A　series　of　classical　absorbing　boundary　conditions　(ABCs),　including　paraxial-approximation　ABCs,　Liao＇s　multi-transmitting　formula　(MTF),　Higdon　ABCs,　and　some　other　related　techniques,　have　the　common　feature　that　the　motion　of　an　arbitrary　artificial　boundary　node　at　each　timestep　is　directly　predicted　from　the　motions　of　some　adjacent　nodes　at　several　previous　timesteps.　They　are　expressed　in　somewhat　equivalent　forms,　contain　similar　control　parameters,　and　have　comparable　accuracy　and　stability　in　numerical　simulations.　This　study　develops　a　theoretical　framework　called　displacement-type　(a　more　exact　name　would　be　＂prediction-type＂　or　＂extrapolation-type＂)　local　ABCs　to　merge　these　boundary　conditions.　The　idea　of　this　theory　mainly　originates　from　the　versatility　of　MTF,　which　uses　a　unified　formula　to　approximate　the　propagation　of　outgoing　waves　through　each　boundary　node.　This　idea　can　be　generalized　to　other　displacement-type　local　ABCs　to　unify　their　expressions　and　to　optimize　their　applications.　These　ABCs　have　two　basic　control　parameters;　one　is　the　boundary　order,　and　the　other　is　adjustable　computational　wave　velocities.　Considering　the　poor　performance　of　paraxial　ABCs　and　the　slight　imperfections　in　MTF　and　Higdon　ABCs,　we　propose　two　new　unified　formulas　to　be　the　starting　points　of　expressing,　evaluating,　and　applying　displacement-type　local　ABCs.　One　formula　is　an　optimized　MTF　by　introducing　various　computational　wave　velocities.　The　other　formula　is　a　generalized　Higdon　boundary　formula,　which　is　established　in　a　unified　local　coordinate　and　uses　the　adjustable　computational　wave　velocities.　The　rule　of　choosing　boundary　parameters　for　the　absorption　of　acoustic　and　elastic　waves　is　discussed　in　detail.　Numerical　tests　validate　the　proposed　theory　and　formulas.　Issues　on　numerical　stability　are　briefly　reviewed　and　tested　in　simulation　examples.　This　is　still　an　active　research　topic　related　to　displacement-type　local　ABCs.