A well known method of non-destructive testing and inspection involves the use of waves propagating at ultrasonic frequencies, which travel into a material and can reflect from cracks or defects.
The most common waves and reflections are P waves, common for localised thickness inspection and local crack identification. Less common are Lamb and Rayleigh waves. The advantage of these are that they can travel over significant distances with relatively little loss of amplitude.
The lowest Lamb modes are A0 flexural waves, which are dispersive and S0 compression waves which are non-dispersive. Research has been carried out into the propagation of Lamb waves through inhomogeneous materials.
Where the material is homgeneous, the solutions are relatively straight-forward, however, when the bending stiffness varies, such as through a change in thickness or density, energy can be exchanged between different wavenumber solutions.
A bi-orthogonality code can be used to match the stress and displacement conditions at each point of a multi-impedance waveguide. As an example, the validation case by Pagneux  is a three part beam (aluminium-copper-aluminium) with an excitation S0 wave induced at a frequency of 50kHzmm from the left hand side and infinite damping on the right hand side.
The illustration shows how the stress changes in the beam with time and distance, especially close to the join of the different materials.