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On the Change of Form of Long
Waves advancing in a Rectangular
Canal, and on a New Type of Long
Stationary Waves
(Diederik Johannes Korteweg (* 31 March 1848, † 10 May 1941)
(Gustav de Vries (* 22 January 1866, † 16 December 1934)

Philosophical Magazine
1895

Contents
1. Introduction

3

dt

6

3. Stationary Waves

11

4. Stationary Periodic Waves (Cnoidal Waves)

14

5. Deformation of Non-Stationary Waves

18

6. Calculation of the Fluid Motion for Stationary Waves to the Higher Order of Approximation

23

7. Calculation of the Equation of the Surface

28

A. List of symbols

31

2. The Formula for

2

1. Introduction
In such excellent treatises on hydrodynamics as those of Lamb and Basset,
we find that even when friction is neglected long waves in a rectangular canal
must necessarily change their form as they advance, becoming steeper in
front and less steep behind1 . Yet since the investigations of de Boussinesq2 ,
Lord Rayleigh3 , and St. Venant4 on the solitary wave, there has been some
cause to donbt the truth of this assertion. Indeed, if the reasons adduced
were really decisive, it is difficult to see why the solitary wave should make
an exception5 ; but even Lord Rayleigh and McGowan6 , who have successfully
and thoroughly treated the theory of this wave, do not directly contradict the
statement in question. They are, as it seems to us, inclined to the opinion
that the solitary wave is only stationary to a certain approximation.
It is the desire to settle this question definitively which has led us into the
somewhat tedious calculations which are to be found at the end of our paper.
We believe, indeed, that from them the conclusion may be drawn, that in a
frictionless liquid there may exist absolutely stationary waves and that the
form of their surface and the motion of the liquid below it may be expressed
by means of rapidly convergent series. But, in order that these lengthy calculations might not obscure other results, which were obtained in a less
elaborate way, we have postponed them to the last part of our paper.
First, then, we investigate the deformation of a system of waves of arbitrary
shape but moving in one direction only, i.e. we consider one of the two
systems of waves, starting in opposite directions in consequence of any disturbance, after their complete separation from each other. By adding to the
motion of the fluid a uniform motion with velocity equal and opposite to the
velocity of propagation of the waves, we may reduce the surface of such a
system to approximate, but not perfect, rest.
If, then, l + η (η being a small quantity) represent the elevation of the surface
above the bottom at a horizontal distance x from the origin of coordinates,
1
2
3
4
5

6

It seems that this opinion was expressed for the first time by Airy, “Tides and Waves”,
Encyclopædia Metropolitana 1845.
Comptes Rendus, 1871, Vol. LXII.
Philosophical Magazine, 1876, 5th series, Vol. 1, p. 257.
Comptes Rendus, 1885, Vol. CI.
Though the theory of the solitary wave is duly discussed in the treatise of Basset, the
inconsistency of his result with the doctrine of the necessary change of form of long
waves seems not to have sufficiently attracted the attention of the author.
Philosophical Magazine 1891, 5th Series, Vol. XXXII.

3

1. Introduction
we have succeeded in deducing the equation


r
2
1 ∂2η
1 2
η
+
αη
+
σ

2
3
3 ∂x2
∂η
g
3
= ·
·
,
∂t
2
l
∂x
where α is a small but arbitrary constant, which is in close connection
with the exact velocity of the uniform motion given to the liquid, and where
σ = 13 l3 − Tρgl depends upon the depth l of the liquid, upon the capillary tension
T at its surface, and upon its density ρ.
On assuming ∂η
= 0 we of course obtain the differential equation for station∂t
ary waves, and it is easily shown that the well-known equation
r
h
η = h sech2 x ·

of the solitary wave is included as a particular case in the general solution
of this equation. But, in referring to this kind of wave, we have to notice the
result that, taking capillarity into account, a negative wave will become the
stationary one, when the depth of the liquid is small enough.
On proceeding then to the general solution, a new type of long stationary
wave is detected, the shape of the surface being determined by the equation
!
r
r
h+k
h
) .
(mod M =
η = h cn2 x ·

h+k
We propose to attach to this type of wave the name of cnoidal waves (in analogy with sinusoidal waves). For k = 0 they become identical with the solitary
wave. For large values of k they bear more and more resemblance to sinusoidal waves, though their general aspect differs in this respect, that their
elevations are narrower than their hollows; at least when the liquid is not too
shallow, in which latter case this peculiar feature is reversed by the influence
of capillarity.
For very large values of k these cnoidal waves coincide with the train of oscillatory waves of unchanging shape discovered by Stokes7 , which therefore
in the theory of long waves8 constitutes a particular case of the cnoidal form.
Indeed the equation9 is obtained by Stokes, when written in our notation,
becomes




3h2 λ2
4πx
2πx

cos
;
η = h cos
λ
64π 2 l3
λ
7
8
9

Transactions of the Cambridge Philosophical Society Vol. VIII. (1847), reprinted in
Stokes, Mathematical and Physical Papers, Vol. I. p. 197.
Stokes’ 8olution is more general in so far as it applies also to those cases wherein the
depth of the liquid is moderate or large in respect to the wave-length.
Stokes, Mathematical and Physical Papers, Vol. I. p. 210.

4

1. Introduction
but, as Sir G. Stokes remarks, in order that the method of approximation
2
adopted by him may be legitimate, λl3h must be a small fraction. Now, when
capillarity is neglected, the wave-length λ of our cnoidal waves is equal to

4K l3
p
,
3 · (h + k)
and therefore
λ2 h
16
16K 2 h
= M 2K 2 .
=
3
l
3 · (h + k)
3
This is a small fraction only when M , the modulus, is small, but the cnoidal
waves then resemble sinusoidal waves; and it is obvious that in this case the
equation of their surface may be developed in a rapidly convergent Fourierseries, of which Sir G. Stokes has given the first two terms.
After some more discussion about these cnoidal waves, concerning their velocity of propagation and the motion of the particles of fluid below their surface, we proceed to a closer examination of the deformation of long waves.
to various types of non-stationary
To this effect we apply the equation for ∂η
∂t
waves, and it will appear that, though sinusoidal waves become steeper in
front when advancing, other types of waves may behave otherwise.

5

2. The Formula for dη
dt
In our investigations (in accordance with the method used by Lord Rayleigh,
Philosophical Magazine 1876, Vol. I. p. 257, whose paper bas been of great
influence on our researches), we start from the supposition that the horizontal and vertical u and v of the fluid may be expressed by rapidly convergent
series of the form
u =f + yf1 + y 2 f2 + . . .
v =yφ1 + y 2 φ2 + . . .
where y represents the height of a particle above the bottom of the canal,
and where f , f1 , . . . φ1 , φ2 , . . . are functions of x and t. Of course the validity
of this assumption must be proved later on by the fact that series of this
description can be found satisfying all the conditions of the problem.
From one of these conditions, viz., the incompressibility of the liquid, which
∂v
+ ∂y
= 0, we may deduce
is expressed by ∂u
∂x
1 ∂fn−1
,
n ∂x
and from another, viz., the absence of rotation in the fluid, expressed by
∂u
∂v
+ ∂x
= 0: –
∂y
1
1 ∂φn−1
∂ 2 fn−2
=−
.
f1 = 0 ; fn =
n ∂x
n(n − 1) ∂x2
φn = −

In this manner we obtain the following set of equations: –
1
∂ 2f
1
∂ 4f
u = f − y2 · 2 + y4 · 4 − . . .
2
∂x
24
∂x
3
∂f
1
∂ f
1 5 ∂ 5f
v =−y·
+ y3 · 3 −
y · 5 + ...
∂x 6
∂x
120
∂x

(2.1)
(2.2)

6

2. The Formula for


dt

and, moreover, if φ be the velocity potential and ψ the stream function:–
1
∂f
1
∂ 3f
f ∂x − y 2 ·
+ y4 · 3 − . . .
2
∂x 24
∂x
2
∂ f
1 5 ∂ 4f
1
y · 4 − ...
ψ = y · f − y3 · 2 +
6
∂x
120
∂x
Z

φ=

(2.3)
(2.4)

which set of equations satisfies for the interior of the fluid all the conditions
of the problem, whilst at the same time it is easy to see that for long waves
these series are rapidly convergent. Indeed, for such waves the state of motion changes slowly with x, and therefore the successive differential-quotients
with respect to this variable of all functions referring, as f does, to the state
of motion, must rapidly decrease.
Passing now to the conditions at the boundary, let p1 (a constant) be the atmospheric pressure, p01 the pressure at a point below the surface where the
capillary forces cease to act, and T the surface tension. We then have, distinguishing here and elsewhere by the suffix (1 ) those quantities which refer
to the surface,
∂ 2 y1
;
∂x2
but, according to a well-known equation of hydrodynamics1 ,
p01 = p1 − T ·


p01
∂φ1 1
= χ(t) −
− · u21 + v12 − gy1 ,
ρ
∂t
2
therefore

dφ1 1
T ∂ 2 y1
p1
= χ(t) −
− · u21 + v12 − gy1 + ·
= L − gy1 + M y12
ρ
dt
2
ρ ∂x2
T ∂ 2 y1
,
+ N y14 + P y16 + . . . + ·
ρ ∂x2

1

(2.5)

The Bernoulli equation.

7

2. The Formula for


dt

where
Z

∂f
1
dx − f 2 ,
∂t
2
 2
2
2
1
∂ f
1 ∂f
1 ∂ f
M= f· 2+

,
2
∂x
2 ∂x∂t 2 ∂x


∂ 4f
1 ∂f ∂ 3 f
1
1 ∂ 2f
1 ∂ 4f
+
·
,
N =− f· 4 −

24
∂x
8 ∂x2
6 ∂x ∂x3 24 ∂x3 ∂t

2
1
∂ 6f
1 ∂ 2f ∂ 4f
1 ∂ 3f
1 ∂f ∂ 5 f
1 ∂ 6f
P =
f· 6+
·
·


+
720
∂x
48 ∂x2 ∂x4 72 ∂x3
120 ∂x ∂x5 720 ∂x5 ∂t
L = χ(t) −

By differentiation with respect to x equation (refequ:5) may be written
∂M
∂N
∂P
∂y1
∂y1
∂L
+ y12 ·
+ y14 ·
+ y16 ·
+ ... − g ·
+ 2M y1 ·
∂x
∂x
∂x
∂x
∂x
∂x
3
∂y
∂y
T

y
1
1
1
+ 4N y13 ·
+ 6P y15 ·
+ ... + ·
= 0.
∂x
∂x
ρ ∂x3

(2.6)

Moreover, a second equation must hold good at the surface, viz.
∂y1
∂y1
+ v1 −
= 0.
(2.7)
∂x
∂t
In order to satisfy equations (2.6) and (2.7) by the method of successive approximations, we put y1 = l + η, f = q0 + β, where l and q0 are supposed to
be constants, and η and β small functions depending upon x and t. Dealing,
then, with the fact that for long waves, whose wave-length is great in comparison with the depth of the canal, every new differentiation with respect to
x gives rise to continually smaller quantities, these equations become as a
first approximation:–
−u1

∂η
∂β ∂β
+
+g·
= 0,
∂x
∂t
∂x
∂η ∂η
∂β
q0 ·
+
+l·
= 0,
∂x ∂t
∂x

q0 ·

and are satisfied by taking


=
= 0;
dt
dt

β=−

q0
· (η + α) ,
l

and
q0 =

p
gl ,

(2.8)

8

2. The Formula for


dt

where α is an arbitrary constant which we will supposed to be small.
It is obvious that this solution coincides with the one usually given for the
case of long waves of arbitrary shape made stationary by attributing to the
fluid a velocity equal and opposite to that of the waves, on the assumption
that the velocity in a vertical direction may be neglected and that the horizontal velocity may be considered uniform across each section of the canal.
But, if we wish to proceed to a second approximation, we have to put
q0
· (η + α + γ)
(2.9)
l
where γ is small compared with η and α. On substituting this in (2.6) and
(2.7) and on writing out the result, rejecting all terms2 which are small compared with any one of the remaining terms, we find respectively:–

 3
q0 ∂η
∂γ g
∂η
1 2
T
∂ η
·
+g·
− (η + α) ·

l g−
· 3 = 0,
(2.10)
l ∂t
∂x
l
∂x
2
ρ
∂x
f = q0 −

and
∂γ g
∂η 1 2 ∂ 3 η
q0 ∂η
·
+g·
− (2η + α) ·
+ l g · 3 = 0.
l ∂t
∂x
l
∂x 6
∂x
In eliminating

∂γ
∂x

(2.11)

from these equations, we have at last
3q0

=
·
dt
2l





1 2
η
2

+ 23 αη + 13 σ ·

∂2η
∂x2

∂x


(2.12)

where
1
Tl 3
σ = l3 −
.
3
ρg

(2.13)

This very important equation, to which we shall have frequently to revert in
the course of this paper, indicates the deformation of a system of waves of
2

The terms for instance with

∂η
∂x

·

∂2η
∂x2

and



∂η
∂x

3

∂η
are rejected in comparison with η ∂x
,
3

3

∂ η

which is retained in the equations, those with ∂γ
∂t and ∂x2 ∂t against dt .
This equation can be differentiated giving as a result:




 3

3 gl 1
∂η
3 gl 2
∂η
3 gl 1
1 3 Tl
∂ η
=
· · 2η ·
+
· α·
+
· ·
l −
·
;
dt
2 l
2
∂x 2 l
3
∂x 2 l
3
3
ρg
∂x3

p
when we set lgl = gl ≡ c̃ and if we factorise 13 l3 in the last term on the right hand side,
we get

 3
∂η
3
∂η
1 3
3
∂ η

= c̃α ·
+ c̃η ·
+ c̃l · 1 −
·
,
dt
∂x 2
∂x 6
Bo
∂x3

where Bo =

ρ g l2
T

is the the Bond number measuring the effects of surface tension.

9

2. The Formula for


dt

arbitrary shape, but moving in one direction only. Before applying it, we may
point out the close connection between the constant α, which may still be
chosen arbitrarily, and the uniform velocity given to the fluid. Indeed it is
easy to see from (2.1) and (2.9) how a variation δα of the constant α corresponds to a change δq = − ql0 δα in this velocity, but, on taking the variation of
(2.12) with respect to α, we obtain

q0
∂η
∂η
=
· δα ·
= −δq ·
,
dt
l
∂x
∂x
which equation may be easily verified geometrically.
δ

10

3. Stationary Waves
For stationary waves must dη
be
dt

1 2

η +
2

zero. Therefore we have from (2.12)

2
1
∂ 2η
αη + σ · 2 = 0 .
3
3
∂x

This gives by integration
2
1
∂ 2η
1
c1 + η 2 + αη + σ · 2 = 0 ;
2
3
3
∂x
and by multiplication with 6dη and further integration,
 2
∂η
3
2
c2 + 6c1 η + η + 2αη + σ ·
= 0.
∂x

(3.1)

(3.2)

If now the fluid be undisturbed at infinity and if l be taken equal to the depth
which it has there, then equations (3.1) and (3.2) must be satisfied by η = 0,
∂η
∂2η
= 0, and ∂x
2 = 0. Therefore, in this case c1 and c2 are equal to zero, and
∂x
equation (3.2) leads to
r
∂η
η 2 · (η + 2α)
=± −
.
(3.3)
∂x
σ
Here, before we can proceed, we have to discriminate between σ positive and
∂η
σ negative. In the first case 2α is necessarily negative because ∂x
must be
real for small values of η. If then, we put it equal to −h, we have
r
p

1

·η· h−η :
dx
σ
from which, supposing x to be zero for η = h, we easily obtain the well-known
equation of the positive solitary wave, viz.:–
r
h
2
η = h sech x ·
(3.4)

In the second case 2α must he positive. In putting it equal to h, and in substituting −η 0 for η we have from (3.3)
r
p
∂η 0
1

· η0 · h − η0 ,
∂x
−σ

11

3. Stationary Waves
or, by integration,
r

h
.
−4σ
This is the equation of a negative solitary wave, and we are able now to draw
the conclusion thatq
whenever σ is negative; that is whenever the depth of the
η = −η 0 = −h sech x ·
2

liquid is less than 3T
, the stationary wave is a negative one. For water at

20◦ C this limiting depth is equal to 0.47 cm. (T = 72, g = 981, ρ = 0.998
B.A.U.).
Now, for a further discussion of equation (3.2), we drop the assumption that
the fluid is undisturbed at infinity. If then l be taken equal to the smallest
∂η
= 0 for η = 0, and therefore in virtue of
depth of the liquid, we must have ∂x
(3.2) c2 = 0. On supposing then σ positive1 , c1 must be negative in order that
∂η
may be real for small positive values of η, but then the equation
∂x
η 2 + 2αη + 6c1 = 0
has a positive root h and a negative −k, and we may get from (3.2)
r
1
∂η

η · (h − η)(k + η) .
∂x
σ

(3.5)

(3.6)

By substitution in this equation (3.6) of η = h cos2 χ and by integration, we find
!
r
r
h
+
k
h
η = h cn2 x ·
M=
,
(3.7)

h+k
which is the equation of a train of periodic waves whose wave-length increases when k deraes.
For k = 0 this length becomes infinite, and the equation may be shown to
coincide with (3.4).
The following figure (fig. 3.1) represents such a train of stationary waves for
9
h, M = 0.8.
the case in which k = 16

1

When σ is negative, let then l be equal to the greatest depth. On substituting σ = −σ 0 ,
η = −η 0 we have again c1 negative,


dη 0
dx

2
=

1 0
η (h − η 0 )(k + η 0 ) ,
σ0

and, finally,
r
0

2

η = −η = −h cn x ·

h+k
,
4σ 0

where h and −k are the roots of η 02 − 2αη + 6c1 = 0.

12

3. Stationary Waves

Figure 3.1.: Train of stationary wave for the case in which k =

9
h,
16

M = 0.8

13

4. Stationary Periodic Waves
(Cnoidal Waves)
Proceeding now to a further investigation of the waves determined by equation (3.7), we calculate from (2.10) and (2.11) the value of γ. From these
equations we get

 3
1
∂η
1 2
T
∂ η

=− η·
+
l −
,
dx
2l ∂x
3
2gρ ∂x3
or by integration,
1
γ = − η2 +
4l



1 2
T
l −
3
2gρ



∂ 2η
,
∂x2

where the constant of integration is rejected because its retention would only
have had the effect of augmenting in equation (2.9) the value of the arbitrary
constant α.
On substituting, then, f from (2.9) in (2.1) and (2.2), observing that in virtue
of (3.1)


1
1
∂ 2η
2
2
=

·

+
4αη
+
6c
·


2
·
(h

k)η

hk
,
=

1
∂x2


these equations are replaced by
r



 
1
η2
1
T
3
g
· η + (k − h) −
+
+
· (h − k)η
l
2
4l
l
2gρσ
3
r 


1
3
1 g
1
3
+ kh − η 2
+
· (h − k)η + hk − η 2 y 2 + . . .
2
2
2σ l
2
2
r
gη · (h − η)(k + η)
·y.
v=


p
u = gl −

(4.1)
(4.2)

When k = 0 they determine the motion of the fluid for a solitary wave.

In the first place we now will endeavour to calculate the velocity of propagation. For the solitary wave this is simple enough. If we consider that the

14

4. Stationary Periodic Waves (Cnoidal Waves)
liquid at infinity is brought to rest when a uniform motion with a horizontal
velocity


p
h
−q = − gl · 1 +
(4.3)
2l
is added to the motion expressed by (4.1) and (4.2), it is clear that this velocity, with reversed sign, must be taken for the velocity of propagation of the
solitary wave.
But for a train of oscillatory waves Sir G. Stokes has shown1 that various
definitions if this velocity may be given, leading at the higher order of approximation to different values. It seemed to us most rational to define it as
the velocity of propagation of the wave-form when the horizontal momentum
of the liquid has been reduced to zero by the addition of a uniform motion.
This definition corresponds to the second one of Sir G. Stokes. According to
it, we have to solve the equation
Z λ Z l+η
dx
(u − q)dy = 0 ,
(4.4)
0

0

where q denotes the velocity of propagation, and where

2K σ
λ= √
h+k
is equal to the wave-length.
If, then,
Z
V =
0

λ

r
σ
· {(h + k)E(K) − kK}
ηdx = 4
h+k


E(K)
= λ · (h + k)
−k
K

(4.5)

(4.6)

denote the volume of a single wave reckoned from above its lowest point, we
get from (4.4), retaining only such terms as are of the first order compared
with η, h, and k:–

1

Mathematical and Physical Papers, Vol. I. p. 202.

15

4. Stationary Periodic Waves (Cnoidal Waves)



R l+η

R λ √

p
η − 12 (k − h) gl (l + η)dx
=
q = R λ R l+η

dx 0 dy
(l + η)dx
0
0




lλ p
gl · 1 − k−h
k−h V
2l
= gh · 1 −

=
lλ + V
2l



p
k + h k + h E(K)
= gh · 1 +

.
2l
l
K
0

dx

udy

0

0

gl −

pg
l

(4.7)

On subtracting this velocity from that expressed by equation (4.1), we obtain
r 
r 


g
E(k)
g
V
0
u =u−q =−
· η + k − (k + h)
=−
· η−
;
(4.8)
l
K
l
l
and it is obvious at once that in this manner we have annulled the velocity
of the particles for which
V
.
l
This last equation has a simple geometrical meaning. It designates those
particles E (fig. ??) whose height above the bottom of the channel is equal to
the height where the surface of the liquid would stand when the waves were
flattened. Therefore for a first approximation we may say that the various
particles of the fluid change the direction of their horizontal motion at the
very moment when one of these points E is passing over them.
η=

We now proceed to the calculation of the path of a single particle of fluid.
Let x0 , y0 denote the coordinates of such a particle at the origin of time, and
x0 = x0 + ξ 0 , y 0 = y0 + ζ 0 its coordinates at the time t, u0 and v 0 its horizontal and
vertical velocity at that time, l+η 0 its elevation above the bottom, then we have

Z t
g
V
0
·
η −
dt ;
ξ =
u dt =
l 0
λ
0
r Z t
Z t
g
∂η 0
0
0
ζ =
v dt =
·
y 0 dt .
l 0 ∂x
0
0

Z

t

0

r

Here η 0 is equal to the value of η for x = x0 + qt; and therefore we have
dx = (u0 − q)dt, or to a first approximation dt = − 1q dx = − √1gl dx; but then
r

Z x0 +√gt·t 
Z x0 +√gt·t
V
t
h
1
V
g
2Kx
ξ0 = − ·
η−
dx =

cn2
dx .
l x0
λ
λ
l
l x0
λ

16

4. Stationary Periodic Waves (Cnoidal Waves)
Or, according to a well-known formula2 ,
"
!


#
2K
x
+
gl
·
t
2Kx
(h
+
k)λ
0
0
· Z·
−Z ·
ξ0 =
2Kl
λ
λ

(4.9)

At the same time we have
1
ζ0 = − · y ·
l

Z


x0 + gl·t

x0

"

 rh + k
p
∂η
h
dx = − · y · cn2 x0 + gl · t
∂x
l

#
r
h+k
.
− cn2 x0 ·


(4.10)

Of course, as all fluid particles with the same y describe congruent paths,
these formulae may be simplified by supposing x0 = 0.

2



Ru
Z(u) = u 1 − E(K)
− M 2 0 sn2 u · du. Compare, for instance, Cayley, “An Elementary
K
Treatise on Elliptic Functions”, 1876, Chap. VI. §187.

17

5. Deformation of Non-Stationary
Waves
In order to study the deformation of non-stationary waves, we will now apply
our formula (2.12) to various types of waves.
Solitary Waves. – As a first example we choose a solitary wave whose surface
is given by
η = h sech2 px .

(5.1)

According to (2.12), the deformation of this wave is expressed by
 
3q0 ph

=−
· 4σp2 − h · − sech2 px
dt
l

2 (α + 2σp2 )
+ ·
· sech2 px · tanh px .
3 (4σp2 − h)

(5.2)

But before we are able to draw any conclusion from this expression, it is
necessary to separate the two parts of dη
, of which the first is due to a true
dt
change of form of the wavesurface, whilst the second may be attributed to
a small advancing motion of the wave,
√ which is left after the addition of the
uniform motion with velocity q0 = gl. To this effect we have still at our
disposal the quantity α, whose close connection with the uniform motion,
which we have added in order to make the wave nearly stationary, has been
indicated above.
One of the best ways to obtain the desired separation is certainly to make
stationary the highest point of the wave, and this is effected by fulfilling the
condition


2 · α + 2σp2 = 3 · 4σp2 − h ,
or

3
α = 4σp2 − h ;
2
for in that case equation (5.2) is simplified to


3q0 ph
=−
· 4σp2 − h · sech2 px · tanh3 px ;
dt
l

(5.3)

18

5. Deformation of Non-Stationary Waves

Figure 5.1.: Change of form of the wave calculated from (6.2), dotted line
and then, for x = 0,
∂η
∂ · ∂x
∂ · ∂η
∂η
∂t
=
is zero together with
.
∂t
∂x
∂x
In discussing this equation (5.3), we see at once that a solitary wave (5.1) is
stationary when h = 4σp2 ; and this is in accordance with the equation (3.4) of
the stationary solitary wave which we have obtained above. When h > 4σp2 ,
the change of form of the wave, calculated from (5.3), is shown by the dotted
line in fig. 5.1.
Here the wave becomes steeper in front1 , whilst for h < 4σp2 the figure would
show the opposite change of form, when, contrary to the opinion expressed
by Airy and others, the wave becomes less steep in front and steeper behind.
If, now, we take account of the fact that, as may easily be inferred from (5.1),
the wave-surface becomes steeper in proportion as p is increased, we are
then justified in saying that a solitary wave which is steeper than the stationary one, corresponding to the same height, becomes less steep in front
and steeper behind, but that its behaviour is exactly opposite when it is less
steep than the stationary one.
Cnoidal Waves. – Applying formula (2.12) to the cnoidal wave,

η = h cn2 px ,

(5.4)



3q0 ph
2 [α − σp2 · (2 − 4M 2 )]

2
=−
·
− cn px
dt
l
3 · (4σM 2 p2 − h)

· 4σM 2 p2 − h sn px · cn px · dn px .

(5.5)

we get

Supposing then


2 · α − σp2 · (2 − 4M 2 ) = 3 · (4σM 2 p2 − h) ,
1

The left side of the figure is the front side of the wave, because the wave has been made
stationary by the application of a positive velocity (i. e. from left to right) to the fluid.

19

5. Deformation of Non-Stationary Waves

Figure 5.2.: Change of form calculated for the case h − 4σM 2 p2 > 0
we have

3q0 ph
=−
· (4σM 2 p2 − h) sn3 px · cn px · dn px .
(5.6)
dt
l
Here fig. 5.2 shows the change of form calculated for the case h − 4σM 2 p2 > 0.
When h − 4σM 2 p2 = 0, the waves are stationary in accordance with (3.7),
whilst for h − 4σM 2 p2 < 0 they become steeper behind; and this last result,
since p is inversely proportional to the wave-length, may be stated by saying
that cnoidal waves become less steep in front and steeper behind when, for
a given modulus and a given height, their length is smaller than the one required for the stationary wave of this modulus and height.
In proportion as M is taken smaller the cnoidal waves more and more resemble sinusoidal waves. They would take the sinusoidal form for M = 0,
but then an infinitely small wavelength would be required for the stationary
case. For this reason sinusoidal waves may always be considered as cnoidal
waves whose length is too large to be stationary, that is, they are always becoming steeper in front.
Sinusoidal Waves. – This last result is easily verified by direct application of
(2.12) to the equation of a train of sinusoidal waves:
η = A · sin

2πx
;
λ

for, supposing
α=

2π 2 σ
,
λ2

we obtain

3q0 πA2
4πx
=
· sin
dt
2lλ
λ

20

5. Deformation of Non-Stationary Waves

Figure 5.3.: Change of form of a sinusoidal wave

Figure 5.4.: Change of form of complicated case
and from this the change of form indicated in fig. 5.3 is easily calculated.
More complicated Cases. – For the sake of curiosity, we represent by means
of the following figures the change of form for some more complicated cases.
Figs. 5.4 and 5.5 refer to the equation
η = A1 · sin

2πx 1
4πx
+ A1 · sin
.
λ
3
λ

2
In fig. 5.4 Al1 is supposed to be small compared with λl , as is the case with
2
waves of extremely small height. In fig. 5.5 we suppose λl to be small in
regard to Al1 . Generally for more complicated forms of waves these two cases
have to be discriminated. When there is a moderate proportionality between
the two fractions the result is still more complicated.
Finally, fig. 5.6 refers to the equation
η = A1 · sin

2πx 1
4πx
− A1 · sin
,
λ
3
λ

2
in case that λl is the smaller fraction.
It is worthy of remark that all these waves grow steeper in front.

21

5. Deformation of Non-Stationary Waves

Figure 5.5.: Change of form of complicated case

Figure 5.6.: Change of form of complicated case

22

6. Calculation of the Fluid Motion for
Stationary Waves to the Higher
Order of Approximation
In order to remove every doubt as to the existence of absolutely stationary
waves, we will show how by development in rapidly convergent series the
state of motion of the fluid belonging to such a wave-motion may be calculated.
Expressing again the horizontal and vertical velocity of a particle by means of
the series (2.1) and (2.2) which fulfil all the conditions for the interior of the
fluid, we have only, neglecting capillarity, to satisfy the surface-conditions,
v1 = u1 ·

∂η
,
∂x

(6.1)

and
u21 + v12 + 2gη = constant.

(6.2)

For the case of cnoidal waves, which is the general one, we have found as a
first approximation,
 2
3
∂η
= 3 η · (h − η)(k + η) .
∂x
l
But now, to obtain higher approximations, we assume, indicating by accents
differentiation with respect to x,
η 02 = aη(h − η)(k + η)(1 + bη + cη 2 + . . .) ,

(6.3)

f = q + rη + sη 2 + tη 3 + uη 4 + . . . .

(6.4)

and

On writing out (6.3), neglecting such terms as are of a higher order than the
fourth compared with η, h and k, which latter quantities are of the same order, we obtain
η 02 = ahkη + {a(h − k) + abhk} η 2 + {−a + ab(h − k)} η 3 − abη 4 ;

(6.5)

23

6. Calculation of the Fluid Motion for Stationary Waves to the Higher Order
of Approximation
and by differentiation,


3
3
1
η = ahk + {a(h − k) + abhk} η + − a + ab(h − k) η 2 − 2abη 3 .
2
2
2
00

(6.6)

From (6.4), by successive differentiations and substitutions, retaining all
th
terms up to the third and the 3 12 order, we deduce:–
f 0 =(r + 2sη + 3tη 2 )η 0 ;
1
f 00 = arhk + {ar(h − k) + abrhk + 3ashk} η
2 

3
3
+ − ar + abr(h − k) + 4as(h − k) η 2 + (−2abr − 5as)η 3 ;
2
2
000
f = [ar(h − k) + abrhk + 3ashk

+ {−3ar + 3abr(h − k) + 8as(h − k)} η + (−6abr − 15as)η 2 η 0 ;


1 2
9 2
iv
2
2
f = a rhk(h − k) + a r(h − k) − a rhk η
2
2
15
15
− a2 r(h − k)η 2 + a2 rη 3 ;
2
2

9
45 2 2 0
v
2
2
2
2
f = = a r(h − k) − a rhk − 15a r(h − k)η + a rη η ;
2
2
where η 0 is a quantity of the order 32 .
Substituting these values in equation (2.1), where y = l + η we have, retaining
terms of the third order:–
1
1
1
1
u1 =f − l2 f 00 − lηf 00 + l4 f iv = q − arl2 hk + a2 rl4 hk(h − k)
24
4
48
2
1 2
1
3
1
+ r − arl (h − k) − abrl2 hk − asl2 hk − arlhk
2
2
2
2

3
1
+ a2 rl4 (h − k)2 − a2 rl4 hk η
24
16

3
3
+ s + al2 r − abrl2 (h − k) − 2asl2 (h − k)
4
4

5 2 4
− arl(h − k) − a rl (h − k) η 2
16


5 2 3
5 2 4 3
2
+ t + abrl + asl + arl + a rl η .
2
2
16

(6.7)

24

6. Calculation of the Fluid Motion for Stationary Waves to the Higher Order
of Approximation
We find in the same way, including terms of the 3 12

th

order:–

1
1
1 5 v
v1 = − lf 0 − ηf 0 + l3 f 000 + l2 ηf 000 −
l f
6
2
120

1
1
1
1 2 5
= η 0 −rl + arl3 (h − k) + abrl3 hk + asl3 hk −
a rl (h − k)2
6
6
2
120

3 2 5
1
1
+ a rl hk + −2sl − r − arl3 + abrl3 (h − k)
80
2
2

4 3
1 2
1 2 5
+ asl (h − k) + arl (h − k) + a rl (h − k) η
3
2
8

 
5 3 3 2
3 2 5 2
3
+ −3tl − 2s − abrl − asl − arl − a rl η .
2
2
16

(6.8)

If now we write, in accordance with (6.1),
u1 =

v1
= A + Bη + Cη 2 + Dη 3 + . . . ,
η0

(6.9)

we have from (6.7) and (6.8):–
1
1
1
A = q − arl2 hk + a2 rl4 hk(h − k) = −rl + arl3 (h − k)
4
48
6
1
1
1
3
+ abrl3 hk + asl3 hk −
a2 rl5 (h − k)2 + a2 rl5 hk .
(6.10)
6
2
120
80
1
1
3
1
B = r − arl2 (h − k) − abrl2 hk − asl2 hk − arlhk
2
2
2
2
3
1
1 2 4
+ a rl (h − k)2 − a2 rl4 hk = −2sl − r − arl3
24
16
2
1
4
1
1
+ abrl3 (h − k) + asl3 (h − k) + arl2 (h − k) + a2 rl5 (h − k) .
(6.11)
2
3
2
8
3
5
3
C = s + al2 r − abrl2 (h − k) − 2asl2 (h − k) − arl(h − k) − a2 rl4 (h − k)
4
4
16
5
3
3
= −3tl − 2s − abrl3 − asl3 − arl2 − a2 rl5 .
(6.12)
2
2
16
5
3
5
(6.13)
D = t + abrl2 + asl2 + arl + a2 rl4
2
2
16
Moreover, since (6.2) may be written in the form
u21 (1 + η 02 ) + 2gη = (A + Bη + Cη 2 + Dη 3 )2 · (1 + ahkη
+ a(h − k)η 2 − aη 3 ) + 2gη = constant,

(6.14)

25

6. Calculation of the Fluid Motion for Stationary Waves to the Higher Order
of Approximation
we readily obtain
2AB + ahkA2 + 2g = 0 ,
2AC + B 2 + a · (h − k)A2 = 0 ,
2AD + 2BC − aA2 = 0 .

(6.15)
(6.16)
(6.17)

From the equations (6.10), (6.11), (6.12), (6.15), (6.16), (6.17), the six quantities q, r, s, t, a, and b may be calculated, and if we had retained everywhere
terms of one higher order, we might have got eight equations with eight unknown quantities, & c.
By a first approximation we readily obtain from (6.10) - (6.13):–
q
r=− ;
l

q
q
1
7
1
1
+ aql ; t = − 3 − aq + abql − a2 ql3 ;
2
l
4
l
3
3
48
q
q
1
A = q ; B = − ; C = 2 − aql ;
l
l
2
q
2
2
1 2 3
D = − 3 + aq − abql + a ql ;
l
3
3
6
s=

and then from (6.15) - (6.17),
3
3
; b= .
3
l
4l
Proceeding to the second approximation, we find
q 2 = gl ;

a=

(6.18)



q
h−k
1
7 q h−k
q
r =− · 1+
; A = q;
; s = 2 + aql + 2 ·
l
2l
l
4
4l
l
q q h−k
q
1
19 q h − k
B=− + ·
; C = 2 − aql −
·
;
l
l
l
l
2
8 l2
l
and then again from (6.15) and (6.16),


h−k
2
q = gl · 1 +
;
l

a=

15 h − k
3

· 4 .
3
l
4
l

(6.19)

26

6. Calculation of the Fluid Motion for Stationary Waves to the Higher Order
of Approximation
Finally, a third approximation leads to:–


q
h−k
9 (h − k)2 93 hk
3 hk
r =− · 1+

; A=q+ q· 2 ;

2
2
l
2l
20
l
80 l
4
l


2
h−k
21 q (h − k)
q q
12 q hk

·
B=− + ·

;
l
l
l
20 l
l2
5 l l2


h−k
1 (h − k)2 33 hk
2
q = gl · 1 +

.

l
20
l2
20 l2

(6.20)

By means of these results we may now readily obtain from (2.1) and (2.2)
th
expressions for u and v including respectively the terms of the 2nd and 2 12
order.
They are:–
 

3 (h − k)2 33 hk
h−k
η
h−k

+
1
+
·
1+

2l
20
l2
40 l2
l
l



2
2

3 hk 3 h − k η 9 η
+ 2 +
· 2 +
− 2 y2
4l
4 l
2 l l
4l
r 


 
g
h−k 7 η
h−k 3 η
0
·
1+
− ·
+ ·
y+ −
y3 ;
v =η
l
l
4 l
2l
2 l

(6.21)





5h−k
3 η
3
(h − η)(k + η) 1 + ·
.
η = 3 · 1−
l
4 l
4 l

(6.23)

p
u = gl ·

where

02



(6.22)

27

7. Calculation of the Equation of the
Surface
We will now show how for the equation of the surface of a stationary train of
waves a more correct expression than (3.7) can be deduced. For this purpose
we have to integrate the differential equation (6.3), or rather we have to prove
that a series can be given which solves this equation to any desired degree of
accuracy. Now such a series may be obtained in the following manner. Let
!
r
p
h1
2 1
(7.1)
η1 = h1 cn x · a · (h1 + k1 ) M =
2
h1 + k1
represent the solution of an equation
η102 = aη1 · (h1 − η1 )(k1 + η1 )

(7.2)

where h1 and k1 have values which are slightly different from those of h and
k in (6.3); then these values and the coefficients α, β, & c., of a series
η = αη1 + βη12 + γη13 + δη14 + . . .

(7.3)

may be determined in such a way1 that this series (7.3) satisfies the equation
(6.3).
Indeed, substituting (7.3) in (6.3) and taking into account (7.2), equation (6.3)
reduces to
(α + 2βη1 + 3γη12 + . . .)2 (h1 − η1 )(k1 + η1 )
=(α + βη1 + γη12 + . . .)(h − αη1 − βη12 − γη13 + . . .)(k + αη1
+ βη12 + γη13 + . . .)(1 + bαη1 + (bβ + cα)η12 + . . .) ,
and it is only necessary to equalize the coefficients of the corresponding terms
of both members of this equation.
If we retain all terms to the fourth order, we find in this way, after some
reductions:–
1

The coefficient a in (7.2) might also have been chosen slightly different in value from a
in (6.3), but this would only have introduced an unnecessary indeterminateness in the
solution.

28

7. Calculation of the Equation of the Surface

αh1 k1 − hk = 0
α (h1 − k1 ) − α2 (h − k) − (bα2 − 3β)hk = 0
− α3 + α4 − (bα4 − 2α2 β)(h − k) − (cα3 − 2bα2 β + 8β 2 − 5αγ)hk = 0
− 4αβ + 3α2 β + bα4 − (cα3 + 3bα2 β − 3β 2 − 4αγ)(h − k) = 0
− 4β 2 − 6αγ + cα4 + 4bα3 β + 3αβ 2 + 3α2 γ = 0 .
2

(7.4)
(7.5)
(7.6)
(7.7)
(7.8)

To a first approximation these equations are satisfied by taking
h1 = h ;

k1 = k ;

α = 1;

1
γ = b2 + c
3

β = b;

(7.9)

If then we substitute in (7.4), (7.5), (7.6), and (7.7)
h1 = h +  ,

k1 = k + æ ,

α = 1 + α1 ,

β = b + β1

where α1 and β1 are quantities of the first,  and æ of the second order, we
find from these equations by second approximation:–


1
2
(7.10)
 = −bhk ; æ = bhk ; α1 = −b(h − k) ; β1 = −2b + c (h − k) .
3
Substituting as a third approximation:–
h1 = h − bhk + 1 ;

k1 = k + bhk + æ1 ,

α = 1 − b(h − k) + α2 ,

we obtain finally,
1
1 = chk(−h + 2k) ;
3

1
æ1 = chk(2h − k) ;
3


α2 =


2
b − c (h2 − hk + k 2 ) . (7.11)
3
2

Hence the equation of the surface of the waves is, including all terms of the
third order:–







2
3
2
2
2
η = 1 − b(h − k) + b − c (h − hk + k ) η1 + b + −2b2
3
3


1
1
+ c (h − k) η12 + (b2 + c)η13 + . . .
3
3

(7.12)

29

7. Calculation of the Equation of the Surface
where
1 p
η1 =h1 cn2 x · α(h1 + k1 ) M =
2
1
h1 = h − bhk + chk(−h + 2k) ;
3

r

h1
h1 + k1

!

1
k1 = k + bhk + chk(2h − k) .
3

Here, according to (6.23),


3
5h−k
+ ... ;
a= 3 · 1−
l
4 l

b=

3
+ ... ;
4l

(7.1)
(7.13)

(7.14)

whereas the value of c and more correct expressions for a and b could only
have been obtained by means of still more tedious calculations, which we
have not executed.
If we confine ourselves to that degree of approximation for which all the calculations have been effected, we may write for the equation of the wavesurface:–

3
3 · (h − k)
η1 + η12
η = 1−
4l
4l

 r
5 · (h − k)
3 · (h + k)
2 1
η1 = h cn · 1 −

2
8l
l3

 r
3k
h
M = 1−
.
·
8l
h+k


(7.15)
(7.16)
(7.17)

For the solitary wave, when k = 0, we have2



3h
3
η = 1−
η1 + η12
4l
4l

 r
1
5h
3h
η1 = h sech2 · 1 −

2
8l
l3

(7.18)
(7.19)

January 1895.

2

Another close approximation of the surface-equation of this wave has been deduced by
McCowan, Philosophical Magazine [5] Vol. XXXII. (1891), p. 48.

30

31

A. List of symbols

A. List of symbols
T

surface tension

η

small quantity of l, amplitude of travelling wave

t

time

l

depth of the liquid

α

constant

ρ

density

σ

1 3
l
3

g

acceleration due to gravity

λ

wave-length

u, u0

horizontal velocity

v, v 0

vertical velocity

y

height of particle above bottom of channel

φ

velocity potential (function of x and t)

ψ

stream-function

p1

atmospheric pressure

p01

pressure at a point below the surface where capillary forces → 0

u1

horizontal velocity referring to the surface

v1
q0

vertical velocity referring to the surface

constant, gl, linear wave speed

β

amplitude of the travelling wave (function of x and t)

γ

constant

−q, q

horizontal velocity, velocity of propagation

x, y

coordinates (of particle or channel)

A

amplitude



Tl
ρg

a, b, q, r, s, t constants
K(M )

complete elliptic integral of the first kind

M

modulus

cn

Jacobian elliptic cosine function

sn

Jacobian elliptic sine function

dn

delta amplitude, complementary modulus,
or complementary modular angle

32


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