The　finite　element　modeling　of　the　dynamic　and　wave　problems　in　unbounded　media　requires　an　artificial　boundary　condition　to　simulate　the　truncated　infinite　domain.　The　Dirichlet-to-Neumann　boundary　condition　has　been　transformed　from　frequency　to　time　domain　by　using　the　rational　function　approximation　and　auxiliary　variable　technique.　It　is　extended　to　three-dimensional　layer　problem　here.　The　resulting　artificial　boundary　condition　is　stable　itself　in　time　domain,　whereas　the　time-domain　instability　of　the　artificial　boundary　condition　coupled　with　the　finite　element　method　is　found　for　the　foundation　vibration　recently　and　for　the　wave　propagation　here.　A　simple　and　effective　method　that　introduces　the　damping　proportional　to　the　stiffness　matrix　in　the　finite　element　method　is　given　to　cure　such　coupling　instability　completely.　The　stabilized　damping　is　so　small　that　it　does　not　affect　the　solution　accuracy　nearly.　The　numerical　examples　show　the　instability　phenomenon　and　indicate　the　effectiveness　of　the　damping　method.　The　time-domain　stability　studies　here　can　be　a　reference　for　the　other　artificial　boundary　conditions.