First, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form (Q(∂′)−a2∂2n)(Q(∂′)−b2∂2n) (where ∂′=(∂1,…,∂n−1) and a>0,b>0,a≠b) with respect to Dirichlet boundary conditions at xn=0. The Green function Gξ is represented by a linear combination of fundamental solutions Ec of Q(∂′)(Q(∂′)−c2∂2n), c∈{a,b}, that are shifted to the source point ξ, to the mirror point −ξ, and to the two additional points −abξ and −baξ, respectively.