We propose two new boundary conditions to regulate coherent reflections from the model boundaries in numerical solutions of wave equations. Both boundary conditions have the common feature that the boundary condition is varied with respect to time. The first boundary condition expands or contracts the computational model during a modeling simulation. The effect is to cause a Doppler shift in the reflected wavefield that can be used to shift energy outside a frequency band of interest. Additionally, when the computational domain is expanding, the range of possible incidence angles on the boundary is restricted. This can be used to increase the effectiveness of many existing absorbing boundary conditions that are more effective for incidence angles close to normal. The second boundary condition is an extension of random boundaries. By carefully changing the realization of a random boundary over time, a more diffusive wavefield can be simulated. We show results with 2D numerical simulations of the scalar wave equation for both these boundary conditions. While the first boundary condition has application to modeling, both of these boundary conditions have potential applications within algorithms that rely upon modeling kernels, such as reverse-time migration and full-waveform inversion.