One dimensional, non-linear wave motion

Int. J. Non-Liwur

Mhanicr.

Vol. 13, pp. 15-20.

Pertamon

Press 1978.

ONE DIMENSIONAL,

Pnntcd

in Great

NON-LINEAR

Britain

WAVE MOTION

C. N. KAUL Department of Mathematics, Indian Institute of Technology, Kharagpur, W.B. 721302, India (Received 20 April 1977)

Abstract-A transformation of the quasilinear wave equation is made to study some of the geometrical properties of its solution. It is also shown that any given solution of it can always be continuously extended across a characteristic in a simple wave. 1. INTRODUCTION

We shall consider here the quasilinear wave equation :

with the assumptions that&C2(u);f’(u) > 0 for all u, but without any restriction on the sign off”(u) (cf. [l] p. 243). The above system describes non-linear, rectilinear, wave-motion in several fields in applied mathematics [2-51 and has been extensively discussed [6, 71. In recent years, weak-solutions of the above system have been studied by Lax [8], Glimm [9], Greenberg [lo], Dafermos [ 1I] and Ballou [ 121 among others. In this paper, we shall consider a transformation of the above system to a single quasilinear equation [13,14], to study some of the geometrical properties of the solution. In the Section 2, the transformed equation is obtained. Some geometrical properties of the solution are derived in the Section 3. In the transformed equation, a class of solution is ‘lost’. This is discussed in the Section 4, and it turns out to be ‘simple-wave’. It is shown that any given solution of the above system can always be continuously extended across a characteristic in a simple-wave. 2. TRANSFORMATION

OF EQUATION

Let C2 =f’(u), where the prime denotes the derivative with respect to u; then C = C(U) is the local wave speed (cf. equation (2.18)). If the partial derivatives with respect to x and t are denoted by a suffix, we have the system u, = U,, v, = c%,.

(2.1), (2.2)

From (2.1), it follows that there exists a function I,+= $(x, t): If+, = u,

lj, = v.

(2.3)

We shall now introduce u, $ as the independent variables, and v as the dependent variable. Let J be the Jacobian: J = (t,x)/(u, t+b),and assume that 0 < (Jl < 00, then $, = t./J,

t+b,= -x JJ, u, = -t*JJ,

u, = x$/J,

(2.4)

where in the above equation and the sequel, u and tj in the suffix stand for the partial derivative with respect to u (keeping $ constant) and rj (keeping u constant). From (2.3) and (2.4) t,=

Ju;

xy=

-Jo,

(2.51, (2.6)

from which it follows that vt, + ux, = 0.

(2.7)

Also from (2.5), (2.6), and the value of J, it follows that = 1.

ux,+vt, 15

(2.8)

C. N.

16

KACI

We have also the relation 2(c, x) q=S(t,r)=-p.

1 ?(c,r)

(2.9)

J SkiI

From (2.2), (2.4) and (2.9), we have (2.10)

L‘,.Y,,, + C’t,,, - 1‘(,,. \‘” = 0, while (2.1), (2.2) and (2.7) give sr = CL’,(.Y~ - cy,)jL.,,(L’.Y.

+ tK2t

(2.1 1)

).

Now (2.10) and (2.11) yield (uu$, + uC’)t,,, = - (uq + U)S$,,

(3.1’)

which with (2.8) gives t,,, = -((uc,+r)i., where A = (u2Cz -c 2) - ‘, is assumed and (2.13) determine

(2.13)

x,;, = (t’v.,+uC’)i.:

to be finite. If we assume that r2 - u’C’ # 0, then (2.10) (3.14)

s, = (l$~,/tl*, where /J’= (I$ - C2), which with (2.7) yields

(2.15)

t, = - (l@E.)/v,,. From (2.13) to (2.15), it follows that t! = v(u, t+k)satisfies c; l’“, - 2L’,q,t..* + PC*,,,-2~uCC,D;(v+ur~,) The case vti = 0 is discussed Theorem

= 0.

(2.16)

in Section 6.

1. Zf v,,, # 0, then the wave motion is governed by c; L’,,- 24 v,,,v,!Jl+ ljv*,,, - 2RCC,t$(v

+ UC,) = 0,

together with the tian.s&nW~ation relation s =

(@Q~,.)du

t=

+ (vu,, + uC2)i. dti,

- (ujU./r, ,) du - (up,, + v)L d$. .’

The characteristic

curves of (2.16) are given by (2.17)

(dv/du) i = ( - v, + C’)/vI. In the t, x-plane,

these directions

transform

to

(dx/dt), A curve $ = tj(u) in the II/, u-plane

(2.18)

= kc.

maps into a curve v = v(u) in the u, u-plane along which (2.19)

(dc/du) = 0, + v,(dt,V/du). Now (2.17) and (2.19) give the characteristic

directions,

(dv/du)* The characteristic

(d$/dt) 3. CURVES

A curve x = x(t) in the t, x-plane (dx/dt)

(2.20)

= k-C’.

in the t, $-plane

directions

in the u, u-plane, as

are therefore (2.21 )

= o+uC.

OF CONSTANT

u AND

v

maps into a curve u = u($) in the $, u-plane along which

= (x,u’+sJ(t~u’+

t+),

u’ = (du/d$).

One dimensional,

non-linear

wave motion

17

This with (2.13) to (2.15) becomes

(dxldt)= -

p.u’ + v,,,(vv, + uC2)

F

puu’+u,,,(uc,+u)

1

’

0.

%!l+

(3.1)

Consider the curve u = constant in the t, x-plane, and denote its slope by R, = (dx/dt),. so defined, gives the ‘shift-rate’ of u. Then (3.1) yields R, = (dx/dt), Again, consider the curve v = constant, v. Since along t’ = constant,

= -[(vu,+

uC2)/(uc,+

v)].

R,,

(3.2)

and let R,. = (dxldt),., which defines the ‘shift-rate’ of

c’ = (do/d+) = -(vi/v,),

(3.3)

R, = (dx/dt),. = - C2[(u+ uu,)/(C2u +vv,)].

(3.4)

we get by (3.1) and (3.3)

From (3.2) and (3.4) R,R, We have thus the following

= C2.

(3.5)

result (cf. [15]).

Theorem 2. At any point in the t, x-plane, C = [f’(u)] ‘I2 is the geometric-mean between the shift-rate of u and v. It is to be noted that in the above result R, # R,., in general. In fact, if R, = R, = C, then (3.2) and (3.4) give (v,+C)(1:+uC)

= 0,

and from it, it follows that v = v(u), i.e. the motion is a simple-wave. then a(v, u)/a(x, t) = 0, and therefore R, = R,., since

Conversely,

if v = v(u),

(dxidt),. = - (c,/c,) = - (u,.u,) = (dxldt),. In contrast possible that line of branch From (3.2)

to the above remark which apertains to R, = R,. in the whole t, x-plane, it is R, = R, only along a certain curve in the t, x-plane. Such a curve is called the points (cf. [14, 16, 171) and it shall be shown to be a characteristic curve. and (3.4) we have R,-R,.

= [(c’-u2C2)(C2-v~)/(C2u+vv,)(v+uv,)].

Thus if R, = R, along a curve in the t, x-plane (hereafter denoted by Z), then if v2 - u2c2 # 0, i.e. the characteristic directions do not coincide with $ = constant along Z, then we have along X the relation 11.’- c2 = 0. In the t, x-plane, (2.13), we have

let s denote the arc-length

(3.6)

along I,$ = constant.

Then from (2.3), (2.4) and

du - = l/J@2 + L?)i’Z. ds Thus if u and u are non-zero, then the restriction < jdu/dsl < cc. But from (2.13) to (2.15), we have

on J:O < I.JI < 00 is equivalent

(3.7) to 0

J = d2~2/v,,, which with (3.7) gives v,,, = i.2(vf - C2)2(u2 +u2)l12(du/ds).

(3.8)

Thus along E, we have from (3.6) and (3.8) p2--2 u NLM Vol. 13. No. I-C

=

v

J

=

0.

(3.9)

C. N. KAUI

18

Now from (2.19) and (3.9), we have along the map of Z in the u, c-plane (dridu)

= c, = f C,

which by (2.20) is the map of a characteristic the following result.

curve in the u, v-plane. We have thus established

Theorem 3. The locus of points in the t, x-plane at brhich the directions qf’curres c?f’constant u and v coincide, is a characteristic curve.

Since u = u(x, t),

v = v(x, t), we have

(du/dv) = (u, dt + u, dx)/(c, dt + I’, d.u). If we consider the curve x = constant (du/dv),, then by (3.10)

in the c, u-plane,

(du/du), Equation

(3.10)

and denote its slope at any point by

(3.11)

= (u,/r:,).

(3.11) with (2.2) gives (du/dv),

= (u,/C2u,)

= - C-2(dx/dt),.

(3.12)

From (3.5) and (3.12) (dc/du), Theorem orthogonal plane.

x (dx/dr),. = - 1.

4. At arly point in the t, x-plane, to the direction

of the curve

of the curve x = constant, at the corresponding

4. THE

In deriving becomes

rhe direction

(3.13)

(2.16), it was assumed

LOST

SOLUTION:

1%= constant,

is

point in the u, v-

r,, = 0

that v,,, # 0. Here we consider

the case t’,!,= 0, then (2.10)

V”X,,,+ c2t,/, = 0,

(4.1)

ux,,, + lx,,, = 1.

(4.2)

while (2.8) is

If 4 = 4(u) = (UC,-uC2)

# 0, then (4.1) and (4.2) give x,,, = (C2/#)Y

r,,, = - (L’“/$).

(4.3)

which with (2.7) gives Ll,‘-

c2

=

(4.4)

0.

Now (2.7) implies

x, = -/lv,

t, = P,

P = A&I//).

(4.5)

From (4.3) and (4.5), we obtain P = where B(u) is an arbitrary

(u)- ‘(vJ4(u)),d$ + B(u), s of u. From (4.3) and (4.5)

function

t = x =

pudu-(v,l4)W, ( -pv)du+(C2/+)d+.

(4.6)

(4.7)

(4.8)

The wave motion for which v,,, = 0, i.e. v = v(u) are governed by (4.4), together with the transformation relation (4.7) and (4.8). These are simple-waves. Let u = u0 and II/ = Ic/,,denote the curves of constant u and II/ through the origin in the t, xplane. In the u, $-plane, the origin of the t,x-plane maps in to the point (u,, IJ~). If (u, $) be

One dimensional,

non-linear

wave motion

19

any point, then it maps into a point in the t, x-glane given by: (U&II) (-pr)du s (Ull.JlO) (U&I,)

x=

+

(U&1 (C*/& dl//, s (Uo.((Io) Ml,JI)

(4.9)

t=

(4.10) s (Ulhid (pu) d” - s (U”,h) w4)dA (4.11)

P = -(U)-l(+-$o)(O#J)“+B(u). In the t, x-plane, let the curve $ = $,, be denoted by x = x,(u), t = to(u), then x0(a) = x(a, $0) = Qu)=

,

s “,I

(4.12)

s

t@,i/QO)= ,::::,,

::z.d”

(4.13)

Thus if an analytic curve x = x,(u), t = to(u) is given, then (4.12) and (4.13) determine B(u) and u(u), while (4.4) gives C; and hence a simple wave is determined. We have thus established the following result. Theorem 5. Given any analytic curve x = x0(u), t = t,,(u), in the t, x-plane, a simple wave with this curve as II/ = $. (const.) can be determined by x = xO(u)+(c*/qW-~~), t = to(u)-

(Q/4) W - $0).

We shall now prove that any wave motion can be patched continuously along a characteristic with a simple-wave. Let Co be a characteristic of a wave motion. Then we shall construct a simple wave patched along Cc,.For the simple wave assign along C,, the values of C and u as for the given wave motion. Then along Cc, by (2.20), u,’ = C* ; which is exactly the equation (4.4), satisfied by the simple wave. Thus along C,, v = u(u) is also identical for both the wave motion. Now for any wave motion, we have along a characteristic (cf. equation (2.21)) v,=

fC,

(drldti) = ll(r + UC),

while along C, for a simple wave, we have (dt/dtj) = t, + t,(du/d$) = - (u,&)+&du/d+). Equating the above two values of (dt/d$), we get P = (d$/da) x (2CM). Substituting the value of ,P in the transformation wave.

relation (4.7) and (4.8), we get the simple

Theorem 6. Any given wave motion can be extended continuously a simple wave.

across a characteristic

into

Acknow/edgements-This work was supported by a research fellowship of the Alexander von Humboldt-Stiftung at the Institute fur Angewandte Mathematik, Hamburg. The author gratefully acknowledges the hospitality extended by Prof. L. Collatz, and the support by the Stiftung. The author also thanks Prof. W. F. Ames for his suggestions on the first draft of the manuscript, which led to considerable improvement. REFERENCES 1. R. Courant and K. 0. Friedrichs, Supersonic Flows and Shock Woues. Interscience (1948). 2. N. Cristescu, Dynaniic Plasticitp. North-Holland (1967). 3. D. P. Reddy and J. D. Achenbach. Simple waves and shock waves in this prestressed elastic rod, Zeit. Angew. Marh. Phy. 19,473 (1968). 4. W. F. Ames, One wave propagation in one-dimensional rubber like materials, J. Math. Anal. 34,214 (1971). 5. W. F. Ames, Non-linear Partial Difirential Equations in Engineering, Vol. II. Academic Press (1972). 6. A. JetTrey and T. Taniuti, Non-linear Wme Propagation. Academic Press (1964). 7. G. B. Whitham, Non-linear Waves. John Wiley (1974).

C. N. KAUL

20

8. P. D. Lax, Hyperbolic system ofconservation laws, II. Commun. Pure Appl. Math. 10, 527 (1957). 9. J. Glimm, Solutions in the large for non-linear hyperbolic system of equations. Comntur~. Pure Appl. .Math. 18. 697 (1965). 10. J. M. Greenberg, On the interaction of shocks and simple waves of the same family, Arch. Rationul Mech. A~ul. 37,136 (1970). 11. C. M. Dafermos. Solutions of the Riemann problem for a class of hyperbolic system of conservation laws by viscosity method, Arch. Rational Mech. Anal. 52. 1 (1973). 12. D. P. Ballou. The structure and asymptotic behaviour ofcompression waves. Arch. Rarionul Me& Amd. 56, 170 (1974). 13. M. H. Martin, The propagation of a plane shock into a quite atmosphere. Curl. J. Mrrth. 3. 165 (1953). . 14. C. N. Kaul, Rectilinear anisentropic gas flows, J. Math. Whys. Sci. 7,183 ( 1974 I. 15. G. S. S. Ludford and M. H. Martin. One dimensional anisentrouic flows. Commun. Pure Am/. . Mark 7.45 (1954). 16. M. J. Lighthill, The hodograph transformation in transonic flow, 1, symmetrical channels. Proc Ro!,. SK. (London) A191,323 (1947). 17. M. H. Martin. Some geometrical properties of plane flows. Proc. Cmh. Phil. Sot. 47. 763 f 1951).

On fait une transformation de 1'6quation d'onde quasili&aire pour 6tudier quelques propri&.& g6om&iques de sa solution. On montre Cgalement qu'une de ses solutions don&e peut toujours ^etre etendue par continuit6 en une onde simple.

Zusamnienfassunq: Eine Transformation der quasilinearen Wellengleichung wird unternotnnen, urneinige geometrische Eigenschaften ihrer Liisunq zu untersuchen. Es wird weiterhin gezeigt, dass es in&r m6glich ist, jede gegebene L6sung der Gieichung kontinuierlich ijber eine Charakteristik in eine einfache Welle auszudehnen.