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Let $S\left(\lambda \right)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪\subset {\mathbf{R}}^n$, $n\ge 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$. The function $s\left(\lambda \right)$, called scattering phase, is determined from the equality $e^{-2\pi is\left(\lambda \right)}=\det S\left(\lambda \right)$. We show that $s\left(\lambda \right)$ has an asymptotic expansion $s\left(\lambda \right)\sim {\sum }_{j=0}^{\infty }{c}_j{\lambda }^{n-j}$ as $\lambda \to +\infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.