Analytical approximations for multiple scattering in one-dimensional waveguides with small inclusions

In this paper, we propose a new model to approximate the wave response of waveguides containing an arbitrary number of small inclusions. The theory is developed for general one-dimensional elastic waveguides to study various types of modes, e.g. longitudinal, flexural, shear, torsional or coupled modes. The precise problem assumes the host material contains small inclusions, with different material and/or sectional properties which behave as scatterers from a wave propagation point of view. The inclusions are modelled through the formalism of generalized functions, with the Heaviside function accounting for the discontinuous jump in different sectional properties of the inclusions. For asymptotically small inclusions, the exact solution is shown to be equivalent to the Green’s function. We hypothesize that these expressions are also valid when the size of the inclusions are small in comparison to the wavelength, allowing us to approximate small inhomogeneities as regular perturbations to the empty-waveguide (the homogeneous waveguide in the absence of scatterers) as point source terms. By approximating solutions through the Green’s function, the multiple scattering problem is considerably simplified, allowing us to develop a general methodology in which the solution is expressed for any model for any elastic waveguide. The advantage of our approach is that, by expressing the constitutive equations in first order form as a matrix, the solutions can be expressed in matrix form; therefore, it is trivial to consider models with more degrees of freedom and to arrive at solutions to multiple scattering problems independent of the elastic model used. The theory is validated with two numerical examples, one with longitudinal waves (classical rod) and the other one with flexural waves (Timoshenko beam). An error analysis is performed to demonstrate the validity of the approximate solutions, where we propose a parameter quantifying the expected errors in the approximation dependent upon the parameters of the waveguide. The approximate solutions were shown to be significantly accurate up to the thresholds of application of each model considered. The approximate expressions were found to be easily applied to consider higher-order models for the waveguide and were simple to implement.