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　Grünwald–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.　©　2021　Elsevier　B.V.