We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number) $ k $ which are getting nearly Lipschitz for large $ k $. Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter $ k $. The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.