  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.
