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

We propose a new model to approximate the wave response of waveguides containing an arbitrary number of small inclusions. The theory is developed to consider any one-dimensional waveguide (longitudinal, flexural, shear, torsional waves or a combination of them by mechanical coupling), containing small inclusions with different material and/or sectional properties. The exact problem is modelled through the formalism of generalised 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, where we perform an error analysis to demonstrate the validity of the approximate solutions, and we propose a parameter quantifying the expected errors in the approximation dependent upon the parameters of the waveguide.