Investigation of the solution of a system of partial differential equations with periodic coefficients

TL;DR

This paper analyzes wave propagation in periodic media using a method that predates modern homogenization techniques by over a century, providing historical insight into early PDE solutions.

Contribution

It presents an early analytical approach to solving PDE systems with periodic coefficients, anticipating modern homogenization methods.

Findings

Early method for PDEs with periodic coefficients

Anticipated multiple-scale homogenization techniques

Historical significance in PDE analysis

Abstract

This is an English translation of a paper by the French physicist Alfred Potier (1840-1905) that originally appeared 150 years ago [A. Potier, ``Recherches sur l'int\'egration d'un syst\`eme d'\'equations aux diff\'erentielles partielles \`a coefficients p\'eriodiques,'' Comptes Rendus de l'Association Fran\c{c}aise pour l'Avancement des Sciences (Bordeaux), vol. 1, pp. 255-272 (1872)]. The paper presents an analysis of wave propagation through a periodic medium by a method that in many ways anticipated the technique of multiple-scale homogenization by more than a century.