The Fourier pseudospectral time-domain (PSTD) method is an attractive method to efficiently model wave propagation through a weakly inhomogeneous fluid medium. The method fails, however, for fluid media with discontinuous properties. This failure is due to the well known Gibbs phenomenon which arises when Fourier transforming a discontinuous function. The extended Fourier PSTD method developed here can accurately and efficiently model wave propagation through weakly inhomogeneous fluid media with discontinuities in the media properties. Rather than using Fourier transforms to calculate the spatial derivatives in the wave equation, the method developed here uses a generalized eigenfunction expansion for which no Gibbs phenomenon arises. Two approaches to solving the resulting time-domain problem are explored and the accuracy of the method is demonstrated. Finally, results from an example calculation of transmission through the water-air interface are shown.