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I find a solution of the following PDE system: $\epsilon^2\left(\Phi_{xx}+\Phi_{yy}\right)+\Phi_{zz}=0$ for $z\lt0$ and $g\Phi_z+\epsilon^2V^2\Phi_{xx}=0$ for $z=0$ with $\epsilon\ll1$ with $V=const$. Is it possible to use the ray method in this case, trying to find a solution in the form: $\Phi(x,y,z)=A(x,y,z)e^{iu(x,y,z)/\epsilon}$ ? Thanks.

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