Influence of parameter mismatch on the convergence of the band structures by using the Fourier expansion method

In the present paper, the convergence of the Fourier expansion method in calculating the band structures of elastic waves propagating in periodic composites is discussed. First, the convergence of the partial sum of a periodic function with discontinuous points is investigated by introducing three parameters to describe the overall error of a Fourier series and the Gibbs oscillation region. Second, the convergence of a product of two discontinuous functions is discussed based on two approximation methods, i.e., Laurent theory and the inverse formula, respectively. Finally, as an application, elastic waves propagating in one-dimensional layered periodic composite are studied. The band structures of the layered composites are obtained and some major influencing factors are discussed. Attentions are concentrated on the influences of the mismatch of mass density, shear modulus and filling ratio on the convergence of dispersion curves.