The propagation of directed scalar waves in D+1 dimensions in a strongly disordered medium is studied. We use the model first proposed by Saul, Kardar, and Read [Phys. Rev. A 45, 8859 (1992)], where unitarity is guaranteed in each step. The beam positions 〈${\mathbf{x}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}〉 and 〈x${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{2}$〉 characterize the transverse fluctuations of a directed wave front, where the overbar means an average over the wave profile for a given realization of randomness, and 〈 〉 means a quenched average over all realizations. We introduce ${\mathit{G}}_{\mathit{q}}^{\mathbf{k}}$(y) as the Laplace-transformed Green function of two free random walkers with center-of-mass momentum k and relative position y. We calculate analytically the mean-square deviation of the beam center, 〈x\ifmmode\bar\else\textasciimacron\fi{} $^{2}\mathrm{〉}$, as a function of time. The results show that, for large t, 〈x${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{2}$〉 behaves as (1/ \ensuremath{\surd}\ensuremath{\pi} )${\mathit{t}}^{1/2}$-1/4+O(${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$) in 1+1 dimensions and as (lnt+4 ln2+\ensuremath{\gamma})/4\ensuremath{\pi}+O(1/t) in 2+1 dimensions and takes the finite value 1/2D[${\mathit{G}}_{\mathit{q}=1}^{\mathbf{k}=0}$(0)- \ensuremath{\surd}(27/4\ensuremath{\pi}) ${\mathrm{t}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{\ensuremath{\delta}}}_{\mathit{D},3}$]+O(1/t) in D+1 dimensions where D\ensuremath{\ge}3, \ensuremath{\gamma} being the Euler constant. We generalize these results to a twofold random walk with any probability-flux-conserving interaction. In all cases the leading term at large t depends solely on the finite value or leading singularity of ${\mathit{G}}_{\mathit{q}}^{\mathbf{k}=0}$(0) at q=1.