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4.3 Linearised Diffusive Wave Hydraulic ModelWe selected the linear diffusive wave as the propagation model in this study.270Under some assumptions, the Saint-Venant system combining continuity equa-tions of mass and momentum can be simplified. The inertia terms of the momentumequation can be neglected if they are small compared to the channelbed slope leading to the diffusive wave approximation. Flow motion is thendescribed by the following equation (Moussa, 1996):275tel-00392240, version 1 - 5 Jun 2009280( ) (∂Q ∂Q∂ 2∂t + C ∂x − q Q− D∂x − ∂q )= 0 (1)2 ∂xwith t the time, x the abscissa along the river reach, Q(x, t) the discharge,q(x, t) the lateral inflows, C the wave celerity and D the diffusion coefficient.If the equation is further linearised around a reference regime, C and D canbe considered to be constant. Hayami (1951) identified an analytical solutionfor this equation in the case where q = 0. This solution takes the form of aconvolution product applied to an input I(t) = Q(0, t) − Q(0, 0) and givingan output O(t) = Q(L, t) − Q(L, 0) as:∫ tO(t) =0K(t) =L2 √ πDI(t − τ)K(τ)dτexp [ CL4D(2 −L− CtCt Lt 3/2)]285where L is the length of the river reach, K the convolution kernel and τa dummy variable. In the case of uniformly distributed lateral inflows (i.e.q(x, t) = q(t)), Moussa (1996) identified the following analytical solution forthe same equation:14