Wave propagation, reflection and transmission in non-uniform one-dimensional waveguides

Waves can propagate freely without reflection in a certain class of non-uniform one-dimensional waveguides even though the properties of the waveguide vary rapidly. In these cases, the amplitude of the wave changes as a function of position but the power associated with the wave is preserved along the waveguide as in uniform waveguides. A generalised wave approach based on reflection, transmission and propagation of waves is used for the analysis of such non-uniform waveguides. The positive- and negative-going wave motions are separated so that the problem is always well-posed. Examples include longitudinal motion of bars and bending motion of Euler–Bernoulli beams, where the cross-section varies as a power of the length. The energy transport velocity, which is the velocity at which energy is carried by the waves in these waveguides, is derived using the relationship between power and energy. It is shown that this energy transport velocity depends on position as well as frequency and differs from the group velocity. Numerical results for wave transmission through a rectangular connector with linearly tapered thickness and constant width are obtained in a straightforward manner without approximation errors and at a low computational cost, irrespective of frequency.