The theory of neutron wave propagation through an interface is investigated with the following models: Model A—One-Speed Diffusion Theory, Model B—One-Speed Transport Theory, Model C—Energy-Dependent Diffusion Theory, and Model D—Energy-Dependent Transport Theory. Numerical results for these four models are given. The wave propagation constants α and β, where k = α + iβ, together with α2 - β2 and 2αβ are compared. In addition, the energy-dependent phase shift θ(E, ω) and amplitude ρ(E, ω) are also computed for Models C, D. The propagation constants compare well with one another. The differences between the four theories, although minor, are enhanced by comparing α2 - β2 as a function of frequency. θ(E, ω) and ρ(E, ω) are identical for Models C and D when plotted. A comparison of the discrete waves written in terms of incident, reflected, and transmitted components is also made. It is concluded that the continuum has a sizeable effect close to the interface. Energy and interface effects were seen to be separable from each other for the models studied. A comparison of the discrete amplitudes was made after neglecting continuum terms. The numerical results show that at the interface, the wave amplitude and phase shifts are almost identical for the two diffusion models but differ substantially from the transport models.