We study the sagittal elastic waves propagating along the ͓100͔ direction in Fibonacci superlattices, formed by the repetition of GaAs and AlAs slabs in A and B constituent blocks following the Fibonacci sequence.We consider infinite repetition of a given Fibonacci generation and the finite Fibonacci generation with free surfaces, as a comparison. We have considered in our study systems ranging from the second to the eighth generation. We employ the surface Green-function matching for N nonequivalent interfaces to obtain the dispersion relation and the density of states of the different systems. In this way it is possible to obtain the spatial localization of the different modes. We have studied the influence of the increasing Fibonacci generations on the dispersion relation of the sagittal elastic modes. ͓S0163-1829͑98͒05822-6͔