The　time　distributed-order　diffusion-wave　equation　describes　radial　groundwater　flow　to　or　from　a　well.　In　the　paper,　an　alternating　direction　implicit　(ADI)　Legendre-Laguerre　spectral　scheme　is　proposed　for　the　two-dimensional　time　distributed-order　diffusion　wave　equation　on　a　semi-infinite　domain.　The　Gauss　quadrature　formula　has　a　higher　computational　accuracy　than　the　Composite　Trapezoid　formula　and　Composite　Simpson　formula,　which　is　presented　to　approximate　the　distributed　order　time　derivative　so　that　the　considered　equation　is　transformed　into　a　multi-term　fractional　equation.　Moreover,　the　transformed　equation　is　solved　by　discretizing　in　space　by　the　ADI　Legendre-Laguerre　spectral　scheme　to　avoid　introducing　the　artificial　boundary　and　in　time　using　the　weighted　and　shifted　Grunwald-Letnikov　difference　(WSGD)　method.　A　stability　and　convergence　analysis　is　performed　for　the　numerical　approximation.　Some　numerical　results　are　illustrated　to　justify　the　theoretical　analysis.　(C)　2021　Elsevier　B.V.　All　rights　reserved.