A sound wave with angular frequency ω and finite amplitude ε, propagating in a rigid waveguide of any non‐rectangular cross section, is considered. The pressure and velocity potential of the wave, corresponding to each mode of the linear theory, are determined as power series in ε. The leading term is just the mode given by the linear theory and further terms are finite‐amplitude corrections to it. The wavenumber or propagation constantknof thenth mode is found to depend upon ε and is also determined as a power series. The fact thatkndepends upon ε, and the expression forkn(ε), is the most interesting consequence of the analysis. This shows that the propagation velocity ω/kndepends upon ε. The results are specialized to a waveguide of circular cross section. Then numerical values ofkn(ε) are given for a circular guide filled with air forn= 1, 2, 3, 4, and 5. The lowest mode (n= 0) is not considered because shocks occur in it. The method of analysis is a perturbation expansion adapted to eliminate secular terms. It has been used before to treat periodic finite‐amplitude sound waves in a closed container and periodic finite‐amplitude vibrations of strings and beams by J. B. Keller and L. Ting [Comm. Pure Appl. Math.19, 371 (1966)], and to treat nonlinear electromagnetic wave propagation [Phys. Rev.181, 1730 (1969)] and other nonlinear boundary‐value problems by the authors [J. Math. Phys.10, 342 (1969)].