Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Sinusoidal
Wave Theory II
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D., Professor Emeritus
Presentation Content
•
Part II of predicting behavior of water waves
•
Algebraic solutions
•
Modeled with sinusoidal wave profiles
Interpreting
• Applying
•
Professor Tom Ravens, UAA College of Engineering, contributed graphics and information to this presentation
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
Sinusoidal Wave Theory
2
1
Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Solutions
•
Velocity potential ∅ , , as affected by wave height (H),
wave period (T), wave length (L), and depth (d)

•
Hg cosh k ( d  z )
sin( kx  t )
2 cosh kd
∅
=−
∅
k
2
L
 
Dispersion Equation: relating period, wavelength, and
depth
 2  gk tanh kd
•
=−
For water surface profile:
L
( , )=
2
T
gT 2
 2d 
tanh

2
 L 
2
cos(
−
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
)
Sinusoidal Wave Theory
3
Sinusoidal wave profile
•
Two wave profile modes:
•
•
Progressive profiles at times t1 = 0 and
t2 = T/2
Standing wave profiles at times t1 = 0
and t2 = T/2
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
1
2
1
0.5
0
 0.5
1
1
2
0
100
200
100
200
x
300
400
500
300
400
500
1
0.5
0
 0.5
1
0
x
Sinusoidal Wave Theory
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Hyperbolic tangent
•
Dispersion Equation:  2  gk tanh kd
•
Deep water (large kd, d > L/2):
•
Lo =
1.56T2 (m)
Lo =
5.12T2
L0 
(ft)
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
gT 2
2
Sinusoidal Wave Theory
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Intermediate and shallow depths
•
Depths between L/2 and about L/20
•
Must apply the full Dispersion Equation
C
L
 2d 

 tanh

C0 L0
 L 
•
L  L0 tanh kd
C
L0
tanh kd
T
Shallow water (small kd, d < L/20):
 2  gk 2 d
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
C  gd
Sinusoidal Wave Theory
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Water particle velocities
•
Velocity potential:   Hg cosh k (d  z ) sin( kx  t )
2 cosh kd
u
 H cosh k (d  z )
 
cos( kx  t )
x 2
sinh kd
w
•
•
 H sinh k (d  z )
 
sin( kx  t )
z
2
sinh kd
/2 out of phase
•
•
u is maximum when w is zero
w is maximum when u is zero
Maximum at surface (z = 0); diminish with depth (-z)
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Orson P. Smith, PE, Ph.D.
Sinusoidal Wave Theory
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Sinusoidal Wave Theory
8
Water particle paths
•
Integrate u and w with respect to time:
•
Horizontal displacement relative to mean:
•
Vertical displacement relative to mean:
 

•
H cosh k ( d  z )
sin( kx  t )
2
sinh kd
H sinh k (d  z )
cos( kx  t )
2
sinh kd
Deep water: circular path radius =
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
H kz
e
2
4
Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Water particle motion decay with depth
cosh k d  z  sinh k d  z 

 e kz
sinh kd
sinh kd
•
•
Exponential decay with increasing depth (-z)
Radii reduced to ~4% of surface at z = -L/2
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
Sinusoidal Wave Theory
9
Motion in intermediate and shallow depths
•
Elliptical, diminishing with depth
•
Flattening to only horizontal motion
at bottom (z = -d)
•
w = 0 at z = -d
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
Sinusoidal Wave Theory
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Water particle accelerations
•
•
Important for estimating wave-induced forces
Neglect 2nd two terms of full differential:
ax 
az 
•
u H 2 cosh k d  z 
 
sin kx  t 
t
2
sinh kd
ax 
u
u
u
u
w
t
x
z
w
H
sinh k d  z 
 2
coskx  t 
z
2
sinh kd
/2 out of phase with each other and with velocities
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
Sinusoidal Wave Theory
11
Potential Energy
•
Vertical displacement of water from SWL (PE = 0)
•
Integrate mgz over a wavelength (m = mass)
•
•
Consider a small column of water beneath a wave
Average potential energy due to waves: PE  1 gH 2
Equals potential energy per unit area
• Depends only on wave height
•
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
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Sinusoidal Wave Theory
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Kinetic and total energy
•
Integrate
+
KE 
•
over L and z:
1
gH 2  PE
16
Total average energy per unit surface area:
KE  PE 
1
gH 2
8
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
Sinusoidal Wave Theory
Group velocity
•
Water particle orbits
Waves transport energy
•
•
Average rate at which work is done on fluid (or by fluid)
Energy flux for one wave period:
=
•
no net mass transport
n
Group velocity:
•
•
1
2 kh 
1 

2
sinh 2 kh 
C g  nC 
d
dk
H1 + H2
•
13
time
Speed at which energy is transported by waves
Deep water, n = ½; shallow water, n = 1
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
Sinusoidal Wave Theory
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Shoaling
•
•
Period treated as constant
L shortens, leading to slower phase speed (C = L/T)
Wave height first diminishes, then increases
•
Shoaling Coefficient:
=
=
=
2
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
5
4.5
Shoaling: Ho = 4 ft; T = 8 s
H
L
328
4
3.5
400
300
4 200
200
150
100
depth (ft)
50
100
0
wave length (ft)
•
Wave characteristics change with decreasing depth
(shoaling), following change in energy flux, i.e., Cg
wave height (ft)
•
Sinusoidal Wave Theory
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Refraction
•
•
•
•
•
Consider straight wave crests approaching shallow water
at an angle
Part of crest slows before rest
Crest bends in toward shore
Snell’s Law:
sin  sin  0

C
C0
cos  cos  0

C
C0
Refraction Coefficient: Kr  cos 0
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
cos
H  H0Ks Kr  H0
Sinusoidal Wave Theory
C0
2 Cg
cos 0
cos
16
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Pressure under progressive waves
•
Substitute  in Bernoulli Equation, neglecting u2 and w2
terms, and applying the DFSBC:
=−
•
•
+
∅
includes a hydrostatic part and a time-varying part due to water
particle accelerations.
Applying solution for :
=−
+
2
+
−
=−
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
+
Sinusoidal Wave Theory
+
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Pressure
•
In deep water:
•
•
+
Exponential decay of dynamic pressure
+
In shallow water:
•
≅
≅1
Pressure is purely hydrostatic beneath the wave form
=
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Spring 2016
−
9
Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
Pressure (continued)

Consider dynamic pressure variation under crest and under trough:
o
•
o
Vertical acceleration, equations of motion (conservation of momentum):
=−
−
or
=−
−
-g term is always negative
o
< 0 under crest; particles are accelerating downward; reduced |dP/dz|
o
> 0 under trough; particles are accelerating upward; increased |dP/dz|
o
|dP/dz| is less under the crest than under the trough
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
Sinusoidal Wave Theory
19
long
waves
[shallow
Water
Waves]
short
waves
[or deep water
waves]
CE A676 Coastal Engineering
Spring 2016
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Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
d = 100 m, T = 10 s, H = 2 m
1/20/2016
Ex. 2.3-1 in Sorenson
What is c, L, H/L, and water particle speed at crest?
Solution:
•
Assume the wave is a deep water wave (we will check this assumption later).
•
If deep water wave,
•
Check deep water wave assumption:
•
Celerity:
•
Steepness:
(Otherwise, we
would need to
use the full
Dispersion Equation)
Ho / Lo = 2 / 156 = 0.013 (small wave steepness)
h = 100 m, T = 10 s, H = 2 m; c = 15.6 m/s
Water is particle speed at crest?
Solution:
•
•
For deep water waves, particle orbits are circular; at the
surface the diameter of the circles is the wave height.
So, what is the nominal velocity?
CE A676 Coastal Engineering
Spring 2016
11
Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
1/20/2016
d = 100 m, T = 10 s, H = 2 m in deep water
•
Suppose the wave has propagated to a depth of 2.3 m
•
Calculate the celerity and wavelength
Solution: assume the wave is in shallow water at d = 2.3 m.
L = C T = 47.5 m
Check shallow water assumption:
d/L = 2.3/47.5 = 0.048 < 0.05
the wave is in shallow water
Note: computing L by applying the full Dispersion Equation will also
confirm the assumption
Calculating wave character from
pressure data
Given: the pressure data below from a bottom pressure transducer in a lake.
Find: water depth, wave height
42500
42000
pressure, Pa
41500
41000
40500
40000
39500
39000
38500
38000
37500
CE A676 Coastal Engineering
Spring 2016
0
0.5
1
1.5
2
2.5
time (sec)
3
3.5
4
4.5
12
Sinusoidal Wave Theory II
Presented by Orson Smith, PE, Ph.D.
Solution:
43000
42000
pressure, Pa
d =?, H = ?
1/20/2016
41000
40000
39000
38000
37000
•
Pavg = 40,000 Pa
0
1
2
time (sec)
3
4
5
(with P = ρgd & ρg  9800 N/m3), d ~ 4 m
•
By inspection, T = 4 sec (time between successive crests
•
Find d/Lo = .183 = transitional wave (< ½, not deep water)
•
•
•
•
Lo = g/2π T2 = 21.9 m, for deep water
From sinusoidal wave table, d/L = 0.208, so L = 19 m.
k = 2 π/L = 0.33 m-1
∆P between crest and SWL (2000 Pa)
•
•
solve for H = 0.8 m
H = (2 ∆P / ρg) cosh(kd)
=
Conclusion
CE A476/676 Coastal Engineering
Orson P. Smith, PE, Ph.D.
CE A676 Coastal Engineering
Spring 2016
Sinusoidal Wave Theory
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