Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability

Abstract

This paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation. More specifically, it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region for dynamics and vibration, but wave-based methods are typically only applicable to homogeneous and uniform waveguides. The WKB approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is also derived and included, so that the internal reflections are taken into account, but this requires a discretization of the waveguide. Moreover, a Fourier like series, the Karhunen–Loeve expansion, is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived. Experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam. It is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the WKB approach is given.