p_\mu=0, \end{equation} where $$=\lim_{M \rightarrow \infty} {1 \over M} \int_0^M q(\xi,\epsilon\xi,\Sigma) d\xi. $$ By integrating (20) with respect to $\xi$ and (21) with respect to $\mu$, we obtain a set of coupled ODEs: \begin{eqnarray} &2 q_{\Sigma}+ q = 0&\\ &2 p_{\Delta} + (1+)p + p^2 = 0& \end{eqnarray} The arbitrary functions of $\Delta$ and $\Sigma$ that result from these integrations are set to zero because the solution has to depend continuously on initial data. The coupled ODEs (23) and (24) have to be solved subject to the conditions that when $t=x$, $q$ equals $g$, the backward-going component in $t$. We now numerically compare the solutions obtained in this paper, (25)-(26), with those obtained in \cite{CE}. Figure 2 plots the $O(1)$ solution at $t=5$, using equations (25)-(26) and the solution derived in \cite{CE} by solving the PDEs resulting from (16)-(17). We used the set of initial-boundary conditions \begin{eqnarray} &u(x,0) = \sin (3 x)&\\ &u_t(x,0) = 0&\\ &u(0,t) = {2t\over 1+t^2} \sin t\,.& \end{eqnarray} In this case $ =0$ in equation (26). The two plots are indistinguishable. Figure 3. plots the same solutions at $x=8$. \vspace{.2in} \psfig{file=2.ps} \vspace{.3in} \psfig{file=3.ps} Figure 4 plots the solutions at $t=5$ for $u(x,0)=\sin^2(3x)$, and the other conditions as in (28)-(29). In this case $\ne 0$. We plotted the solution in the region $x>5$ where interaction between left and right going waves take place. In Figure 5 are solutions at $x=8$. Note that each of figures 2-5 contains two plots which are indistinguishable except in figure 5. \vspace{.2in} \psfig{file=4.ps} \vspace {.3in} \psfig{file=5.ps} \section{Conclusion} We presented a method to develop uniformly valid asymptotic approximations to weakly nonlinear wave equations that is considerably simpler than previously available methods. In principle, the procedure could be carried out to higher order, although we presented only first order approximations. The new method offers a way to study the nonlinear interaction of left and right running waves through a set of coupled ordinary differential equations. This should be compared with previous methods that required solving coupled sets of partial differential equations. Our method used a pair of ``slow'' characteristic variables $\Delta=\epsilon (t+x)$ and $\Sigma=\epsilon (t-x)$ and certain assumptions on the averages of forward and backward going wave amplitudes to simplify the analysis. This simplification will hopefully help analyze a variety of weakly nonlinear wave equations that occur in applications such as gas dynamics and acoustics (\cite{KC},\cite{Nay}), in which the boundary and initial data are non-zero. %\newpage \begin{thebibliography}{1000} \bibitem{CE} S. C. Chikwendu and C. V. Easwaran, {\em Multiple-scale solution of initial-boundary value problems for weakly nonlinear wave equations on the semi-infinite line}. SIAM J. Appl. Math. {\bf 52}(1992), pp. 964 -- 958. \bibitem{KC} J. Kevorkian and J. D. Cole, {\em Multiple Scale and Singular Perturbation Methods}. Springer Verlag, New York, 1996. \bibitem{Mye} C. J. Myerscough, {\em A simple model for the growth of wind-induced oscillations in overhead lines}. J. Sound and Vibration {\bf 39}(1975), pp. 503 -- 517. \bibitem{Nay} A. H. Nayfeh, {\em A comparison of perturbation methods for nonlinear hyperbolic waves.} In {\em Singular Perturbations and Asymptotics}, R.E. Mayer and S.U.Parter, Eds, Academic Press, NY 1980, pp 223-276. \bibitem{TS} A. N. Tychonov and A. A. Samarski, {\em Partial Differential Equations of Mathematical Physics}. Holden-Day, San Francisco, 1964. \bibitem{Gbw} G.B. Whitham, {\em Linear and Non-Linear Waves}. Wiley-Interscience, New York, 1974. \end{thebibliography} \bigskip {\sc Chirakkal V. Easwaran}\\ Department of Mathematics and Computer Science\\ State University of New York\\ New Paltz, NY 12561 USA\\ E-mail address: easwaran@mcs.newpaltz.edu\\ http://www.mcs.newpaltz.edu/$\sim$easwaran \end{document}