# Spring 2016

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.

### Date: March 11, 2016

## Speaker: Ivan Christov

School of Mechanical Engineering,

Purdue University

#### Title: "**Multiple-Scale Asymptotics of Plane Waves in Media with Variable Phase Speed**"

**Abstract:**

The mathematical problem of linear wave propagation in a medium with slowly spatially-varying phase speed is examined by means of the multiple-scale perturbation technique. Specifically, two canonical problems are considered: (a) the propagation of electromagnetic waves in a gravitational field or through a photonic material, and (b) the propagation of infinitesimal mechanical waves emitted from a source with variable velocity and arriving at a stationary observer. In both cases, a governing second-order wave equation with slowly spatially-varying coefficients can be derived. A multiple-scale asymptotic analysis of such wave propagation reveals novel features, while capturing the cumulative effects on a plane wave's frequency as it traverses the medium with (slowly) varying phase speed. In particular, it is shown that, unlike the classical Doppler effect, frequency is now a function of time (i.e., there is the possibility of frequency modulation) and, additionally, amplitude modulation of the signal is also possible. As a consequence, our closed-form leading-order (with respect to a proper small parameter) solutions make rigorous a heuristic approach in some of the physics literature of inserting a time-varying speed directly into the classical Doppler formula. The mathematical results will be illustrated for some representative electromagnetic and mechanical wave examples.