Understanding of wave phenomena underlies many scientific and technological advancements, since waves can transmit information without loss over distances of many wavelengths and for long times. While free wave propagation is well-understood in a background medium (or vacuum), this work concerns the problem of wavefield (e.g. acoustic or electromagnetic) scattering in the presence of a bounded obstacle, viewed from both analytical and computational perspectives. We outline a hybrid frequency-time computational method that promises efficient [O(1) cost to sample the scattered solution at any time] and high-order dispersionless long-time transient solutions to the time-dependent obstacle scattering problem. It is embarrassingly parallel by design and its effectiveness is demonstrated with comparison to other popular methods that also utilize boundary integral equation techniques (convolution quadrature and time-stepping).
Certain aspects of the method motivate analysis of temporal decay of wave solutions (including in scenarios where the obstacle “traps” waves), a classical question that has connections with the well-known Lax-Phillips scattering theory. We develop (computationally amenable) “domain-of-dependence” bounds on solutions to wave-scattering problems and establish rapid decay estimates of the solutions to scattering problems using only Helmholtz resolvent estimates on the real frequency axis (which are known for a variety of obstacle classes), including for geometries that lead to wave trapping and which have previously posed as barriers to proving rapid decay.
