A method is given of calculating second-order singularities related to inflection points of slowness curves. An approximated formula is derived for the effective permittivity function in the neighbourhood of the corresponding singular points. A numerical analysis of several piezoelectrics is peformed, and crystal cuts are calculated for which the function has such singular points. The analysis shows that inflection points may appear for almost every crystal cut, as in the case of lithium niobate and langasite.