Wave Propagation on Multiple Scales along Rough Thin-Elastic Solids

Waves attenuate with distance travelled into rough and randomly disordered media. This is reminiscent of the wave localisation phenomenon, which is a product of wave scattering and occurs in many branches of wave science. Wave localisation refers here to exponential attenuation (on average) of waves in rough media. In this work, the attenuation of waves travelling along rough solids is studied in the setting of strings, beams and floating plates. Two methods for computing the complex-valued effective wavenumber of a rough beam in the context of linear time-harmonic theory are presented. The roughness of the beam is modelled as a continuous random process of known characteristic length and root-mean-square amplitude for either the beam properties (beam mass or beam rigidity) or beam thickness. The first method is based on a random sampling method, with the effective wave field calculated as the mean of a large ensemble of wave fields for individual realisations of the roughness. The individual wave fields are calculated using a step approximation, which is validated for deterministic problems of mass and rigidity variations via comparison to results produced by an integral-equation approach. The second method assumes a splitting of the length scale of the fluctuations and an observation scale, employing a multiple-scale approximation of the beam deflection to derive analytical expressions for the effective attenuation rate and phase change. Numerical comparisons show agreement of the results of the random sampling method and the multiple-scale approximation for a wide range of parameters in the small roughness-amplitude regime. It is shown that the effective wavenumbers only differ by a real constant between the cases of varying beam mass and rigidity. The numerical and semi-analytical methods are extended to describe the attenuation of water waves in a two-dimensional fluid domain, which has its surface covered by a rough thin elastic plate and is of finite depth. The plate roughness is modelled with similar properties as for the in-vacuo beam problem. The numerical method is based on an approximation of the full-linear solution in the fluid domain, the semi-analytical method uses a multiple-scale expansion of the velocity potential, from which an equation can be derived describing the attenuation of the effective wave field. The results obtained via the numerical method validate the multiple-scale approach for small-amplitude plate roughness. However, individual wave fields attenuate significantly slower that effective wave fields for the in-vacuo beam and floating plate problems. Localisation of individual wave fields is shown for strings with continuous density variations for large roughness amplitudes, for which multiple-scale approaches are not valid anymore. Individual and effective wave fields show similar attenuation behaviour in this large roughness-amplitude regime, and a connection between strings with continuous roughness profile and beaded strings is established. Finally, a beam with periodically located notches is presented and our numerical method is modified to simulate wave propagation in the time domain. It is shown that introducing disorder into the notch depths leads to localisation in the audible frequency range