Abstract: Up to now there have been a dazzling number of artificial boundary conditions (ABCs) in the field of numerical simulation of wave propagation. In order to choose the most appropriate ABCs and assess their performance in complicated wave problems, the related systems of theory and formula that can be used to classify or merge these ABCs still need to be improved. In this work we develop the theory of extrapolation-type artificial boundary condition which can merge a series of classical ABCs that have the common feature that the motion of each artificial boundary node is extrapolated from the motions of a set of adjacent nodes at several previous time steps. These ABCs include Liao's multi-transmitting formula (MTF), paraxial-approximation absorbing boundary conditions, Higdon boundary conditions, the auxiliary-variable-based ABCs of Givoli-Neta, Hagstrom-Warburton or AWWE, et al. Due to the fact that the existing boundary formulas usually have somewhat imperfections, thus we propose two new basic formulas for the extrapolation-type ABCs. One formula is an optimized MTF which incorporates a set of adjustable artificial wave velocities as the control parameters. The other formula is a unified Higdon boundary formula which is defined in a local coordinate system centered at the boundary node and uses various artificial wave velocities as control parameters. The two basic boundary formulas are the most simple and effective extrapolation-type ABCs. Other local ABCs of this type can mostly be transformed from the two basic boundary formulas, or have connections with them via some kind of equivalent intermediate formulas. Numerical experiments are conducted to validate the effectiveness of the proposed theory and boundary formulas. As compared to traditional ABC employing a single artificial wave velocity, the superiority of using ABCs with adjustable artificial wave velocities is preliminarily revealed in this work. It can be expected that the superiority will be more remarkable in simulating complicated wave problems that have several distinct physical wave velocities, such as elastic waves in soft soils with large ratio of longitudinal and transversal wave velocities, dispersive waves in ocean acoustics or meteorology and so forth.