Acoustic sensors and acoustic measurements receive much attention in various applications. Because waveguides are commonly used in sensor design, theoretical means to study acoustic propagation and interaction in waveguides are necessary. However, current methods for elastic wave coupling, including the transfer matrix method and coupled mode theory in planar 2D waveguides, are not satisfactory. In this work, a coupled mode analysis for acoustic waves in 3D stress-free elastic waveguides is proposed. Similar to the coupled mode theory in optical waveguides, the analysis is presented by the evolution of modal amplitudes. It can solve various modal conversion and scattering problems in elastic waveguides with small changes of cross sections and stress-free boundaries. To demonstrate the practicability, the coupled mode analysis is used to calculate the reflection spectrum of the newly proposed structure, the acoustic fiber Bragg grating. In a notch-based grating fabricated on a thin cylindrical waveguide, the results from coupled mode analysis are in good agreement with those from the transfer matrix method, which has been already validated experimentally. The coupled mode analysis is a promising method to solve various scattering problems.