Communications in Mathematical Sciences

Volume 13 (2015)

Number 8

Global geometrical optics approximation to the high frequency Helmholtz equation with discontinuous media

Pages: 1949 – 1974



Chunxiong Zheng (Department of Mathematical Sciences, Tsinghua University, Beijing, China)


The global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed by the author in [33]. In this paper, we rederive it in a more concise way. It is shown that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, distinct from the WKB function used in the classical geometrical optics approximation. A new and the main contribution of this paper is an interface analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident rays include caustic points.


high frequency waves, global geometrical optics approximation, caustics, WKB analysis, discontinuous media

2010 Mathematics Subject Classification

65M25, 78M35

Published 3 September 2015