file - UPMC

KdV
waves”. Another famous student of Korteweg is A. Moens. They proposed the
velocity in arterial flow ”Moen-Korteveg” equation (arterial flows and water
flows are very similar ; the elasticity of arteries is the gravitation in water flows).
The KdV equation was not studied much after this until Fermi- Pasta- Ulam
and Zabusky & Kruskal (1965). They wanted to study the heat transfer in a
solid consisting in a crystal modeled by masses en springs in a periodic domain
(one dimensional lattice). They discovered traveling waves which were not damped.
P.-Y. Lagrée
CNRS & UPMC Univ Paris 06, UMR 7190,
Institut Jean Le Rond ∂’Alembert, Boı̂te 162, F-75005 Paris, France
[email protected] ; www.lmm.jussieu.fr/∼lagree
12 novembre 2015
Résumé
The inviscid equations of water flow are presented. First, the case of small amplitude perturbations in small depth is presented, so linearized, this leads to the
”d’Alembert equation”. Second, he case of small amplitude perturbations in any
depth, this allows to explain the dispersive behavior of water waves. This is called
”Airy wave theory.” Third the ”shallow water theory” is presented, this corresponds to significant perturbations of the height of water, which is small compared
to the length of the waves ; the equations are non linear. Then, if one considers
in this latter framework small amplitude waves, and the first correction due to
depth, one may have a balance between non linearities and dispersion. This leads
to the solitary wave solution of ”KdV Equation.” The KdV (Korteweg–de Vries)
equation is presented as an application of multiple scale analysis
1
See Dauxois, Newell [11], Remoissenet [14] and Maugin [13] for other historical
details and a view of the fields of application. The fields are very large, from
hydrodynamics, lattice waves, waves in electric lines, light waves in optic cables
etc.
Introduction
First observed by John Scott Russell in 1834, on the Glasgow-Edinburgh
channel. Generated by a boat (drawn along a narrow channel) stopping suddenly.
A wave appeared, and moved with constant shape. He took his horse to follow it
on several miles (see the text from Remoissenet [14], on page 319 of the Report on
Waves, section I ”The wave of translation”). He did after that a lot of experiments
to reproduce it in a 30’ wave tank in his back garden.
It was followed by theoretical investigations by Joseph Boussinesq ( Théorie
de l’intumescence liquide, appelée onde solitaire ou de translation, se propageant
dans un canal rectangulaire , dans Comptes rendus de l’Académie des Sciences,
vol. 72, 1871, p. 755–759) and ( Théorie des ondes et des remous qui se propagent
le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu
dans ce canal des vitesses sensiblement pareilles de la surface au fond , dans
Journal de Mathématique Pures et Appliquées, Deuxième Série, vol. 17, 1872, p.
55–108) and Rayleigh (1876) and, finally, Korteweg and De Vries in 1895.
De Vries was the student of Korteweg, the title of the thesis Bijdrage tot
de kennis der lange golven, in dutch ”Contributions to the knowledge of long
2
Scope of the chapter : heuristical point of view
Waves in fluid are classical in mechanics of fluid courses, what we want to do
here is to restart from scratch this study with a uniform point of view in order to
recover the ∂’alembert Equation, the Saint Venant and the Airy Wave theory. For
instance, the ∂’Alembert Equation, is heuristically proved as follows, we suppose
a plug flow u(x, t), its acceleration is ρ∂u/∂t, the forces exerted are the pressure
ones. Due to the elevation of the wave η (over the free level y = 0), the pressure
is simply ρgη for hydrostatic reason. Then
ρ
∂u
∂η
= −ρg ,
∂t
∂x
The thickness of the fluid layer is h0 , and η is smaller than h0 , the other needed
equation is the conservation of mass, as there is no v in the equations, changes of
- MHP KdV PYL 2.1-
P.-Y. Lagrée, KdV
Figure 1 –
The Soliton reproduced in 1995 on the very same place
than Scott Russell ’first’ observed a solitary wave on the Union Canal
near Edinburgh in 1834. (Photo from Nature v. 376, 3 Aug 1995, pg 373)
http://www.ma.hw.ac.uk/solitons/press.html
the flux of u :
∂h0 u
∂x
Figure 2 – The original sketches of Scott Russel. ”The great wave of translation” http://www.ma.hw.ac.uk/∼chris/Scott-Russell/SR44.pdf. The wave may be
generated by a moving wall, top left or a falling weight top right, or opening a
gate. The final result is a unique wave which translates without change in shape
on very long distance.
are compensated by the displacement of the interface :
∂η ∂h0 u
+
=0
∂t
∂x
eliminating u gives the wave equation :
∂2η
∂2η
−
gh
=0
0
∂t2
∂x2
√
with a celerity c0 = gh0 . We can put some non linearities in these equation :
∂u
total derivative ∂t +u ∂u
∂x , and flux will be (h0 +η)u, this will give the Saint Venant
(Shallow Water) equations. Or, we can imagine a linear small perturbation, with
a flow which is not a plug flow, but depends on x and y : u(x, y, t), this will give
the famous dispersion relation :
ω 2 = gk tanh(kh0 ).
for long waves, this equation gives again the ∂’Alembert one with c0 = ω/k =
√
gh0
Then the solitary wave will be presented as a mix of all this non linearity (η/h0
small) and dispersion, kh0 small. There are simple argument to settle at this point
KdV such as :
• from the wave equation the solution is u = gη/c0 , with c20 = gh0
from the wave equation the wave which travels from left to right, this solution
satisfies
∂η
∂η
+ c0
=0
∂t
∂x
p
•
from
the
linear
Wave
Theory
ω
=
gk tanh(kh0 )., this is expanded in ω =
q
√
1
3
2
4
gk h0 − 3 gk h0 + ... for long waves, and ω = gh0 k(1 − 61 k 2 h20 + ... then for a
- MHP KdV PYL 2.2-
P.-Y. Lagrée, KdV
∂
∂
wave eiωt−ikx we identify ω with −i ∂t
, and k with i ∂x
therefore
ρ(
∂
∂
1
∂3
= −c0 (
− h20 3 ) + ...
∂t
∂x 6 ∂x
so
∂
∂t
ρ(
∂v
∂v
∂v
∂p
+u
+v )=−
− ρg,
∂t
∂x
∂y
∂y
notice the irrotational hypothesis which is very strong, and which is not so much
discussed in the literature. These equation are for −h0 < y < η. Boundary conditions are the pressure p(x, y = η) = p0 at the surface (we neglect here surface
tension) and the relation linking the perturbation of the moving interface and the
velocity of the water just at the surface :
∂η
∂η
1 ∂3η
+ c0
+ c0 h20 3 + ... = 0
∂t
∂x
6 ∂x
• from the Shallow water theory, the time derivative is
∂u
∂u
∂p
∂u
+u
+v )=− ,
∂t
∂x
∂y
∂x
∂
+ u ∂x
, so :
v(x, η, t) =
∂η
∂η
η ∂η
is in fact
+ c0
∂t
∂t
h0 ∂x
∂η(x, t)
∂η(x, t)
+ u(x, η, t)
∂t
∂x
and slip conditions at y = −h0 .
• Then the final equation is estimated as :
3.2
∂η
∂η
η ∂η
1 ∂3η
+ c0
+ c0
+ c0 h20 3 = 0
∂t
∂x
h0 ∂x
6 ∂x
Equations with the potential
From irrotational flow, one usually define a potential :
But here we will present the complete theory with all the expansions, we will
see that the effective equation is
∂η
3η ∂η
1 ∂3η
∂η
+ c0
+ c0
+ c0 h20 3 = 0
∂t
∂x
2h0 ∂x
6 ∂x
u=
∂φ
∂φ
; v=
∂x
∂y
and from incompressibility :
We will use the longitudinal scale, say λ such that the dispersive term is small, it
is (h0 /λ)2 1 and the non linear term is small as well η/h0 1, but :
∂2φ ∂2φ
+ 2 = 0,
∂x2
∂y
η/h0 = (h0 /λ)2 1
this Laplacian must be solved with boundary conditions at the free surface and at
∂η
the bottom. At the free surface v(x, y = η, t) = ∂η
∂t + u ∂x which is
We will present the complete theory with small parameters (h0 /λ)2 1 and
η/h0 1, (we will define δ = h0 /λ and ε = η0 /h0 the small parameters) and
dominant balances and multiple scale. This, unfortunately, makes the ∂’Alembert
and Shallow Water description less clear. that is the price to catch the Soliton.
∂φ
∂η ∂φ ∂η
|y=η =
+
∂y
∂t
∂x ∂x
and writing the momentum with irrotationality
Let us do the reset.
ρ(
3
3.1
Equations
∂u
∂u
∂u
∂u 1 ∂u2
1 ∂v 2
∂u ∂v
+u
+ v ) = ρ(
+
+
+ v(
−
)))
∂t
∂x
∂y
∂t
2 ∂x
2 ∂x
∂y
∂x
gives a kind of Bernoulli equation :
Equations with dimensions
Write Navier Stokes, without dimension, put Re = ∞ come back with dimensions, here is Euler incompressible and irrotational (remember conservation of
vorticity in ideal fluids) :
∂u ∂v
∂u ∂v
+
= 0, and (
−
)=0
∂x ∂y
∂y
∂x
ρ(
∂p
∂u 1 ∂ 2
+
(u + v 2 )) = − ,
∂t
2 ∂x
∂x
with the pressure at the surface p(x, y = η) = p0 . Note here that the real jump
relation in inviscid flow between two media 1 and 2 may be written with the surface
tension :
→
− −
p1 (x, η(x, t), t) − p2 (x, η(x, t), t) = σ ∇ · →
n 12
- MHP KdV PYL 2.3-
P.-Y. Lagrée, KdV
3.3
the normal to the surface
→
−
n 12
Equations without dimension
It is here a good idea to distinguish between scale in x and scale in y, so we
write x = λx̄ and y = h0 ȳ, the surface elevation is η = η0 η̄. The potential is not
known, as the time : φ = ϕφ̄, t = τ t̄. We define the ratio of scales δ = h0 /λ and
ε = η0 /h0 . Then incompressibility :
p
= (−∂η/∂x, 1)/ (1 + (∂η/∂x)2
Here we neglect σ/(ρU02 h0 ) (the inverse of the Weber number).
At equilibrium, when there is no flow
p = p0 + ρg(−y)
δ2
Then p0 + ρg(−y) is the ”hydrostatic” pressure, we look at P the departure from
it. p0 + ρg(−y) + P Pressure is then
∂ 2 φ̄ ∂ 2 φ̄
+ 2 =0
∂ x̄2
∂ ȳ
(1)
momentum at the surface
2
p = p0 + ρg(−y) + P.
εη̄ +
At the interface
(2)
velocity at the surface
p(x, y = +η) = p0 . so p(x, y = η) = p0 = p0 + ρg(−η) + P (x, y = η)
∂ η̄ ϕτ δ 2 ∂ φ̄ ∂ η̄
ϕτ ∂ φ̄
=
+ 2
εh20 ∂ ȳ
∂ t̄
h0 ∂ x̄ ∂ x̄
We hence deduce
P (x, y = η) = ρgη.
∂ φ̄
=0
∂ ȳ
∂p
∂P
∂η
=
= ρg
∂x
∂x
∂x
We will first explore two simplifications of this system.
The reader may skip to read 4.4 and come back here.
At the free surface, the previous momentum with irrotationality and the pressure :
∂ ∂φ 1 2
ρ (
+ (u + v 2 ) + gη) = 0,
∂x ∂t
2
so that the final system (far usptream every thing is 0) :
∂φ 1 ∂φ 2 ∂φ 2
+ (
+
) + gη = 0
∂t
2 ∂x
∂y
∂η
∂t
∂η
+ u ∂x
which is :
∂φ
∂η ∂φ ∂η
=
+
∂y
∂t
∂x ∂x
4
This is the full system we have to solve. First we write it without dimension.
Solutions of the Equations
4.1
Fully linearised waves
The first simple case is the case of fully linear waves in a shallow water, this leads
to the ∂’Alembert equation as we just will see. First we start from the Laplacian
Eq. 1
∂ 2 φ̄
∂ 2 φ̄
= −δ 2 2 ,
2
∂ ȳ
∂ x̄
2
so ∂∂ ȳφ̄2 ' 0 at leading order φ̄ is only a function of x̄ a first integration gives (and
this is the transverse velocity)
On bottom y = −h0 v = 0 which is
∂φ
= 0.
∂y
(3)
at the bottom ȳ = −1
The active part of the pressure is related to variations of interface η :
at the free surface v =
2
ϕ ∂ φ̄
ϕ2
∂ φ̄
∂ φ̄
+
+
)=0
(δ 2
gτ h0 ∂ t̄
2gh30
∂ x̄
∂ ȳ
∂ 2 φ̄
∂ φ̄
= A(x̄) − δ 2 2 ȳ
∂ ȳ
∂ x̄
2
from the value at the bottom −1 we have ∂∂φ̄ȳ = −δ 2 ∂∂ x̄φ̄2 (ȳ + 1). We have to look
at the surface, it is in ȳ = εη, as ε is small, this is ȳ = 0. This is ”flattening”
- MHP KdV PYL 2.4-
P.-Y. Lagrée, KdV
of boundary conditions. The domain is −1 ≤ ȳ ≤ 0. At the surface ȳ = 0, the
gradient is the variation of interface, hence
∂ φ̄
∂ 2 φ̄
εh2 ∂ η̄
= −δ 2 2 (0 + 1) = 0
∂ ȳ
∂ x̄
ϕτ ∂ t̄
so that ϕτ = εδ −2 h20 and
∂ 2 φ̄
∂ η̄
=
2
∂ x̄
∂ t̄
The momentum at the surface (Eq. 2), with ϕ = εgτ h0 , gives :
η̄ +
∂ φ̄
= 0.
∂ t̄
2
2
∂ φ̄
∂ φ̄
∂ φ̄
∂ φ̄
∂ η̄
∂ φ̄ ∂ η̄
+
) = 0 and as well at surface
+ ε2 (
=
+ε
∂ t̄
∂ x̄
∂ ȳ
∂ ȳ
∂ t̄
∂ x̄ ∂ x̄
at the bottom
∂ φ̄
ȳ = −1,
=0
∂ ȳ
the solution of
 2
∂ φ̄ ∂ 2 φ̄



+ 2
=0

2

∂ ȳ

 ∂ 2x̄
∂ φ̄
∂ φ̄
|0 +
|0 = 0
2
 ∂ t̄
∂ ȳ



∂ φ̄



|−1
=0
∂ ȳ
η̄ +
So eliminating ϕ and using the relevant scales λ = h0 /δ h0 , gives
p
λ/τ = gh0
√
√
where gh0 is a velocity. Hence ϕ = ελ gh0 . Then, eliminating φ̄ gives :
This problem as solution in ei(k̄x̄−ω̄t̄) , with this ansatz, the solution of the Laplacian,
∂ 2 φ̄
∂ φ̄
∂ φ̄
−k̄ 2 φ̄ + 2 = 0 with B.C.
|−1 = 0,
|0 = ω̄ 2 φ̄(0).
∂ ȳ
∂ ȳ
∂ ȳ
∂ 2 φ̄
∂ 2 φ̄
∂ 2 η̄
∂ 2 η̄
=
and
=
∂ x̄2
∂ t̄2
∂ x̄2
∂ t̄2
√
the ∂’Alembert wave equation of unit velocity ( gh0 ). This is valid for waves of
small amplitude ε 1 in shallow water δ 1. In the next paragraph, we will
study waves of small amplitude ε 1 in deep water δ = 1. In the paragraph after,
we will study waves in not small amplitude ε = 1 in shallow water δ 1. Finally
we will study waves of small amplitude ε 1 in shallow water δ 1 but no so
shallow.
For those various waves, we will follow the same way : do some dominant balance
to take some terms, integrate the Laplacian from bottom (slip) to top (perturbation
of free surface), which gives in fact the small transverse velocity and use of the
momentum/ Bernoulli equation.
This gives a solution in φ̄(0) cosh(k̄ ȳ + k̄)/ cosh(k̄) which preserves the
4.2
Linear dispersive solution, Airy 1845 solution
boundary condition. So
dispersion :
∂ φ̄
∂ ȳ |0
∂ φ̄
∂ ȳ |−1
=0
= φ̄(0)k̄ tanh k̄ then, it gives the famous relation of
ω̄ 2 = k̄ tanh(k̄)
with dimensions
ω 2 = gk tanh(kh0 ).
Looking a small depth
ω 2 = k(kh0 −
(kh0 )3
+ ...)
3
3
This (kh30 ) will be important for solitons, we will discuss it after. Then, the phase
velocity c = ω/k, gives at smalll kh0 ie very small depth,
p
c = gh0 .
We look here at the swell (houle in French) in open sea. this is a very classical
solution (Lamb Chap IX, Landau Lifshitz §12) The length and the depth are of
same order. So we consider δ = 1, we take the same scales in x and y, it is more
simple to use h0 here, so that the bottom will be in ȳ = −1. The equation 1 is then a
full Laplacian. Taking both scales equals is of course the first simple possibility that
we have to explore before looking at different scales. We consider small amplitude
waves so that ε 1 then from the same balance
at the surface
p of the velocityp
√ (Eq
.3) and from the momentum 2 we have τ = h0 /g then ϕ = ε gh30 = η0 gh0
This is again the shallow water velocity we have already seen in the previous
paragraph.
If depth is too small one has to take into account the surface tension : σ/(ρU02 h0 )
2
is here σ/(ρgh
p0 ). So if the depth is of same order of magnitude than the capillary
length λc = σ/(ρg) then one has to use the surface tension jump. This gives the
correct dispersion relation
∂ 2 φ̄ ∂ 2 φ̄
+ 2 =0
∂ x̄2
∂ ȳ
Notice here that for surface tension wave, small wave length travel faster. But, the
effect is reversed for Airy waves in shallow water.
ω 2 = g(1 + k 2 λ2c )k tanh(kh0 ).
- MHP KdV PYL 2.5-
P.-Y. Lagrée, KdV
2
η̄ + (
∂ φ̄ 1 ∂ φ̄
)=0
+
∂ t̄
2 ∂ x̄
1 ∂ φ̄
∂ η̄ ∂ φ̄ ∂ η̄
=
+
.
2
δ ∂ ȳ
∂ t̄
∂ x̄ ∂ x̄
Coming back with ū = φ̄x̄ we have for momentum at surface (after derivation by
x̄), a balance between the total derivative of the velocity (which is a ”plug” flow,
ū(x̄, t̄)) influenced by the variations of pressure due to the changes of surfaces η̄ :
Figure 3 – Waves on a sloping beach with Gerris. Code it at the end.
4.3
∂ ū
∂ ū
∂ η̄
+ ū
=− .
∂ t̄
∂ x̄
∂ x̄
Non linear waves : Shallow water
There is another relevant simplification of the full problem which corresponds
to flows in a small depth of water. Or, flow with changes at a scale much larger
than the depth. So, if δ = h0 /λ 1, the length wave is long compared to the
depth, and now we can allow ε = 1. The latter means that the wave may be large
in height, so non linear phenomena appear. Let us look at this : a balance of terms
with non linearities. At first order
Then working on the continuity equation, we first we integrate the Laplacian
∂ φ̄
∂ 2 φ̄
= −δ 2 2 (ȳ + 1)
∂ ȳ
∂ x̄
2
and at ȳ = η, this gives ∂∂φ̄ȳ at the interface which is δ 2 ∂∂ x̄φ̄2 (η̄ + 1). Using the
definition of ū, we inject this in relation of the surface velocity, δ disappears :
2
∂ 2 φ̄
2 ∂ φ̄
=
−δ
∂ ȳ 2
∂ x̄2
∂ η̄
∂ η̄
∂ ū
(η̄ + 1) =
+ ū .
∂ x̄
∂ t̄
∂ x̄
but non linearities in Eq. 3 give the balance τ ϕδ 2 /h20 = 1 which is τ ϕ/λ2 = 1
∂ η̄ ∂ φ̄ ∂ η̄
ϕτ ∂ φ̄
=
+
2
h0 ∂ ȳ
∂ t̄
∂ x̄ ∂ x̄
In the momentum Eq.2, we have εη̄ which is η̄, then
ϕ ∂ φ̄
gτ h0 ∂ t̄
the next two which are, with τ ϕδ 2 /h20 = 1, the non linear term
Hence, as δ is small :
2
ϕ ∂ φ̄ 1 ∂ φ̄
η̄ +
(
+
)=0
gτ h0 ∂ t̄
2 ∂ x̄
that we compare to
2
ϕ
∂ φ̄ 2
2 ∂ φ̄
2gτ h0 ( ∂ x̄ +δ ∂ ȳ ).
This system gives advection, shocks, one example is on figure 4 were we see a
moving hydrolic jump.
As we have τ ϕ/λ2 = 1 from the non
√ linearities of 3 and ϕ = gτ h0 from the non
linear terms of 2, this gives λ/τ = gh0 and ϕ = λ2 /τ .
in front of ∂∂φ̄ȳ the velocity (of Eq. 3) is λ2 /h20 ,
With these scales, the scale ϕτ
h2
0
which is large : δ −2 1. This is the subtle part, as φ̄ does not change so much
trough the layer (from the Laplacian, the variation in ȳ are of order δ 2 ), the small
variation is magnified by δ −2 , so√that the result is of order one.
√
We have
shown that λ/τ = gh0 and ϕ = λ2 /τ , so that ϕ = λ gh0 and
√
τ = λ/ gh0 , so with ε = 1 and δ 1, the final system reads :
2
∂ 2 φ̄
2 ∂ φ̄
=
−δ
∂ ȳ 2
∂ x̄2
We can write this system in a more readable way, as we obtain the famous Shallow
Water system (Saint Venant, in French, GfsRiver in Gerris) :

∂ η̄

 ∂ ū + ū ∂ ū
=−
∂ t̄
∂ x̄
∂ x̄
∂ η̄
∂


+
((1 + η)ū) = 0
∂ t̄
∂ x̄
4.4
Equations with ε and δ
The two previous subsection were in the case δ = 1, ε 1 and δ 1, ε = 1.
These are two fundamental points of view. We turn now to a case in between with
some dispersion and some nonlinearities in the set of equations (1-3). So we are
removing the dimensions in these equations at the light of the two previous cases.
The system we wrote is in fact with ϕ = εh20 /(τ δ 2 ) to preserve the balance
−2 ∂ φ̄
δ ∂ ȳ ∼ ∂∂η̄t̄ of the surface velocity Eq. 3 and Eq. 1.
√
With this ϕ taking τ = √
λ/ gh0 preserves the balance η̄ ∼ ∂∂φ̄t̄ of the momentum
Eq. 2. We have ϕ = ελ gh0 . Note that this is consistant with the scales we
obtained in the fully linearized and Shallow Water cases.
- MHP KdV PYL 2.6-
P.-Y. Lagrée, KdV
4.5
Non linearity balances dispersion ε equals δ 2
2
As δ 2 ∂∂ x̄φ̄2 +
says
h1
∂ 2 φ̄
∂ ȳ 2
= 0, by integration we have
∂ φ̄
∂ ȳ ,
The velocity at the surface (3)
1 ∂ φ̄
∂ η̄
∂ φ̄ ∂ η̄
=
+ε
,
δ 2 ∂ ȳ
∂ t̄
∂ x̄ ∂ x̄
h2
at the bottom ȳ = −1
∂ φ̄
= 0.
∂ ȳ
Solving r the Laplacian (1)
δ2
h1
∂ 2 φ̄ ∂ 2 φ̄
+ 2 = 0,
∂ x̄2
∂ ȳ
with a Poincaré expansion φ̄ = φ̄0 + δ 2 φ̄1 + δ 4 φ̄2 + ... at first oder φ0 = f (x) so
the various orders solve the recurrence
h2
∂ 2 φ̄i+1
∂ 2 φ̄i
=− 2
2
∂ ȳ
∂ x̄
Figure 4 – here example of a moving hydrolic jump computed by Gerris solution
of GfsRiver . Code it at the end.
The solution for φ̄ is then a polynom in ȳ with coefficient functions and its derivatives of x̄,
1
1
1
1
φ̄ = f (x̄) − δ 2 f 00 (x)(ȳ + ȳ 2 ) + δ 4 f 0000 (x)( ȳ 3 + ȳ 4 − ȳ) + ...
2
6
24
3
The final system with ε and δ reads :
δ2
2
so that if we substitute in
2
∂ φ̄ ∂ φ̄
+ 2 =0
∂ x̄2
∂ ȳ
1 ∂ φ̄
δ 2 ∂ ȳ
=
∂ η̄
∂ t̄
+ ε ∂∂φ̄x̄ ∂∂ x̄η̄ we have in ȳ = εη̄
−f 00 (x)(1 + εη̄) + δ 2 f 0000 (x)
momentum at the surface
2
−1
∂ η̄
∂ φ̄ ∂ η̄
=
+ε
.
3
∂ t̄
∂ x̄ ∂ x̄
Let us define ū = f 0 (other choices are possible, not the value at surface but
the mean value, they lead to the various Boussinesq systems, see after) then, this
equation is
∂ η̄
∂ ū 1 2 ∂ 3 ū
∂ η̄ ∂ ū
+
= −εū
− εη̄
− δ
,
∂ t̄
∂ x̄
∂ x̄
∂ x̄ 3 ∂ x̄3
the momentum is follows from the scale of φ̄, we notice that
2
∂ φ̄
1 ∂ φ̄
∂ φ̄
η̄ +
+ε (
+ δ −2
)=0
∂ t̄
2 ∂ x̄
∂ ȳ
velocity at the surface
1 ∂ φ̄
∂ η̄
∂ φ̄ ∂ η̄
=
+ε
2
δ ∂ ȳ
∂ t̄
∂ x̄ ∂ x̄
at the bottom ȳ = −1
∂ φ̄
=0
∂ ȳ
If, in this set of equations we put ε = 1 and δ 1 we have the shallow water
again, and if, in this set of equations we put ε 1 and δ = 1 we have Airy linear
dispersive solution again, we just look at what happens with a small wave in a not
so shallow river.
∂ φ̄
∂ φ̄1
= 0 + δ2
+ ..
∂ ȳ
∂ ȳ
so we can again neglect it in momentum at the surface (2)
- MHP KdV PYL 2.7-
2
∂ φ̄
∂ φ̄
η̄ +
+ (ε/2)(
+ δ −2 (O(δ 4 ))) = 0.
∂ t̄
∂ x̄
P.-Y. Lagrée, KdV
4.6
Which is at leading order
2
η̄ +
KdV
Let us no present the final canonical form of KdV with the final balance of
unsteadiness, non linearity and dispersion, written in a moving frame. This system
has ε terms, so we look at an expansion :
∂ φ̄
∂ φ̄
+ (ε/2)
= 0.
∂ t̄
∂ x̄
It is the usual inertial-pressure balance
ū = ū0 + εū1 + ...
∂ ū
∂ ū
∂ η̄
+ εū
=−
.
∂ t̄
∂ x̄
∂ x̄
v̄ = v̄0 + εv̄1 + ...
At this point, as we want a dominant balance of the perturbative terms, wet take
ε = δ 2 , this is η0 /h0 = (h0 /λ)2 , the ratio :
η0 λ
h30
The solution at order 0 will clearly imply ∂’Alembert equation...
∂2
∂2
ū
−
ū0 = 0,
0
∂ x̄2
∂ t̄2
2
say that ξ = x̄ − t̄ and ζ = x̄ + t̄ to classically solve the wave equation so
is called the the fundamental parameter in the theory of water waves.
So the final system of interest is :

∂ η̄
∂ ū 1 ∂ 3 ū

 ∂ η̄ + ∂ ū
= −εū
− εη̄
− ε
∂ t̄
∂ x̄
∂ x̄
∂ x̄ 3 ∂ x̄3

 ∂ ū + εū ∂ ū = − ∂ η̄
∂ t̄
∂ x̄
∂ x̄
∂
∂
∂
=
+
,
∂ x̄
∂ξ
∂ζ
then at order 0
∂
∂
(−η̄0 + ū0 ) +
(ū0 + η̄0 ) = 0
∂ξ
∂ζ
or by sum and substraction

3

 ∂ η̄ + ∂ ((1 + εη̄)ū) = − 1 ε ∂ ū
∂ t̄
∂ x̄
3 ∂ x̄3
∂ η̄

 ∂ ū + εū ∂ ū
=−
∂ t̄
∂ x̄
∂ x̄
∂
∂
(−η̄0 + ū0 ) = 0,
(ū0 + η̄0 ) = 0
∂ξ
∂ζ
Before looking at KdV, we see that this system has the non linear shallow water
terms, plus an extra term which comes from the depth which is not so shallow.
Considering linear waves, gives
∂ η̄ ∂ ū
1 ∂ 3 ū
∂ ū
∂ η̄
+
= − ε 3 and
=−
∂ t̄
∂ x̄
3 ∂ x̄
∂ t̄
∂ x̄
the plane wave solution ei(k̄x̄−ω̄t̄) gives
εk̄ 4
3
so that
−η̄0 + ū0 = F (ζ) and ū0 + η̄0 = G(ξ)
we will focus on a wave going to the right, and follow it with our horse. We deduce
that η̄0 = ū0 and that this is a function of ξ, a moving wave to the right. We prefer
now to be in the moving frame, so that ξ = x̄ − t̄.
If we specify only the right moving wave and do not care about the other, then
∂
∂
in this moving frame ∂∂x̄ = ∂ξ
, and ∂∂t̄ = − ∂ξ
. But, due to the ε terms, we guess
that it will create cumulative terms. So we do a multiple scale analysis, with a
slow time τ = εt̄ we then have for the time an extra slow term,
∂
∂
=
,
∂ x̄
∂ξ
remember that Airy wave gave the dispersion relation
ω 2 = k̄ tanh k̄
3
∂
∂
∂
=−
+
∂ t̄
∂ξ
∂ζ
∂
∂
(−ū0 + η̄0 ) +
(ū0 + η̄0 ) = 0
∂ξ
∂ζ
We may write it :
ω̄ 2 = k̄ 2 −
∂2
∂2
η̄
−
η̄0 = 0,
0
∂ x̄2
∂ t̄2
which gives for long wave expansion ω 2 = k̄(k̄ − k̄3 + ....), so we find without
surprise the same relation (after change of scale which implies the δ 2 ).
∂
∂
∂
=−
+ε
∂ t̄
∂ξ
∂τ
so the momentum equation is after expansion ū = ū0 +εū1 +... and v̄ = v̄0 +εv̄1 +...
gives
∂
∂
∂
∂
∂ ū0
(−ū0 + η̄0 ) + ε( ū0 −
ū1 +
η̄1 + ū0
)=0
∂ξ
∂τ
∂ξ
∂ξ
∂ξ
- MHP KdV PYL 2.8-
P.-Y. Lagrée, KdV
∂η
∂u
depending on θ and notice that ∂h
∂t = ∂t = −c0 ∂x . These are other common
Shallow Water equation with a dispersive term (see literature)

3 3

 ∂h + ∂(hu) = h ∂ u
∂t
∂x
6 ∂x3
∂u
∂u
∂h h2 ∂ 3 u


+u
= −g
+
∂t
∂x
∂x
2 ∂x2 ∂t
whereas the mass conservation gives
∂
∂
∂
∂
∂ η̄0
∂ ū0
1 ∂ 3 ū0
)=0
(ū0 − η̄0 ) + ε( η̄0 −
η̄1 +
ū1 + ū0
+ η̄0
+
∂ξ
∂τ
∂ξ
∂ξ
∂ξ
∂ξ
3 ∂ξ 3
from the first, we have
∂
∂
∂
∂ ū0
ū1 −
η̄1 =
ū0 + ū0
∂ξ
∂ξ
∂τ
∂ξ
They give all KdV so that they are a bit more universal.
we substitute in the second
4.8
3
(
∂
1 ∂ ū0
∂
∂ ū0
∂ η̄0
∂ ū0
η̄0 +
ū0 + ū0
+ ū0
+ η̄0
+
)=0
∂τ
∂τ
∂ξ
∂ξ
∂ξ
3 ∂ξ 3
but the wave solution
(
∂
∂ξ (ū0
Coming back with variables, assuming that the fundamental parameter in the
theory of water waves is of order one :
η0 λ2
= O(1),
h30
− η̄0 ) = 0 gives ū0 = η̄0 + F (τ ) so
∂
∂
∂ η̄0
∂ η̄0
1 ∂ 3 η̄0
F + 2 η̄0 + η̄0
+ 2(η̄0 + F (τ ))
+
)=0
∂τ
∂τ
∂ξ
∂ξ
3 ∂ξ 3
the KdV equation reads with scales :
so that F (τ ) is a secular term, it must be always 0, hence we finally obtain the
KdV equation :
3 ∂ η̄0
1 ∂ 3 η̄0
∂
η̄0 + η̄0
+
=0
∂τ
2 ∂ξ
6 ∂ξ 3
”Le chemin qui y conduit semble long car nous avons détaillé chaque étape. On
peut trouver des solutions élégantes pour établir rapidement l’équation KdV mais
le prix à payer est que l’on ne contrôle pas les approximations” as says [12], who
takes another tortuous path.
4.7
∂
3c0 ∂η
c0 h20 ∂ 3 η
∂
η + c0 η +
+
=0
η
∂t
∂x
2h0 ∂x
6 ∂x3
√
with c0 = gh0 . this equation represent the dominant balance between non
linearities that will create a hydraulic jump and dispersion that destroys the wave
in several waves. When there is balance, a special wave, the ”soliton”, exists. It
does not change in shape.
The solution with no elevation of the surface up and down stream needs some
algebra. We look at traveling waves f (x − Ct) = f (s), so the solution of :
3
∂f (x, t) 1 ∂ 3 f (x, t)
∂
f (x, t) + f (x, t)
+
=0
∂t
2
∂x
6 ∂x3
Boussinesq Equation
From the point of view we developed :


 ∂h + ∂(hu)
∂t
∂x

 ∂u + u ∂u
∂t
∂x
One famous solution
so that
f 00 = 6Cf − 9/2f 2 + cste
h3 ∂ 3 u
=
3 ∂x3
∂h
= −g
∂x
for s → ±∞ perturbations of f are zero, then multiply by f 0 and integrate again :
(f 0 )2 = 6Cf 2 − 3f 3
Depending on the choice of the horizontal fluid velocity given at some definite
height in the fluid column, see [1], in fact different types of Boussinesq equations
have been introduced,

3
3

 ∂h + ∂(hu) = h (θ2 − 1 ) ∂ η
∂t
∂x
2
3 ∂x2 ∂t 2 3
∂u
∂u
∂h
h ∂ u


+u
= −g
+ (1 − θ2 )
∂t
∂x
∂x
2 ∂x2 ∂t
a function like α cosh(βs)γ works ! ! with α = 2C and β = (3C/2)1/2 , γ = −2 so
that :
η0
q
η=
η0
cosh2 2h1 0 3ηh0 (x − c0 (1 + 2h
)t)
This is the ”Soliton” or Solitary Wave solution. It has a lot of properties. Other
solutions exist such as cnoidal waves (see literature, Whitham Lighthill, Debnath)
- MHP KdV PYL 2.9-
P.-Y. Lagrée, KdV
5
The ondular bore or Mascaret
The simple bore is solution of Shallow water equation, but if the river is with
enough water, then the bore breaks and create an ”ondular bore” (Mascaret, Poroqua, see Chanson [2] definite book : ”Tidal Bores, Aegir, Eagre, Mascaret, Pororoca : Theory and Observations”). It is present in some rivers in the world and it
is due to the high tide. The mascaret on the ”Severn River” (Lighthill book [10]),
is famous. But the mascaret on the Dordogne in Saint Pardon is spectacular (of
course the ondular bores of China and Amazonia are the largest in the world). see
figure 7. Far upstream, it breaks in solitary waves.
Figure 5 –
Soliton in a flume at Palais de la Découverte, photo
PYL http://www.lmm.jussieu.fr/ lagree/SIEF/SIEF97/solitongd.mov , right the
1/cosh2 solution.
Figure 7 – Mascaret or Ondular Bore at Saint Pardon on the Dordogne, Photo PYL See other photos : http://www.lmm.jussieu.fr/∼
lagree/SIEF/SIEF97/sieft97m.html
According to Whitham [15] the ondular bore equation is then the kdV,
√
1
Figure 6 – 3 solitons of various selfsimilar shape 0 / cosh2 2 30 x , they are
in the moving frame. The larger the height, the thinner the width, the faster the
wave. Time increases from bottom to top.
∂
3c0 ∂η
c0 h20 ∂ 3 η
∂2η
∂
η + c0 η +
η
+
−ν 2 =0
3
∂t
∂x
2h0 ∂x
6 ∂x
∂x
√
with c0 = gh0 and ν an ad hoc viscous coefficient necessary for the model but
without real significance. The boundary conditions are different from KdV as teh
levels are not the same downstream and upstream.
- MHP KdV PYL 2.10-
P.-Y. Lagrée, KdV
3.0
2.5
Figure 10 – ondular bore in a channel ENSTA experimental lab. Batterie de l’Yvette, photo PYL. See other films :
2.0
1.5
http://www.lmm.jussieu.fr/∼lagree/SIEF/SIEF97/MAQUETTE/mascaret.html
1.0
0.5
-40
-30
-20
-10
Figure 8 – A ”Mascaret” with Gerris.
3
“
2
Figure 11 – A hydraulic jump is metamorphosed in a undular bore due to a small
increase in depth. photo PYL, Baie de la Fresnaye (22) Port à la Duc. [click to
launch the movie, Adobe Reader required]
1
0
-1
-15
-10
-5
0
5
Figure 9 – A ”Mascaret” in the moving frame, numerical solution of the reduce
equation y 00 (x) − my 0 (x) − y(x) + y(x)2 = 0, ”Model bore structure” see figure 13.6
page 484 of Whitham [15]
Figure 12 – Some meters down stream, the hydraulic jump changes ... into an
undular bore photo PYL, Baie de la Fresnaye (22) Port à la Duc.
- MHP KdV PYL 2.11-
P.-Y. Lagrée, KdV
6
Conclusion
http://en.wikipedia.org/wiki/Korteweg-de_Vries_equation
http://en.wikipedia.org/wiki/Airy_wave_theory
In this chapter, we observed waves in water. First, we study waves of small
amplitude ε 1 in shallow water δ 1. This gives the ∂’Alembert wave
equation. Second, we study waves of small amplitude ε 1 in deep water δ = 1.
This is Airy wave theory. Third, we study waves in not small amplitude ε = 1
in shallow water δ 1. This is shallow water. Finally we study waves of small
amplitude ε 1 in shallow water δ 1 but no so shallow, with ε = δ 2 1. This
is Boussinesq KdV theory.
up to date 12 novembre 2015
This course is a part of a larger set of files devoted on perturbations methods,
asymptotic methods (Matched Asymptotic Expansions, Multiple Scales) and
boundary layers (triple deck) by P.-Y . L agrée .
The web page of these files is http://www.lmm.jussieu.fr/∼lagree/COURS/M2MHP.
The Soliton and the Ondular Bore are nice examples of waves.
Références
[1] Magnar Bjørkavåg, Henrik Kalisch Wave breaking in Boussinesq models for
undular bores Physics Letters A 375 (2011) 1570–1578
[2] H. Chanson (2011), ”Tidal Bores, Aegir, Eagre, Mascaret, Pororoca : Theory
and Observations”, Word Scientific googlebook
[3] F. Coulouvrat, notes de Cours de DEA de Mécanique LMM/UPMC, Introduction aux Méthodes Asymptotiques en Mécanique.
[4] Debnath L. (1994) ”Nonlinear Water Waves” Academic Press.
[5] Lokenath Debnath Nonlinear Partial Differential Equations for Scientists and
Engineers googlebook
[6] J.E. Hinch Perturbation Methods, Cambridge University Press, (1991)
[7] J. Kevorkian & J.D. Cole, Perturbation Methods in Applied Mathematics,
Springer (1981)
[8] H. Lamb, hydrodynamics
[9] Landau Lifshitz
[10] Lighthill J. (1978) ”Waves in fluids” Cambridge Univ. press
[11] Newell A. C. (1985) : ”Solitons in mathematics and physics”, SIAM RCSAM
n˚48
[12] Peyrard Dauxois Physique des solitons
[13] Gérard A. Maugin Solitons in elastic solids (1938–2010) Mechanics Research
Communications 38 (2011) 341–349
[14] Michel Remoissenet (1999) Waves Called Solitons : Concepts and Experiments
googlebook
Figure 13 – From the book Sir James Lighthill and Modern Fluid Mechanics
Lokenath Debnath
[15] Gerald Beresford Whitham, ”Linear and Nonlinear Waves” Wiley-Interscience
1974 googlebook
- MHP KdV PYL 2.12-
P.-Y. Lagrée, KdV
/Users/pyl/
...
/kdv.pdf
- MHP KdV PYL 2.13-
P.-Y. Lagrée, KdV
Some Gerris code for water waves
mkdir SIM
rm SIM/sim*
InitFraction T ((h0 - y))
Init { } {U = Uhoul*0 V = Vhoul*0
}
# air/water density ratio si T1=0 RATIO si T1=1 1
PhysicalParams { alpha = 1./VAR(T1,RATIO,1.) }
gerris2D -m hydrolicjump3Bv.gfs | gfsview2D
AdaptGradient { istep = 1 } { cmax = 0.0 maxlevel = LEVEL1 } U*T
ProjectionParams { tolerance = 1.e-3 }
# Title: Airy waves
ApproxProjectionParams { tolerance = 1.e-3 }
#
# Description:
#
# Author: PYL
Define Uhoul 0.25*sin(omega*t + 2*pi*x/lambda)*cosh(2*pi*y/lambda)/cosh(2*pi/lambda)*(y<1.1)
Define Vhoul -0.25*cos(omega*t + 2*pi*x/lambda)*sinh(2*pi*y/lambda)/cosh(2*pi/lambda)*(y<1.1)
Define LEVEL2 ((LEVEL-2) *(y<h0+.3)+(LEVEL-4)*(y>=h0+.3))
Define LEVEL1 (((LEVEL-2)*(y<=h0-.3))+(LEVEL*(y>(h0-.3)&&(y<h0+.4)))+(LEVEL-3)*(y>=h0+.4))
Define Nraf 9
# suffit 8 pour houle simple
3 2 GfsSimulation GfsBox GfsGEdge {
# shift origin of the domain
x = 0.5 y = 0.5 } {
Global {
#define LEVEL Nraf
#define h0 1
#define RATIO (1.2/1000.)
#define VAR(T,min,max) (min + CLAMP(T,0,1)*(max - min))
#define pi 3.141516
#define eps 1.e-6
#define lambda 4.0
#define omega sqrt(2*pi/lambda*tanh(2*pi/lambda))
}
PhysicalParams { L = 10 }
Refine LEVEL2
VariableTracerVOF T
VariableFiltered T1 T 1
Time {end = 100 }
RefineSolid Nraf
Solid ( y + 0.1*(x-30./2))
Source V -1.
Source U 0.0
RemoveDroplets { istep = 1 } T -2
OutputTime { step = 2 } stderr
OutputSimulation { istep = 25} stdout
# noter le format 000
OutputSimulation { step = 0.25 } SIM/sim-%06.2f.gfs
}
GfsBox {
left = Boundary {
BcNeumann U 0
BcNeumann T 0
}
top = Boundary
bottom = Boundary {
BcDirichlet V 0
}}
- MHP KdV PYL 2.14-
P.-Y. Lagrée, KdV
GfsBox {
top = Boundary
bottom = Boundary
BcDirichlet V 0
}}
{
GfsBox {
top = Boundary
bottom = Boundary {
BcDirichlet V 0
}
right = Boundary {
BcDirichlet U Uhoul
BcDirichlet V Vhoul
BcNeumann T 0
}
}
1 2 right
2 3 right
Improve this code, verify that the dispersion relation works, try to do a solitary
wave and a mascaret.
Here code for Saint Venant hydraulic jump (the bore)
#
#
#
#
#
#
Title: Steady Hydraulic Jump
Description:
Author: PYL
Command: gerris2D dam.gfs
# Required files: dam.plot
# Generated files: jump.gif
##
# F1^2 0.375
# F2^2 0.
# h1 1
# h2 0.5
# W 1.22474
#
#Define L0 10
#
# Use the GfsRiver Saint-Venant solver
1 0 GfsRiver GfsBox GfsGEdge {} {
PhysicalParams { L = 10 }
RefineSolid 9
# Set a solid boundary close to the top boundary to limit the
# domain width to one cell (i.e. a 1D domain)
Solid (y/10. + 1./pow(2,9) - 1e-5 - 0.5)
# Set the topography Zb and the initial water surface elevation P
Init {} {
Zb = 0
U = 0.387632*(x<-3)+(-.22474*0.5)*(x>-3)
P = {
double p = x < -3 ? 1 : 0.5;
//
p = 1+(1.30277563773199-1)*(1+tanh(x))/2;
return MAX (0., p - Zb);
}
}
PhysicalParams { g = 1. }
# Use a first-order scheme rather than the default second-order
# minmod limiter. This is just to add some numerical damping.
AdvectionParams {
# gradient = gfs_center_minmod_gradient
gradient = none
}
Time { end = 7}
OutputProgress { istep = 10 } stderr
# Save a text-formatted simulation
OutputSimulation { step = 0.1 } sim-%g.txt { format = text }
# Use gnuplot to create gif images
EventScript { step = 0.1 } {
time=‘echo $GfsTime | awk ’{printf("%4.1f\n", $1);}’‘
- MHP KdV PYL 2.15-
P.-Y. Lagrée, KdV
cp sim-$GfsTime.txt sim.txt
montée de marée
cat <<EOF | gnuplot
load ’dam.plot’
set title "t = $time"
set term postscript eps color 14
set output "sim.eps"
ut
ressa
h(x)= 1-(0.5)*(x>-3+1.*$time)
plot [-5.:5.][0:2]’sim-$GfsTime.txt’ u 1:7:8 w filledcu lc 3, ’sim-0.txt’ u 1:7 w l lw 4 lc 1 lt 1,h(x)
rivière
EOF
time=‘echo $GfsTime | awk ’{printf("%04.1f\n", $1);}’‘
convert -density 300 sim.eps -trim +repage -bordercolor white -border 10 -resize 640x282! sim-$time.gif Ressaut mobile
rm -f sim.eps
}
# 1:x 2:y 3:z 4:P 5:U 6:V 7:Zb 8:H 9:Px 10:Py 11:Ux 12:Uy 13:Vx 14:Vy 15:Zbx 16:Zby
# Combine all the gif images into a gif animation using gifsicle
h1
h2
EventScript { start = end } {
gifsicle --colors 256 --optimize --delay 25 --loopcount=0 sim-*.gif > mjump.gif
U1 = 0.3876
U2 = -0.22474 h1 = 1 h2= 0.5 W=1
rm -f sim-*.gif sim-*.txt
}
Figure 14 – bore at Port à la Duc, baie de la Fresnaye. Photo PYL and with
}
Gerris
GfsBox {
left = Boundary { BcNeumann U 0 }
right = Boundary { BcNeumann U 0 }
}
..
- MHP KdV PYL 2.16-
P.-Y. Lagrée, KdV
Raymond Subes ”Sans Titre” 1961 (entrée de Jussieu Quai Saint Bernard)
- MHP KdV PYL 2.17-
P.-Y. Lagrée, KdV