Multiple-Scale Solution of Initial-Boundary Value Problems for Weakly Nonlinear Wave Equations on

Author(s): S. C. Chikwendu and C. V. Easwaran

Source: SIAM Journal on Applied Mathematics, Vol. 52, No. 4 (Aug., 1992), pp. 946-958Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2102183Accessed: 07/11/2008 07:00

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SIAP 728 SIAM J. APPL. MATH. Vol. 52, No. 4, pp. 946-958, August 1992 ? 1992 Society for Industrial and Applied Mathematics 003 MULTIPLE-SCALE SOLUTION OF INITIAL-BOUNDARY VALUE PROBLEMS FOR WEAKLY NONLINEAR WAVE EQUATIONS ON THE SEMI-INFINITE LINE* S. C. CHIKWENDUt AND C. V. EASWARANt Abstract. A multiple-scale perturbation solution of initial-boundary value problems for weakly nonlinear wave equations on the semi-infinite line is derived. The presence of both initial and boundary data requires the introduction of two new variables, a slow time and a slow space variable. It is shown that when the nonlinearities involve only first-order derivatives of the dependent variable, the leading-order solution is governed by two coupled first-order partial differential equations (PDEs). Examples are given to show that these PDEs can sometimes be solved to determine the leading-order solution. Key words. multiple scale, perturbation, nonlinear, waves AMS(MOS) subject classifications. 35C20, 35L20 1. Introduction. Weakly nonlinear wave equations occur in many areas of science and engineering. In this paper, we consider the initial-boundary value problem for a weakly nonlinear hyperbolic wave equation on the semi-infinite line (1.la) (l.lb) (1.c) utt-uXX + rh(u, ut, ux) = 0, 0 < x < oo, t > 0,0 < << 1 u(x, 0) = a(x), Ut(x, O) = u(0, 0 0, t > 0; and twice continuously differentiable in this region, except possibly on the line x = t. We also assume that u and its derivatives are bounded for all t, a(x) and p(t) are twice continuously differentiable, b(x) is continuously differentiable, and h is analytic in its arguments. The solution of problem (1.1) is assumed to depend continuously on the initial and boundary conditions, in the sense that small changes in the initial-boundary data lead to small changes in the solution. We seek a formal perturbation series solution in powers of E, the small parameter, using multiple scales. What makes this problem challenging is the presence of both initial and boundary data. The key to the approach used in this paper is that the solution of problem (1.1) can be written as two separate equations, one valid for 0 ' t x and the other valid for t '-x _ 0, as follows (see, for example, Tychonov and Samarski [13, p. 60]): ' 1 1 lX+t 2 ? 2 t u(x, t)= [a(x+t)+a(x-t)]+2J (1.2a) CX+(t-) b(A) dA ~~~~~x-t UA(A, 7)) +- J dr J 2 0 x-(t--z) (u(A, r), u(A, 7), dA, 0-' t 'x, * Received by the editors October 1, 1990; accepted for publication (in revised form) August 13, 1991. t Department of Mathematics and Computer Science, State University of New York, College at New Paltz, New Paltz, New York 12561. 946 MULTIPLE-SCALE SOLUTION OF NONLINEAR WAVE EQUATION 947 U (X, t) = p(t t-x) + ( a(x + tl )-a(t t-x)) + - x+(t-,r) b (A) dA t ? x X _ . (1l.2b) + 2 Jr o t 'd h(u(A, r), u,(A, T), uA(A, 7)) dA, lX -( t--)j Equation (1.2a) shows that, in the region t < x, the boundary condition p(t) has no effect on the solution, and the problem is purely an initial value problem. Thus the multiple-timescale method previously introduced by Chikwendu and Kevorkian [3], Lardner [8], Chikwendu [2], and others, for initial value problems, can be used to describe the evolution of the solution for 0? t _ X. This method constructs asymptotic van der Burgh [14] has studied approximations by introducing a slow timescale T = Et. the asymptotic validity of this perturbation method, and, most recently, van Horssen [15] has established the wellposedness and the asymptotic validity of the formal perturbation approximations. The boundary condition comes into effect for t > x, as seen from (1.2b). For zero initial conditions (a(x) = 0, b(x) = 0), (1.1) is a signalling problem, with the boundary data p(t) radiating into the region x > 0. Chikwendu and Kevorkian [3, p. 248] treated this signalling problem by introducing a slow space variable X = Ex instead of a slow time, and found the far-field cumulative effect of the nonlinearity h(u,, ux) on the leading term of the perturbation solution. Thus when the initial conditions are nonzero, we expect that the evolution of the solution in the region t > x will be described by both a slow time T = Et and a slow spatial variable X = Ex. The multiple-scale perturbation procedure is developed in ? 2, and the equations governing the slow variation of the leading-order approximation are obtained in ? 3 (for 0-' t _ x) and ? 4 (for t x> X?_0). In the region t < x, the equations governing the slow variation of the solution have previously been obtained (see [3]). However, for t _ x _ 0, the evolution of the solution, depending on two slow variables, is shown to be given by a pair of coupled first-order PDEs with interesting \"initial\" data. A linear example is presented in ? 6 and a nonlinear example is given in ? 7, showing that, in some cases, these equations can be solved explicitly. Equations of the form (1.la) describe many physical problems. Examples may be found in [15] and [9] for the galloping oscillations of overhead powerlines and wave propagation in a weakly nonlinear elastic medium. It is pointed out in ? 5 that the method developed in this paper applies to a wider class of problems than given in (1.1). Some comments are made in ? 8 on the possible utility of the method developed in this paper for the asymptotic solution of physical problems, such as those arising in one-dimensional gas dynamics and acoustics. 2. Multiple-scale perturbation procedure. In applying the method of multiple scales, slow variables are usually introduced. As explained in ? 1, we anticipate here that the solution will involve both a slow time and a slow space variable (2.1a) T = Et, X = Ex. -=t In addition, we introduce new fast variables = x, xc (2. 1b) so that we have a change of independent variables from (x, t; ?) to (x, t, X, T; E). We then seek a solution of (1.la) in the form of a perturbation series expansion in powers of ?, N (2.2) u(x, t; ?)= E?u, n =O t,X, T) + O( N). 948 S. C. CHIKWENDU AND C. V. EASWARAN The derivatives now become = Ax + 6x, Ax- at =at+ E8T, =xx =axx + 2 8E3x + E2aXX, . adU+ 2EacT + 82 att = When these derivatives and expansion (2.2) are introduced in (1.la)-(1.ld) and the , a hierarchy of initial-boundary coefficient of E' is equated to zero for n = 0, 1, 2, value problems is obtained (see, for example, Kevorkian and Cole [6]). In the following, we drop the bars over the fast variables for convenience. The first two problems are (2.3) (2.4) ul, ul, = uO -uOy = 0, 2UOTT +2uoyx h(uo, uO, uo_), with the initial conditions (2.5a) (2.5b) (2.6a) (2.6b) u(x, 0, X, 0) = a(x), un(x, 0, X, O) =0, un(x, 0, X, O) = uo(x, 0, X, 0) =b(x), U(n-1)T(X, O, X, O), n ' 1 and boundary conditions t, 0, T) =p (t), uo(O, Un(0, t, 0, T) = 0, n _ 1. 3. First-order solution for t < x. It is clear from (1.2a) that the boundary data have no effect on the solution when 0< t x x FIG. 1. Forward- and backward-going characteristics of the wave equation, showing reflection at the boundary. MULTIPLE-SCALE SOLUTION OF NONLINEAR WAVE EQUATION 949 = x - t and e = x + t are the characteristic variables of the hyperbolic equation where o- (1.la). From (1.2a) we see that when T =0 (corresponding to =0), we must have the initial conditions (3.1b) 4a(cr)-J f(lo, 0) = j b(A) dA] =F(o\"), (3. 1c) (3.2) afle, -4u1sne = +Jb (A) dA] G() In this case, (2.4) becomes 2foT -2gT -h(f + g, ge -f, ge +f,). To eliminate \"secular\" terms that would lead to nonuniformities and inconsisten- cies and to determine the slow time (T) variation of f, we integrate (3.2) with respect In this to e from 0 to M, with o- held fixed, divide by M, and take the limit as M -> oo. limit, the left-hand side goes to zero, since ul and its derivatives are bounded, and the resulting equation is (3.3) 2f,T - M-ooo M. 0 lim 1 M +f ,) de 0. h (f + g, gf -f,, gf The corresponding equation for g is (3.4) 2g9T + lim M 1 M = 0. h(f + g, gf -f,, gf +f,) do- f and f, explicitly. does not depend on g and gf explicitly. However, this term may involve the slowly (T) varying e-average of a function of g and g~. Similarly, the second term in (3.4) may involve the slowly varying o--average of a function of f and f,, but does not contain The second term in (3.3) is an average in e, and so is independent of e. Thus (3.3) Equations (3.3) and (3.4) are a pair of coupled integrodifferential equations for f and g. These equations can sometimes be solved subject to some special initial conditions. (See, for example, Lardner [9], Pearson [12], Chikwendu and Kevorkian [3].) When f and g are periodic in o- and e, respectively, the terms involving limits in (3.3) and (3.4) give the average of h over a period. 4. First-order solution for t > x. It can be seen from (1.2b) that, in the region t> x, the solution of (1.1) is influenced by both the initial and boundary data. We therefore must use both the slow time T and the slow space variable X. In terms of the variables t> x, is At = t - x and ( = t + x, the solution of the linear wave equation (2.3), for (4.1) (4.2) uo (x, t, X, T) =p (,u,X, T) +q (e,X, T). 4u1, = -2P,,T -2P,x + 2qx - In this region, (2.4) becomes -2qeT h (p + q, pZ + qe, qf -p~1). The slow (X and T) dependence of p and q can be determined from (4.2) if the secular terms are eliminated in the same manner as was done with (3.2). The resulting equations are (4.3a) I2X+ 2PI1T+2 2qeT-2qex fi(X, T, p, p,)=0, + +(X, (4.3b) T, q, q) = 0, 950 S. C. CHIKWENDU AND C. V. EASWARAN where (4.4a) 1 M I 0=lim hr(pn+ M->CoM q, p,+ q, q- , Jde 1 M M M->00 The averaging in (4.4a) eliminates the e-dependence of 0, and hence / does not contain q and qf explicitly. However, / may involve the slowly varying (-average of a function of q and qe. Similarly, qf does not contain p and pZ, explicitly but may contain the o--average of a function of p and p,,. In general, (4.3a), (4.3b) form a pair of rather complicated, nonlinear, coupled, second-order PDEs for p and q, and may be as difficult as (or even more difficult than) the original equation (1.la). However, there are many cases where these equations can, in fact, be solved. To simplify the equations, we will now consider the special case when the nonlinearity h involves only the first derivatives of u, that is, h(u,, ux). The simplification results from the fact that (4.3) now takes the form (4.5a) T + 2p,X + 0 (X, T, p~,) = O, 2P~, 2qeT-2qx (4.5b) where (4.6a) (4.6b) = + qj(X, T, qe) = O, Mj0 M -~.0 lim 1M -p)d(, de h(p + qe, qe h(p +qe, q -p)pc. q= lim Equations (4.5a), (4.5b) form a pair of coupled, nonlinear, first-order PDEs for pl, and q?. The theory of such equations is quite well developed, and we now proceed to solve (4.5a), (4.5b), subject to appropriate initial conditions. 4.1. Initial data for the first-order PDEs. Figure 1 shows the right and left running characteristics and illustrates the geometry of the method used in this paper. At this stage, we assume that, for the region t x, the leading approximation (3.la) has already been determined and that f(o-, T) and g(e, T) are obtained from (3.3) and (3.4). We now move to the region t ' x and first solve (4.5b) for qf on the left running characteristic (e). As can be seen from Fig. 1, the initial data for this semilinear first-order PDE (4.5b) will be specified on the line X = T and must match with the already-determined value of g(e, T) when T =X. Then we solve (4.5a) for pl, on the right running characteristic (tL). Figure 1 shows that the incoming wave q hits the boundary and is reflected as the outgoing wave p. Thus the initial data for p must be specified on the boundary (X =0) and must match with a combination of the incoming value of q(e, 0, T) and the imposed boundary condition p(t) from (2.7a). Thus the appropriate \"initial\" conditions for q(e. X, T) and p,,(A, X, T) are (4.7a) (4.7b) qe((e T, T) = g (e T), p,_(A, 0, T) = p'(,u) - q( 0, T), where we use the fact that e = A = t on the line x =0, and (4.7b) is written (using (2.7a)) such that, on this line, uo = p = p + q. MULTIPLE-SCALE SOLUTION OF NONLINEAR WAVE EQUATION 951 5. Solution of the first-order PDEs for t > x. 5.1. Solution for the incoming wave. To solve (4.5b), we let v = qf and introduce the subcharacteristic variables (r, s), so that we have T(r, s), X(r, s), and v(e, r, s). The subcharacteristic equations of this first-order PDE are thus (see, for example, Garabedian [4]) (5.1a) (5. 1b) dT d= 1, a-=-2+(X s 2 a dX =1 (r, s), T(r, s),. v(g r, s)). On the initial line X = T, we set s = 0, and the initial data are (5.2a) (5.2b) T(r, O)= X(r, O)=r, r, O)-=g,(f, r)- v(et, T=r+s, X=r-s. The solutions of (5.1a) subject to (5.2a) are (5.3) Equation (5.1b) is a nonlinear first-order ordinary differential equation (ODE) whose We solution, subject to the initial condition (5.2b), will depend on the nature of 4'. note that (5.1b) is independent of ,u and does not contain p,t explicitly, but depends on p,, only through the ,u-average given in (4.6b). 5.2. Solution for the outgoing wave. To solve (4.5a) we let w(r, s) = p, and introduce the subcharacteristic variables (r, s) from (5.3), so that (4.5a) becomes (5.4) aw --4(X(r. 2 ar S), T(r, s), w(,u, r, s))- The initial line X =0 becomes the line r = s, and the initial condition (4.7b) on this line becomes (5.5) w(%,u s, s) = p'(u) - v(, s, s). Equation (5.4) is independent of e and does not explicitly contain v, but depends on v through the e-average given in (4.6a). 5.3. Combined solution. The determination of the leading approximation in the solution of problem (1.1) in the region t _ x has now been reduced to the solution of two equations-(5.1b) for the incoming wave and (5.4) for the outgoing wave. These equations are coupled, since from (4.6a), (4.6b), (5.6a) (5.6b) + (X, T, w)= lim - +f(X, T, w)-=lim-( M-*cxo 1 M 0c)m M ?? M h(w+v, v-w) de, h(w +v, v -w) d,u. M 0 Thus, in general, (5.1b) and (5.4), together with (5.6a), (5.6b), form a rather daunting pair-of coupled first-order PDEs for v and w, with the initial data (5.2b) and (5.5) specified on two different lines. However, there are many nonlinearities h for which these equations can be uncoupled and therefore much simplified. 952 S. C. CHIKWENDU AND C. V. EASWARAN Finally, we note that, although the method has been developed for nonlinearities of the form h(ut, u_), it is, in fact, applicable to a more general class of nonlinearities of the form (5.7) h = &h1(u, ut, u,.) Ul, U.) + &h2(U, at ax 5 which occur in some application areas. 6. A linear example: h = u,. In this linear case, (3.3a) and (3.4a) become (6.1) 2fT +fcr=O, + g9 ? 2g9T These linear ODEs are easily solved, and the resulting leading approximation (3.1a) for t'-x is (6.2a) u0(x, t, T) = e-T/2F(oj) + e-T/2G(() + K(T), t _ x, where K(T) is an arbitrary function of T with K(O) =0, and F(o-) and G(e) are obtained from the initial conditions (3.1b), (3.1c). If we assume that the solution of the original problem (1.1) depends continuously on initial data, then small departures from the initial data (F(x) and G(x)) must lead to small departures from the approxi- function K(T), which is completely indepen- mate solution u0. Therefore an arbitrary dent of the initial conditions, cannot occur in u0. Thus we must require that (6.2b) For t'- x, (5.6a), (5.6b) give K( T)--O. 4 = -w, +- = v, so that (5.1b) and (5.4) become, respectively, (6.3a) (6.3b) and their solutions are (6.4) (6.5) av=v, (V,r aw= 1 w fixed), s fixed), (jL, r) e s/2, v(e, r, s) = VO(f, w(A,u r, s) = WO(p. s) e /r, where V0 and W0 will be obtained from the initial conditions (5.2b) and (5.5). The result is (6.6) s) = G'(e) e-(r+s)/2 v(e, r, s) = [p'(A) e+s/2- w( r, es/2] G'(p-) er/2 (6.7) These equations can be integrated with respect to e and ,u, respectively, and with the aid of (5.3) and (5.7) can be written in terms of X and T, leading to the equation MULTIPLE-SCALE SOLUTION OF NONLINEAR WAVE EQUATION 953 This linear example can be solved exactly using Laplace transformation or other transform methods. The exact solution is U(X, t; ?) 1 =- 2 e-E/ E1 [a(x + t) + a(x -t)] / t2 x(x x?t (6.9a) -6e9E)2 ~ u(x t;? ~ )= + II,E x-t 2e-?t2 4 [at _x+t- ~ E0 xtot-x,) E ( eX2p(tx 2 dp2 ( ) a (/) d +e( tx)+ ( Io(2(x- AE+ dA0? t x; 2 2 t?X V At/(X_/ 2_2() (6.8) o taie The (1)soltio usin te- perturbfax tion aethod.+e obEtaie by exadng(.) is idni A al t th soluion 12p( 62 n ~~~t 2 2 4 ~ Numerical calculations were performed to check the accuracy of the perturbation method. A finite difference method of characteristics scheme modified by a local linearisation technique was used to numerically solve the perturbation problem (see 2~~~~~~~~~~ [ 1], [5], [ 10]). For the perturbation function h =u, and ? -0. 1, Figs. 2 and 3 show t = 5 -1x 1.5 ,' 0.5 / Xt+X X / H -1.5 0 2 4 6 8 10 12 14 16 x FIG. 2. comparison dash-dot =0.1, and the solution for ? =0 (labelled h-u. line, ---) for E initial-boundary (6.10) were used. conditions of the numerical solution (solid line, solution (dashed line, ), the perturbation -) at t=5. Here and the 954 2 S. C. CHIKWENDU AND C. V. EASWARAN x 8 1.5 30.5 -1 0 1 2 3 4 5 t 6 7 8 9 10 solution (dashed line, FIG. 3. Comparison of the numerical solution (solid line, - ), the perturbation = 0 at x = Here h = ut and the dash-dot ) 8. and the solution for -) for E 0.1, ? (labelled line, conditions (6.10) were used. initial-boundary comparisons of the numerical and perturbation solutions and the solution correspond- ing to E = 0, for the initial and boundary conditions (6.1 Oa) (6.1 Ob) (6. 1lc) u (x, 0) = sin (3x), (x, 0) = 0, ut p(t)= 2sin (t). For this case, the numerical and perturbation solutions were virtually indistinguishable. 7. A nonlinear example: h = 2u, + u(u (7.1) - uJ). For t < x, (3.2) in this case reduces to -f) + 2(f + g)f, -4ul, = 2fT -2geT-2(ge while (3.3a) and (3.4a) become (7.2a) (7.2b) 2fT + 2(1 + (g))fa + 2ff = 0, = 0, 2geT + 2g& where (7.3) (...)=lim ---dF 1frM The solution of (7.2b), subject to the initial condition (3.1c), is (7.4) g(e, T) = G(e) eT (g) =e-T(G) and (7.2a) can be reduced to the following Bernoulli differential equation: (7.5) 2fT + 2(1 + (g))f+f2 = 0, where, because of the assumption that the solution depends continuously on the initial data, just as we did in (6.2b), the arbitrary functions of T that would occur on the right side of (7.4) and (7.5) have been eliminated. MULTIPLE-SCALE SOLUTION OF NONLINEAR WAVE EQUATION 955 Equation (7.5) has the solution (7.6) where (7.7a) A(T) = exp [-f f(o-, T) = 1 + F(a)A(T) (1 +(g)(T)) dr] =exp (-T+ (e T 1)(G)) and JA (r) di-= (7.7b) = 1 {- exp ( e(-T)(G))}, (l - - e-T) ~~~~(I (G) = 0. (G)# O, Since F(o-) occurs in the denominator of (7.6), for the solution to remain bounded, we must require that F((T)> I-2e(G)I _2(G<), (G):A0, (7.7c) > -2, (G) = 0. For t> x, (4.3a), (4.3b) become, respectively, (7.8) 2= O, 2PT +2pX +2(1 +(q))p+ p qT- qX + q = . (7.9) (7.10) (7.11) When written in terms of the (r, s) variables, these equations reduce to the form \"P=-(l+q))p_-p1, ds = g((, r) = The solution of (7.11) satisfying the initial condition q(e, r, O) G(e) e-r is (7.12) (q) &e-(r+s)(G). q(e, r, s)= G(e) e-(r+s) The Bernoulli differential equation (7.10) has the solution (7.13) p(/, r, s) = A(r, s) where A(r, s) = exp [-{ (7.14) and 0(r,s)=-I (7.15) lrr 2J5 (1 +(q)(r', s)) dr'] = exp {-(r - s) - (G) e-s(e-s - e-r)} esr A(r',s) dr'= 2 2 The initial conditions on r = s are (7.16) (7.17) 1 -(1- 2(G) I{1-exp((er es-r), es)(G))}, (G) = 0. (G)o0 A(s, s) = 1, 0(s, s) = 0, P(,u, s) =p(pg, s, s) = p(,t) - q(,u, s, s) = p(,u) - e-2sG(,). 956 s. C. CHIKWENDU AND C. V. EASWARAN Combining (7.12) and (7.13)-(7.17) and using (5.3) and (4.1), the 0(1)-approxima- tion in the region t _ x can be written as - e(X-T)G(t-x)] u0(x, t, X, T) = e TG(()? +1+a(X, T)[p(t-x) t'x 1+f3(X, T)ptx- where a (X, T) = A(r, s) = exp [-X-(G) and /3(X, T) = (rs)e( [(r, s) = T-X )/2 XT ( -)] e T(eX -1)] I2) 2(G) - exp [-2(G) (G)=0. e-T/2 sinh (X/2)]], (G) # 0, 1 -(1-e-X), 2 For this nonlinearity, Figs. 4 and 5 show comparisons of the numerical and perturbation solutions for e = 0.1, as well as the solution corresponding to e = 0, using initial and boundary conditions (6.10). The perturbation solution is observed to agree with the numerical solution to well within 0(E). 8. Concluding remarks. In this paper, we have presented a multiscale perturbation method to construct a formal asymptotic approximation of nonlinear wave equations involving both initial and boundary conditions on the semi-infinite line. The well- posedness of problem (1.1) on an e-dependent timescale and the asymptotic validity of the formal approximations are not investigated in this paper. These remain open problems. Only the leading-order term of the formal perturbation expansion is dis- cussed, but the method readily extends to higher orders. In the region t_? x, two slow variables are necessary to describe the wave propagation. For the backward-going 2 1.5 ,,\"\\E=0 t= 5 1 A -0.5 0 2 4 6 ~~~~~8 10O x 12 1'4 16 solution (dashed line, FIG. 4. Comparison of the numerical solution (solid line, ), the perturbation ) at t = 5. Here h = 2u, + u(u, - uj) -) for E = 0.1, and the solution for E = 0 (labelled dash-dot line, conditions (6.10) were used. and the initial-boundary MULTIPLE-SCALE SOLUTION OF NONLINEAR x =8 WAVE EQUATION 957 EQ0 1.5- 0.5- 0 -0.5- 0 1 2 3 4 5 t 6 7 8 9 10 FIG. 5. Comparison of the numerical solution (solid line, ~) the perturbation solution (dashed line, ) at x =8. Here h =2u, + u(u, - u.,) -) fore- = 0.1, and the solution for - = 0 (labelled dash-dot line, conditions (6.10) were used. and the initial-boundary wave, the continuity of the solution across x =t 15 imposed, while, for the forward-going wave, the initial condition involves both the boundary data and the contribution from the incoming wave at x = 0. The success of the method depends on the solution of the PDEs (5.1b) and (5.4). As the examples in ?? 6 and 7 show, explicit solutions can be obtained for some nonlinearities and special initial and boundary conditions. For the two special examples investigated in this paper, comparison between numerical sol- utions and perturbation solutions show good agreement. Even when the PDEs (5.1b) and (5.4) cannot be solved exactly, we expect that approximate solutions to the first-order PDEs would provide valuable insight into the interplay between initial and boundary conditions for many nonlinear wave propagation problems. It is pointed out in ? 5 (5.7), that the method developed in this paper is applicable to a wider class of nonlinear hyperbolic wave equations. Such equations occur, for example, in gas dynamics or acoustics, often in the form (see, for example, Nayfeh [11]f) Ut rX=(2uxuxx+ (y- 1)u,uxx) = ((y- 1)(u,ux)x+3 2 and zero initial conditions u(x, 0) = ur(x, 0) =0, as, for example, the waves generated by a piston [6, Chap. 5]. The more general initial-boundary value problem (on xrtb ?0) in which both the boundary data at x = 0 and the initial data at t = 0 are nonzero can be approached where u (x,, t) is the velocity potential and y is the ratio of specific heats for the gas. For such equations, the following two types of problems have traditionally been considered: (i) Initial value problems on -co< x <00o, for example,, in [ 11], [7]; (ii) Signalling problems or simple wave problems on x: (0, t) = p (t) ?0, with u using the methods developed in this paper, not only for acoustics but also in other application areas. 958 S. C. CHIKWENDU AND C. V. EASWARAN REFERENCES [1] W. F. AMES, Numerical Methods for Partial Differential Equations, Academic Press, New York, 1977. [2] S. C. CHIKWENDU, Nonlinear wave propagation by Fourier transform perturbation, Internat. J. Non- Linear Mech., 16 (1981), pp. 117-128. [3] S. C. CHIKWENDU AND J. KEVORKIAN, A perturbation method for hyperbolic equation with small nonlinearities, SIAM J. Appl. Math., 22 (1972), pp. 235-258. [4] P. R. GARABEDIAN, Partial Differential Equations, Chelsea, New York, 1986. [5] J. B. KELLER AND P. D. LAX, The initial and mixed boundary value problems for hyperbolic systems, Los Alamos Sci. Lab. 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