What is fw a function of:

bed roughness, ks

orbital excursion, ab

R*

977.5)(213.5 194.0

b

s

a

k

w ef

In rough turbulent region,

Alternatively, we can write the Shield’s entrainment function using fw:

Plot with the uni-directional threshold curve

gD

uf

s

bw

w

2max

~2

1

Suspended sediment concentration profile under waves:

(combined waves and currents)

Rouse Equation:

c

cw

u

Ws

cw

a

z

u

Ws

a

a

z

zc

c

z

z

c

c

*

*

for z < cw

for z > cw