What does it mean for a 3D star-shaped scatterer to be small in the time domain?

TL;DR

This paper investigates the concept of smallness for 3D star-shaped scatterers in the time domain, challenging the idea of an additional temporal scale and proposing stable asymptotic models for sound-soft obstacles.

Contribution

It demonstrates that the smallness of obstacles in the time domain can be characterized without an extra temporal scale, providing stable asymptotic models for 3D star-shaped scatterers.

Findings

Asymptotic models with time-independent error are possible.

The notion of smallness in the time domain differs from the frequency domain.

Stable models are developed for sound-soft star-shaped obstacles.

Abstract

In the frequency domain wave scattering problems, obstacles can be effectively replaced by point scatterers as soon as the wavelength of the incident wave exceeds significantly their diameter. The situation is less clear in the time domain, where recent works suggest the presence of an additional temporal scale that quantifies the smallness of the obstacle. In this paper we argue that this is not necessarily the case, and that it is possible to construct asymptotic models with an error that does not deteriorate in time, at least in the case of a sound-soft scattering problem by a star-shaped obstacle in 3D.