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We consider the equation ut =uxx + uyy + b(x)f(u) + g(u), (x, y) ∈ R≠ with monostable nonlinearity, where b(x) is a nonnegative measure on R that is periodic in x.In the case where b(x) is a smooth periodic function, it is known that, for each θ ∈ [0, 2π), there exists a "planar" travelling wave in the direction θ-more precisely a pulsating travelling wave that propagates in the direction (cos θ, sin θ)-with average speed c if and only if c ≥ c*(θ, b), where c*(θ,b) is the so-called "minimal speed".We extend his theory by showing the existence of the minimal speed c*(θ,b) for any nonnegative measure b with period L.We also verify the known formula for the "spreading speed" for this case.We then study the question of maximizing c* (θ, b)under the constraint f[0,L)b(x)dx =αL, where α is an arbitrarily given positive constant.We prove that the maximum is attained by periodically arrayed Diracs delta functions h(x) =αL Σk∈Z δ(x + kL) for any direction θ.Based on these results,for the case that b =h we also show the monotonicity of the spreading speedsin θ and study the asymptotic shape of spreading fronts for large L and small L.Finally, we show that for general 2-dimensional periodic equation ut =uxx + uyy + b(x, y)f(u) +g(u), (x, y) ∈ R≠, where b ≥ 0 is a periodic measure on R≠.the above variational problem has no solution as there is no upper bound on the speed c*(θ, b).