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The Korteweg-de Vries Equation for Wave Propagation in an Infinitely Long Thin Walled Circular Cylinder Obtained via the Lagrangian Method K. Murawski Department of Physics, Akademicka 13, 20-033 Lublin, Poland

Z. Naturforsch. 40 a, 955-956 (1985); received April 18, 1985

On the basis of the Lagrangian method a nonlinear wave theory is developed to obtain the Korteweg-de Vries equation for the incompressible, one-dimensional, weak-nonlinear, and dis-persive motion of fluid in an infinitely long circular cylinder (with thin walls of elastic rings). This equation has first been obtained by Lamb [1] via the reductive Taniuti-Wei's method [2].

The nonlinear partial different ia l equat ions which govern wave phenomena in fluid filled tubes have resisted analysis for many years. It has been shown that in the presence of dissipat ion the fundamen ta l set of equat ions for the irrotat ional mot ion of waves can be reduced to Burgers equat ion [ 3 - 5 ] . In 1969 Johnson introduced the so-called Korteweg-de Vries-Burgers equat ion for a wave propagat ion on an elastic tube filled with a viscous fluid, which may be regarded as a simple model of an artery [6], Also wave propagat ion and shock fo rmat ion in nonl inear elastic and viscoelastic fluid filled tubes has been discussed for a Mooney-Rivi l in material [7], Some t ime ago, L a m b [1] has used the reductive Taniut i -Wei's method to obtain the Korteweg-de Vries equat ion for the one-dimensional , i rrotat ional , weak-nonlinear, and incompressible mot ion of fluid that is confined within an infinitely long circular cylinder (with walls of elastic rings). For this case the governing equat ions are the conservation of mass equat ion given by

A, + (A V)x = 0 ,

Euler's equat ion,

V,+ VVx + —px = 0, Qo

and Newton 's equat ion

E A,,+

, 2 n a (.A - na2) - p = 0 .

(1)

(2)

(3) Qm a- Qm h

The symbols used have the following meanings: A section field, V velocity of f luid, Q0 density of f luid,

Reprint requests to K. Murawski, Department of Physics, Akademicka 13, 20-033 Lublin, Poland.

p pressure of f luid, E Young's modulus , gm density of the tube mater ia l , h thickness of a tube, a radius of a tube at the undis turbed un i form state. The sub-scripts .Y and t indicate part ial different iat ion.

Introducing new coordinates as follows

A -*• (na2)~l A , p pö1p , t T~l t ,

T Eh L~ x , K- — F, PO = — ,

L 2 a

Ti Qma" r l Qm a h , L z

=

2 £>o

dimensionless equat ions are obta ined, i.e.,

A, + (A V)x = 0 ,

V, + vvx + Px = 0,

Att + A - p = 1 .

(4)

(5)

(6)

(7)

Appl icat ion of the reductive Taniut i -Wei ' s me thod to ( 5 ) - ( 7 ) corresponds to the usual way for con-structing the Korteweg-de Vries equat ion.

However, our way is ra ther different . Using a Lagrangian for the fundamen ta l set of equat ions ( 5 ) - ( 7 ) , we derive the Korteweg-de Vries equat ion via the e expansion method for the Lagrangian only. This me thod has been presented in [8].

Wi thout loss of generality we take V to have the form

V= V x . (8)

Subst i tut ing this def ini t ion and (7) into (5) and (6), we obta in

A, + (A fx)x = 0 ,

rt + iV? + A„ + A = l .

(9)

(10)

0340-4811 / 85 / 0900-0955 $ 01.30/0. - Please order a reprint rather than making your own copy.

956

A Lagrangian for the above set of equat ions can be found and written as follows:

L = \AV2x+AV,-{A2 + \A2-A + \ . (11)

We introduce the stretched coordinates

Y=eU2(x + t), (12a)

T =eV2t (12b)

together with the following order ing for A, f , and L

A = \ + sAx + ... ,

W = £W2 WX + £3/2 *F2 + ... ,

L = L0 + E Lx + ... . (13)

All quanti t ies are considered to be funct ions of Y and i only. In the new coordinates the Lagrangian becomes

L={eAV2y+ EU2A Vy + £3/2 A f r + ±A2-A-hj

-j(£A2y + 2£2ArAr+£3A2z). (14)

Inserting expressions (13) into the Lagrangian and equat ing coefficients of the same powers of e°, e1, e2, and e3, we obta in

L 0 = 0 , L\ = {F\y ,

L2 = {V2y + AxYxy+Vu + \A2x,

Li = ±A x V 2 y + A x V u - { A 2 Y , (15)

K. Murawski • The Korteweg-de Vries Equation

respectively. So the Euler-Lagrange equat ion for L2

leads to the expression

ÖL2 — ^ = f /

] Y + A] = 0 . (16) 0AX

Introducing this equat ion into L 3 , the Euler-Lagrange equa t ion takes the following form:

| ^ = 2<F lTy + y ( < P , 2K ) y - ^ y r y y = 0 . (17)

Bearing t ransformat ion (8) in mind, the Korteweg-de Vries equat ion can be obta ined ,

VU + ±VXVXY-\VXYYY = 0 . (18)

Based on the r igorous me thod developed in the Lagrangian per turba t ion theory, we have derived the Kortweg-de Vries equat ion as a first-order of approx imat ion which describes the weak-nonlinear propaga t ion of waves in an infinite thin walled circular cylinder. It d i f fers f rom the corresponding equat ion as obta ined by L a m b by the sign of the last term. But since we have introduced a different sign for the t ime t in the coordinate Y, see the for-mula (12a) , our result is not at variance with the one of [1].

The au thor would like to thank Dr. Ernst F. Hef ter for his valuable remarks.

[1] G. L. Lamb, Jr., Elements of Soliton Theory, John Wiley, New York 1980.

[2] T. Taniuti and C. C. Wei, J. Phys. Soc. Japan 24, 941 (1968).

[3] P. L. Bhatnagar, Nonlinear Waves in one-dimensional Dispersive Systems, Claredon Press, Oxford 1979.

[4] K. Murawski, Acta Phys. Pol. A69/1, 1986, to be pub-lished.

[5] K Murawski, Z. Naturforsch., to be published. [6] R. S. Johnson, Doctoral thesis, Univ. London, London

1969. [7] R. J. Tait and T. B. Moodie, Wave Motion 6, 197

(1984). [8] E. Infeld, Acta Phys. Pol. A60,623 (1981).