Long-period seismograms are generated by a spectral method for body waves interacting with the upper mantly discontinuities that define the low-velocity and transition zones of the spherically symmetric but radially inhomogeneous earth models, PEM-C and T7. A complex velocity profile incorporates the SL8 Q model. Models PEM-C and T7 predict distinct differences in displacement waveforms for P waves that interact with the 400- and 670-km discontinuities. The waveforms demonstrate that later arrivals such as interference head waves can be important constraints upon earth models. Theoretical seismograms were computed by a spectral method that modeled the velocity profile with arbitrarily thick inhomogeneous layer (full wave theory). The seismograms agreed well with those generated by methods that approximate the velocity profile with thin homogeneous layers (the Cagniard and reflectivity methods) provided the displacements were convolved with source-time functions and a long-period instrument response. Minor differences among the results of the methods do not affect the interpretation of observed waveform data but do help define the limitations of the methods. For example, the thin-layer methods suffer from numerical limitations on layer thickness and the umber of layers needed to approximate the earth model. The full wave theory may occasionally also require the use of thin layers when the zero-order asymptotic theory it uses fails to describe wave propagation properly in layers having rapid velocity variation with depth.