Copyright c  2005 by ABCM November 6-11, 2005, Ouro Preto, MG ASYMPTOTIC WAVE PROPAGATION ALONG CHAINS OF REPETITIVE SYSTEMS Karl Peter Burr Department of Mechanical Engineering, Escola Politêcnica, USP karl.burr@poli.usp.br Abstract We consider wave propagation along chains of non-uniform repetitive systems. For harmonic waves, the governing equation for this class of systems is a second order difference equation with non-constant coefficients. When the ratio between the sub-systems inertia force and the sub-systems coupling force is small, we developed an asymptotic theory for harmonic wave propagation along this class of systems. We assume an asymptotic expansion for the dependent variable based on the small parameter defined above. Its zero order term is the solution of a second order differential equation with variable coefficient. To solve it, we use a change of the independent variable, which maps the original equation to a model equation with known solution in terms of special functions. The choice of the model equation is guided by the topology of the turning points of the original equation. By assuming an asymptotic expansion for the change of independent variable, and incorporating it into the asymptotic expansion for the dependent variable, the strategy used to solve the zero order problem is applied to the higher order problems. Each set of turning points is modeled by a transfer matrix. As a result, quantities of interest are given along the system as a product of transfer matrices. We apply the asymptotic theory to case with known solution in terms of Bessel functions. We illustrate how to apply the asymptotic theory to wave propagation problems. Keywords: repetitive system, asymptotic method, difference equation, wave reflection, comparison equation method 1. Introduction. We consider wave propagation along one-dimensional repetitive systems. These type of systems are chains of inter- connected subsystems which can be identical to each other (uniform repetitive system) or they may vary among each other (non-uniform repetitive system). In this work, only single-degree-of-freedom (d.o.f) subsystems are considered. Examples of repetitive systems are chains of coupled oscillators, crystal lattices and any non-uniform one-dimensional continuous medium discretized by pieces where at each piece the medium is assumed to have uniform properties. A chain of coupled pendula is an example of a chain of coupled oscillators. The subsystem is a pendulum coupled to its two nearest neighbors only through springs. Linear gravity waves propagating along a shallow channel with non- uniform bottom, which is discretized by steps is an example of a continuous one-dimensional medium with non-uniform properties discretized by pieces where at each piece the medium has uniform properties. The subsystem is a shelf between two consecutive steps, and the coupling between the two nearest shelves is given by the continuity of the free surface displacement and of the horizontal velocity (Devillard et al., 1988). For harmonic wave propagation, the governing equation for such class of repetitive systems is a second order difference equation (2nd order d.e.) which can be reduced to the canonical form y x+1 − 2y x + y x-1 + β 2 Q(x)y x =0 (1) where β 2 represents the ratio between the subsystem inertia and the coupling force between subsystems and Q(x) is the 2nd. order d.e. coefficient function. For the case of a chain of coupled pendula, β 2 is the ratio between the pendulum inertia and the coupling spring force and Q(x)=1/(1 + ǫ(x)) − ω 2 , where (1 + ǫ(x)) and ω are, respectively, the non-dimensional pendulum lenght and non-dimensional wave frequency. The objective of this work is to develop an asymptotic theory for wave propagation along long non-uniform one- dimensional repetitive systems when the parameter β is small. The asymptotic theory reveals to some degree how the non-uniformity in the system parameters affect wave propagation in such class of systems. This could be an useful tool to design the non-uniformity in the system parameters such that the repetitive system behaves, for example, as wave filter, selecting the range of wave frequencies that waves are allowed to pass or reflected back. These design problems are one motivation for this work. The asymptotic theory developed here is outlined in the paragraph below. We use the concept of pseudo-differential operators to show that 2nd order d.e.s are equivalent to infinite dimensional ordinary differential equations (i.d.o.d.e.s). We adapted the comparison equation method (see chapter 2 of Froman & Froman, 1997) to the infinite dimensional o.d.e. equivalent to 2nd order d.e.s. In the comparison equation method, the original second order ordinary differential equation (2nd order o.d.e.) with an assumed small parameter is reduced by a change of the independent variable to a model equation with known solution in terms of special functions. The choice of the appropriate model equation is guided by the topology of the turning points of the original equation. Once we decided on the form of the model equation, we obtain a differential equation