Global, uniform, asymptotic wave-equation solutions for large wavenumbers (Q1099735)

The author considers a class of wave equations \[ ik^{-1}\frac{\partial \phi}{\partial z}(x,z)={\mathcal H}(-ik^{-1}\frac{\partial}{\partial x},x,z\quad)\phi (x,z). \] Here \(z\in {\mathbb{R}}\), \(x=(x_ 1,...,x_ n)\in {\mathbb{R}}^ n \)denotes transverse coordinates, k is a large parameter. For an artificial paramete \(\Omega\) he introduces the coherent-state transformation \[ \psi (p,x,z)\equiv (\frac{k\Omega}{\pi})^{n/4}\int \exp (-k\Omega y^ 2/2-kp\cdot y)\phi (x+y,z\quad)dy \] under which the wave equation is transformed to \[ ik^{-1}\frac{\partial \psi}{\partial z}(p,x,z)={\mathcal H}(-ik^{-1}\partial /\partial x,x+ik^{-1}\quad \frac{\partial}{\partial p},z)\psi (p,x,z) \] being the starting point for asymptotic approximations in the spirit of the geometrical theory of diffraction. In the limiting case \(\Omega\to \infty\) (\(\Omega\to 0)\) the proposed method reduces to the ray method (resp. Maslov's method). But in contrast to them the amplitude factor never diverges. Thus the method is a global and uniform one giving an appropriate description for caustic dominated fields and wave propagation in random media as well.