Mathematical study of sound propagation in complex materials

Tuesday 11 June 2024, 12:30 pm
PhD candidate
M. Meliani MSc.
Promotor(s)
prof. dr. G.J. Lord
Co-promotor(s)
dr. V. Nikolic
Location
Aula

This PhD thesis investigates non-local partial differential equations describing acoustic waves. Non-locality refers to the fact that the future behaviour of the acoustic field cannot be predicted from its current state alone, but also requires information about its entire history. The equations studied in this research are motivated by the propagation of sound in complex materials such as biological tissue. The main emphasis lies on understanding the solvability and asymptotic behaviour of these equations, with non-locality playing a crucial role. This research provides a framework for studying and generalising results from local wave equations to the context of non-local locomotion. The construction of solutions requires studying suitable functional spaces and their compactness properties, which we discuss rigorously. An asymptotic analysis using energy methods also shows that in certain cases, local acoustics can still provide a valid approximation, even in the presence of non-local effects. We also establish the long-term behaviour of the acoustic field of some non-local wave equations.

Mostafa Meliani was born in Casablanca, Morocco on October 24th 1994. Since a very young age he has shown a strong penchant for mathematics and physics. After obtaining his high school diploma in Mathematical Sciences, he enrolled for ``Classes Prepas", a two-year advanced undergraduate coursework in Mathematics and Physics. After finishing the coursework in 2014, he joined ISAE-Supaero to study aerospace engineering in Toulouse, France. During his time in France, particularly through internships at the research centre IRT-Saint Exup\'ery and at the University of Michigan, he discovered a passion for research. He graduated with a master's degree from ISAE-Supaero in 2018 with a thesis on optimization for aircraft design under the supervision of Dr. Nathalie Bartoli, Prof. Joseph Morlier, and Prof. Joaquim Martins.
After a short experience at Sweden’s KTH Royal Institute of Technology in 2019, he has been since August 2020 a PhD student at Radboud Universty under the supervision of Prof. Gabriel Lord and Dr. Vanja Nikolić.