# Time-dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half-plane

## TL;DR

This paper demonstrates that the classical Sommerfeld diffraction solution for a half-plane is the limiting amplitude of a unique, time-dependent diffraction problem, established through a novel approach avoiding initial radiation conditions.

## Contribution

It introduces a new proof of the Sommerfeld solution's uniqueness by linking it to a time-dependent problem and employing Sobolev space techniques without initial radiation assumptions.

## Key findings

- The Sommerfeld solution is the limiting amplitude of a unique time-dependent solution.
- Uniqueness is proved via reduction to a stationary problem with complex wavenumber.
- The approach naturally derives radiation and regularity conditions.

## Abstract

We consider the Sommerfeld problem of diffraction by an opaque half-plane with a real wavenumber interpreting it as the limiting case, as time tends to infinity, of the corresponding time-dependent diffraction problem. We prove that the Sommerfeld formula for the solution is the limiting amplitude of the solution of this time-dependent problem which belongs to a certain functional class and is unique in it. For the proof of uniqueness of solution to the time-dependent problem we reduce it, after the Fourier-Laplace transform in $t$, to a stationary diffraction problem with a complex wavenumber. This permits us to use the proof of uniqueness in the Sobolev space $H^1$. Thus we avoid imposing the radiation and regularity conditions on the edge from the beginning and instead obtain it in a natural way.