The course aims to introduce several classical methods of mathematical modeling of guided wave propagation, namely ray theory, parabolic equation theory and normal modes. Students are invited on an exciting journey that begins with boundary value problems and Cauchy problems for classical wave equations and ends with concrete applications in ecosystem acoustics. In the course of this journey, we will encounter a bestiary of asymptotic methods that help transform original equations into something much easier to solve (for example, ray equations and parabolic equations). Numerical methods for dealing with the resulting equations are also discussed, and mini-projects that would help students master these tools by solving realistic practical problems are offered. Some of the classics will take place in the fluid dynamics lab, and students are expected to write some not-so-complicated programs in MATLAB or any other language of their choice. As a reward, beautiful images depicting acoustic waves will be produced, and an understanding of how to use mathematics in some real-world situations will emerge.

References:
1. Computational ocean acoustics / F. B. Jensen, W. A. Kuperman, M. B. Porter, H. Schmidt. — Springer Science & Business Media, 2011.
2. Brekhovskikh L. M., Godin O. A. Acoustics of layered media I: Plane and quasi-plane waves. — Springer Science & Business Media, 2012. — Vol. 5.
3. Maslov, V. P., Fedoriuk, M. V. Semi-classical approximation in quantum mechanics (Vol. 7). Springer Science & Business Media, 2001.
4. Kravtsov Y. A., Orlov Y. I. Geometrical optics of inhomogeneous media. — Spring-Verlag, Berlin, 1990.
5. Petrov P. S., Antoine X. Pseudodifferential adiabatic mode parabolic equations in curvilinear coordinates and their numerical solution // Journal of Computational Physics. — 2020. — P. 109392.
6. A review of transparent and artificial boundary conditions techniques for linear and nonlinear schr¨odinger equations / X. Antoine, A. Arnold, Ch. Besse et al. // Commun. in Comput. Physics. — 2008. — Vol. 4, no. 4. — P. 729-796.
7. Collins M. D. A split-step pad’e solution for the parabolic equation method // The Journal of the Acoustical Society of America. — 1993. — Vol. 93, no. 4 — P. 1736-1742.