Microsoft PowerPoint - 2 Linear wave theory PRT..through wave shoaling..
Linear Wave Fundamentals
Wave steepness: H/L
Wave Classification: Spectrum
Fig. 2-1
5
Wind Direction
Wind Direction friction
Assumptions (p.2-6):
• uniform and constant pressure at the surface
• ideal fluid (no viscosity)
• long-crested waves (2D approach)
L T
L
tanh x x
L
Particle motion vs. Water depth
9
L
L
2 tanho
(iterative solution required) 2

For a simple harmonic wave train, the wave period is independent of depth!
1 2 1 2n n T T
10
Example, page 2-11:
GIVEN: A wave with a period T=10 sec propagates shoreward over a uniformly sloping shelf from a depth d=200 m to a depth d=3 m.
FIND: The wave celerities C and lengths L corresponding to depth d =200 m and d=3 m.
Ex 2-11: Solution Using Eq. (2-8a) gives
2 21.56 1.56(10) 156 oL T m
For d= 200 m:
From Table C-1 it is seen that for values of
0 0
L L L
Therefore, L = L0 = 156 m (logical because d/L > 0.5 means deep water)
By Eq. (2-1) C = L0/T = C0 = 156/10 = 15.6 m/s
For d= 3 m:
By Eq. (2-1) C = L/T = 53.2/10 = 5.32 m/s
from column 2.
2 cosh(2 / )
L d L L T

2 cosh(2 / )
L d L L T

cosh(2 / )x
L d L L T

cosh(2 / )z
L d L L T

2 sinh(2 / )
d L L T
2 sinh(2 / )
d L L T
d d
Example, page 2-13:
GIVEN: A wave with a period T=8 sec, in a water depth d=15 m, and a height H=5.5 m.
FIND: The local horizontal and vertical velocities u and w, accelerations ax and az at an elevation z=-5 m below the still water level (SWL) when θ=2 πx/L - 2πt/T = π/3 (= 60 deg).
13
L d L
L d L
L d L
L d L
Ex 2-13: Solution
First (as usual) we need L, but also cosh(2πd/L) seems to be in all Eqs.
2 21.56 1.56(8) 99.8 ~ 100 oL T m
0
L d L
L d L
L d L
L d L
H gT K
L d L
cosh(2 / ) 81.7 1.742
(in front of u och w)
(in front of ax och az)
(in front of u och ax)
(in front of w och az)
1 3 cos60 1.515 1.3106 0.5 0.99 /ou K K m s
1 4 sin60 .... 1.11 /ow K K m s
2 2 3 sin 60 ... 1.35 /o
xa K K m s
2 2 4 cos60 ... 0.5 /o
za K K m s
Thus:
L d L
cosh(2 / ) 2 a
z d L H x t p g gz p
d L L T
Water surface profile =
(2-26)

Example, page 2-22:
GIVEN: An average maximum pressure p=124 kN/m2 is measured by a subsurface pressure gage located in salt water (ρ = 1025 kg/m3) 0.6 m above the bed in water depth d=12 m. The wave period is 15 sec.
FIND: The height of the wave H assuming that linear theory applies and the average wave period corresponds to the average wave amplitude.
16
d L
2 21.56 1.56(15) 351oL T m 0
12 0.0342
cosh 2 ( 11.4 12) / 156.8 0.8949
1.117zK
=>
Wave Energy
Potential energy:

( )
2 cos
21
(per unit surface area)


1 x L
d L T
Distance from Wavemaker
t6 10 9 8 7 6 5
C
Cg
> Cg
d L T
g


1 2 1 1 2 2
2 2 2 2 cos cos
2 2
L T L T
L L TT
HIGHER-ORDER WAVE THEORIES
2 3 2 3cos ( , ) cos 2 cos3 ....a a B L d a B (2-46)
20
8 sinh (2 / )
H x t
L d L L T


DEEP WATER – 1st order
L d L
- 3D waves
- Stokes 2nd
Fig. 2-7
Solitary Wave
dz dx
u w
3 3 E gH d (2-66, 2-70)
Oscillatory Wave
d
22
L2-13