We are concerned with a 2D time harmonic wave propagationproblem in a medium including a thin slot whose thickness εis small with respect to the wavelength. In a previous article, we derivedformally an asymptotic expansion of the solution with respect to εusing the method of matched asymptotic expansions. We also proved theexistence and uniqueness of the terms of the asymptotics. In this paper,we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.

In this article, we derive a complete mathematical analysis of acoupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.

Raphe-like vestiges have been found regularly in two new taxa which presumably belong to Fragilaria, one of them livingin several lakes of the Kerguelen Islands, the other in the Yellowstone National Park, Wyoming, U.S.A. The latter onepossesses septa-like elements as integrated parts of a girdle band. Other septa-like elements can occur in different speciesof Fragilaria. They represent appendices of the valvocopula, located beneath the transapical costae. Structures like linkingspines on marginal parts of the valve face occur in an Achnanthes species. Most of these characters seem to havebeen unknown in these taxa until recently.