Class Objectives

The course aims at providing Ph.D. students with analytical tools useful in applied research on general wave phenomena. The unifying theme is that of complex analysis, of which a compact, self-contained introduction is presented. Fundamental techniques for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering from half planes and spheres. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method and the Watson transformation will be introduced.

Class Schedule

The course will be held from 19 September to 26 October 2023 in the seminar room at the second floor of the DIET department, Via Eudossiana 18, 00184 Rome, Italy, with the following schedule:

Tuesday              10:00-13:00

Thursday             10:00-13:00

In case of necessity, the lessons may also be offered at distance via Google Meet at the link: https://meet.google.com/hmw-agon-ihm

Syllabus

1.            Fundamentals of complex function theory

1.1   Elementary holomorphic functions, Cauchy-Riemann equations, elementary Riemann surfaces

1.2   Complex integration, Cauchy theorem and consequences, residue calculus

2.            Asymptotic expansions and ray optics

2.1   Introduction, asymptotic sequences, and elementary examples

2.2   The Luneburg-Kline asymptotic expansion; ray optics.

3.            Asymptotic evaluation of integrals

3.1   Integration by parts, Watson lemma, Laplace method, stationary-phase method

3.2   The method of steepest descents (saddle-point method)

4.            Applications: Time-harmonic waves in layered media

4.1   Vertical dipole above a single interface: space waves, lateral waves, plasmon waves, Zenneck waves

4.2   Vertical dipole above a grounded slab: surface waves, leaky waves

5.            Applications: Plane-wave scattering from half planes

5.1   PEC half plane: elementary solution and Wiener-Hopf approach.

5.2   Resistive half plane: Wiener-Hopf solution and uniform asymptotic evaluation of the field.

6.            Applications: Scattering from spheres

6.1   Spherical wave functions; dipole on a PEC sphere, Watson transformation, creeping waves.

6.2   Plane-wave scattering from PEC and dielectric spheres; the rainbow and the glory.

Final Examination

Discussion of a scientific paper related to the course.

Learning and teaching support materials

Class slides.