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Dispersion in Shallow Water
John D. CarterSeattle University
in collaboration with
Harvey SegurUniversity of Colorado at Boulder
David GeorgeU.S. Geological Survey
Diane HendersonPenn State University
John D. Carter Dispersion in Shallow Water
Outline
I. Experiments
II. St. Venant equations
III. KdV equation
IV. Serre equations
V. Whitham equation
VI. Summary
John D. Carter Dispersion in Shallow Water
Experimental Set Up
Figure not to scale!
h0=10cm
Gauge a
x= 61cm
Gauge b
x= 561cm
Gauge c
x= 1061cm
Gauge d
x= 1561cm
Gauge e
x= 2061
Experiments conducted by Joe Hammack.
John D. Carter Dispersion in Shallow Water
Experimental Initial Conditions
Figure not to scale!
h0=10cm
Gauge a
x= 61cm
Gauge b
x= 561cm
Gauge c
x= 1061cm
Gauge d
x= 1561cm
Gauge e
x= 2061
A0
Experiments conducted by Joe Hammack.
John D. Carter Dispersion in Shallow Water
Parameter Definitions
I ε = Hh0
is a (dimensionless) measure of nonlinearity
I δ = h0λ is a (dimensionless) measure of shallowness
I H represents a typical wave height
I h0 represents the undisturbed water depth
I λ represents a typical wavelength
John D. Carter Dispersion in Shallow Water
Experimental Measurements: A0 = 0.5cm344 J . L. H a m m k and H . Segur
0
- 0.03
- 0.06
- 0.09
0
- 0.03
- 0.06
-0.09 . mE 0
II i - 0.03
- 0.06
- 0.09
0
- 0.03
- 0-06
- 0.09
0
- 0.03
- 0.06 0 25 50 75 100 125 150 175 200 225
- X = t ( g / h ) f - x / h
FIGURE 2. Experimental wave systems: h = 10 cm, L, = 122 cm, A , = 0.5 cm, M = 0.0335, xo = - 3.66. (a) z/h = O or 37 = 0, (b ) z / h = 50 or 37 = 25, (c ) s /h = 100 or 37 = 50, (d) z /h = 150 or 37 = 75, (e) z / h = 200 or 37 = 100. -+, trajectory baaed on average wavenumber between two stations and linear dispersion relation; - +, extrapolation of previous trajectory.
3. Experiments 3.1. Equipment and procedure
The experimental equipment used in this study was described briefly in part 2 and in greater detail by Hammack (1973). The wave maker consists of a rectangular piston 61 cm long at the end of a wave tank. The rectangular wave propagating out of the generation region following a sudden downthrow of the piston has a length of twice the piston length and an amplitude of one-half the piston stroke. By varying the piston stroke while the quiescent water depth is fixed at h = 10cm, the nonlinearity of the initial wave is varied. For all of the experiments presented herein the length of the initial wave is constant at L, = 122 cm or L = 12.2. The initial wave amplitudes are A* = 0-5cm, 1.5cm and 2-5cm.
Waves are measured using parallel-wire resistance gauges and an oscillograph recorder a t the following positions: x = 0, 50h, 100h, 150h and 200h, where x = 0 is the downstream edge of the piston. Some of these measurements, especially those at x = 200h, are incomplete because the wave reflected from the downstream end of the tank returned prior to the complete passage of the rightward-running wave system. In the comparisons with theory, wave traces at xlh = 0, 50, 100, 150 and 200 are assumed to represent the spatial wave a t the times 37 = 0, 25, 50, 75 and 100, respec- tively, according to the normalization given by (2). Consequently, a rightward-
From J.L. Hammack and H. Segur, TheKorteweg-deVries equation and water waves.Part 3. Oscillatory waves, Journal of FluidMechanics, 84:337-358, 1978.
ε =0.5
10= 0.05
δ =10
2 ∗ 61= 0.08
John D. Carter Dispersion in Shallow Water
Experimental Measurements: A0 = 1.5cmKorteweg-de Vries equation and water waves. Part 3 346
0.15
0
-0.15
-0.30
0.15 5
I1 z 0
-0.15
- 0.30
0.15
0
-0.15
-0.30
0.15
0
-0.15 0 25 50 75 100 125 150 175 200 225
- X = t (g/h)* - .y/h
FIG~RE 3. Experimental wave systems: h = 10 cm, L* = 122 cm, A , = 1.5 cm, M = 8.36 x 10-6, xo = -2.108. (a) z/h = 0 or 37 = 0, (a) z/h = 50 or 37 = 25, (c) z/h = 100 or 37 = 50, (d) z/h = 150 or 37 = 75, (e) z/h = 200 or 37 = 100. +, trajectory based on average wavenumber between two stations and linear dispersion relation ; - +, extrapolation of previous trajectory.
running wave necessarily appears leftward-running, i.e. the wave front appears a t the left in each figure. Moreover, a point moves to the left or right in succeeding measurements depending on whether its velocity is greater or less than (gh)B, respectively.
In the comparison of theory and experiment for the trailing wave region, wave- numbers k are required; however, only wave frequencies w = (h/g)Bw* are directly measurable from the experiments. To compute wavenumbers from the measured frequencies, the complete (all k) dispersion relation for linear water waves
w2 = k tanh k
is used. This procedure is adopted since measured frequencies and the water depth h = 10cm indicate that many of the oscillatory waves which evolve are not ‘long’ and considerable error is introduced by replacing (37) by its long-wave approximation.
(37)
3.2. Observed wave evolution
Figures 2-4 show the downstream wave measurements for three experiments with initial wave amplitudes of A , = 0.5 cm, 1.5 cm and 2.5 cm, respectively. The nor- malized wave amplitude f (or #y/h) is shown as a function of the non-dimensional co-ordinate - x (or t(g/h)t - x/h) . Wave traces are presented in this manner to empha- size tha t they are in fact temporal measurements at a fixed spatial location. which are
From J.L. Hammack and H. Segur, TheKorteweg-deVries equation and water waves.Part 3. Oscillatory waves, Journal of FluidMechanics, 84:337-358, 1978.
ε =1.5
10= 0.15
δ =10
2 ∗ 61= 0.08
John D. Carter Dispersion in Shallow Water
Variable Definitions
I g represents the acceleration due to gravity
I h0 represents the undisturbed water depth
I h(x , t) represents the local water depth
I η(x , t) represents the free surface displacement
I Note: h(x , t) = h0 + η(x , t)
I u(x , t) represents the horizontal fluid velocity
I u(x , t) represents the depth-averaged horizontal fluid velocity
John D. Carter Dispersion in Shallow Water
Initial Conditions for Numerics
h0=10cm
A0
x= 61cm
John D. Carter Dispersion in Shallow Water
St. Venant Equations
The dimensional St. Venant (a.k.a. the classic shallow-waterequations) are
ht + (hu)x = 0
(hu)t +(1
2gh2 + hu2
)x
= 0
These equations are nondispersive.
John D. Carter Dispersion in Shallow Water
St. Venant Equations Experiment #2: A0 = 0.5cm
St. Venant simulations computed by David George.
John D. Carter Dispersion in Shallow Water
St. Venant Equations Experiment #3: A0 = 1.5cm
St. Venant simulations computed by David George.
John D. Carter Dispersion in Shallow Water
KdV
The dimensional Korteweg-deVries equation is
ηt +√
gh0 ηx +3
2h0
√gh0 ηηx +
1
6h20√
gh0 ηxxx = 0
The linear dispersion relation is
ω =√
gh0(k − 1
6h20k
3)
John D. Carter Dispersion in Shallow Water
Dispersion in KdV
0 1 2 3 4 5 6k h0
0.2
0.4
0.6
0.8
1.0
Ω
k c0
KdVEuler
John D. Carter Dispersion in Shallow Water
KdV Experiment #2
10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
HaL10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
KdV
20 40 60 80 100 120 140
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
50 100 150 200
-0.06-0.04-0.02
0.000.020.04
HdL
50 100 150 200 250
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
KdV Experiment #2, Larger t Interval
50 100 150 200
-0.08
-0.06
-0.04
-0.02
0.00
HaL100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
KdV
100 200 300 400 500
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
200 400 600 800 1000
-0.06-0.04-0.02
0.000.020.04
HdL
200 400 600 800 1000
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
KdV Experiment #3
10 20 30 40 50
-0.3
-0.2
-0.1
0.0
0.1HaL
20 40 60 80 100
-0.2
-0.1
0.0
0.1
HbLExpt
KdV
20 40 60 80 100 120 140
-0.15-0.10-0.05
0.000.050.100.15
HcL
50 100 150 200 250
-0.15-0.10-0.05
0.000.050.10
HdL
50 100 150 200 250
-0.10
-0.05
0.00
0.05
0.10HeL
John D. Carter Dispersion in Shallow Water
KdV Experiment #3, Larger t Interval
100 200 300 400 500
-0.3
-0.2
-0.1
0.0
0.1HaL
100 200 300 400 500 600 700
-0.2
-0.1
0.0
0.1
HbLExpt
KdV
200 400 600 800 1000
-0.15-0.10-0.05
0.000.050.100.15
HcL
200 400 600 800 1000
-0.15-0.10-0.05
0.000.050.10
HdL
200 400 600 800 1000
-0.10
-0.05
0.00
0.05
0.10HeL
John D. Carter Dispersion in Shallow Water
Serre
Serre story
John D. Carter Dispersion in Shallow Water
Serre
The dimensional Serre equations are
ht + (hu)x = 0
ut + uux + ghx −1
3h
(h3(uxt + uuxx − (ux)2
))x
= 0
The linear dispersion relation is
ω2 =3gk2h0
3 + k2h20
John D. Carter Dispersion in Shallow Water
Dispersion in Serre
0 1 2 3 4 5 6k h0
0.2
0.4
0.6
0.8
1.0
Ω
k c0
SerreKdVEuler
John D. Carter Dispersion in Shallow Water
Serre Experiment #2
10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
Serre
20 40 60 80 100 120 140
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
20 40 60 80 100 120 140
-0.06-0.04-0.02
0.000.020.04
HdL
20 40 60 80 100 120 140
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
Serre Experiment #2
10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
20 40 60 80 100 120 140
-0.08
-0.06
-0.04
-0.02
0.00
0.02
HbLExpt
KdV
Serre
20 40 60 80 100 120 140
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
20 40 60 80 100 120 140
-0.06-0.04-0.02
0.000.020.04
HdL
20 40 60 80 100 120 140
-0.06
-0.04
-0.02
0.00
0.02
0.04
HeL
John D. Carter Dispersion in Shallow Water
Serre Experiment #2, Larger t Interval
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
Serre
200 400 600 800
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
200 400 600 800
-0.06-0.04-0.02
0.000.020.04
HdL
200 400 600 800
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
Serre Experiment #2
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02
HbLExpt
KdV
Serre
100 200 300 400 500
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
100 200 300 400 500
-0.06-0.04-0.02
0.000.020.04
HdL
100 200 300 400 500
-0.06
-0.04
-0.02
0.00
0.02
0.04
HeL
John D. Carter Dispersion in Shallow Water
Serre Experiment #3, Larger t Interval
100 200 300 400 500
-0.3
-0.2
-0.1
0.0
0.1HaL
200 400 600 800
-0.2
-0.1
0.0
0.1
0.2HbL
Expt
Serre
200 400 600 800
-0.15-0.10-0.05
0.000.050.100.15
HcL
200 400 600 800
-0.15-0.10-0.05
0.000.050.10
HdL
200 400 600 800
-0.10
-0.05
0.00
0.05
0.10HeL
John D. Carter Dispersion in Shallow Water
Whitham
The dimensional Whitham equation is
ηt +3
2h0
√gh0 ηηx +
∫ ∞−∞
∫ ∞−∞
√gk tanh(kh0)
2πkeik(x−ξ)η
ξdk dξ = 0
The linear dispersion relation is
ω =√
gk tanh(kh0)
John D. Carter Dispersion in Shallow Water
Whitham Experiment #2
10 20 30 40 50
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
20 40 60 80 100 120 140
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
Whitham
20 40 60 80 100 120 140
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
20 40 60 80 100 120 140
-0.06-0.04-0.02
0.000.020.04
HdL
20 40 60 80 100 120 140
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
Whitham Experiment #2, Larger t Interval
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HaL
100 200 300 400 500
-0.08
-0.06
-0.04
-0.02
0.00
0.02HbL
Expt
Whitham
100 200 300 400 500 600
-0.08-0.06-0.04-0.02
0.000.020.04
HcL
100 200 300 400 500 600
-0.06-0.04-0.02
0.000.020.04
HdL
100 200 300 400 500 600
-0.06-0.04-0.02
0.000.020.04
HeL
John D. Carter Dispersion in Shallow Water
Whitham Experiment #3, Larger t Interval
100 200 300 400 500 600
-0.3
-0.2
-0.1
0.0
0.1HaL
100 200 300 400 500 600
-0.2
-0.1
0.0
0.1
0.2
HbLExpt
Whitham
100 200 300 400 500
-0.15-0.10-0.05
0.000.050.100.15
HcL
100 200 300 400 500 600
-0.15-0.10-0.05
0.000.050.100.15
HdL
100 200 300 400 500 600
-0.10
-0.05
0.00
0.05
0.10
HeL
John D. Carter Dispersion in Shallow Water
Summary
1. Dispersion can be important in shallow water
2. “Any old form” of dispersion is not necessarily sufficient
3. Nonlinear effects are important in shallow water
4. None of the examined models get the amplitudes correct
5. Dissipation likely plays an important role
John D. Carter Dispersion in Shallow Water
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