Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media R. V. Craster a , L. M. Joseph a , J. Kaplunov b a Department of Mathematics, Imperial College London, London, SW7 2AZ, U.K. b School of Computing and Mathematics, Keele University, Keele, Staffordshire, ST5 5BG, UK Abstract This article explores the deep connections that exist between the mathematical represen- tations of dynamic phenomena in functionally graded waveguides and those in periodic media. These connections are at their most obvious for low-frequency and long-wave asymptotics where well established theories hold. However, there is also a complemen- tary limit of high-frequency long-wave asymptotics corresponding to various features that arise near cut-off frequencies in waveguides, including trapped modes. Simultaneously, periodic media exhibits standing wave frequencies, and the long-wave asymptotics near these frequencies characterise localised defect modes along with other high-frequency phenomena. The physics associated with waveguides and periodic media are, at first sight, apparently quite different, however the final equations that distill the essential physics are virtually identical. The connection is illustrated by the comparative study of a periodic string and a functionally graded acoustic waveguide. Keywords: asymptotic, long-wave, low-frequency, high-frequency, homogenisation, waveguide, functionally graded, thin structure, trapped mode 1. Introduction For long waves within a waveguide, at low-frequency, intuition suggests that (for waveguides governed by Helmholz equation with Neumann boundary conditions) the waveguide behaves effectively as a string; when viewed from afar the guide is long and thin. Similarly, a string composed of periodic elements where the length scales associated with the periodicity are much less than the wavelength of the excitation also intuitively behaves as some effective string. In both cases the word “effective” is rather vague, but as we shall see this can be made precise through an asymptotic approach involving two- scales; the thickness of the guide or periodicity scale and the length-scale of the guide or overall string. As we shall see an application of multiple scales leads rapidly to an effective equation for these two problems that can be simultaneously treated. Perhaps less obviously one can also consider high-frequency wave propagation which almost immediately equates to short wavelength, as one typically thinks of waves within a bulk medium, and asymptotic techniques for waveguides based upon the WKBJ ansatz are popular and versatile [1, 2, 3, 4]. However, the imposition of boundaries such as those of a waveguide can alter this intuitive viewpoint and long-wave solutions are also possible. Taking a straight, constant thickness, isotropic waveguide a natural approach Preprint submitted to Elsevier August 20, 2013