Large-Time Series Expansion of the Wave Front Length in the Euclidean Disk

Abstract

In the paper [4], the second author proves that the length $\left|S_t\right|$ of the wave front S_t at time t of a wave propagating in an Euclidean disk $\mathbb{D}$ of radius 1, starting from a source q, admits a linear asymptotics as $t\to +\infty$: $\left|S_t\right|=\lambda (q)t+\mathrm{o}(t)$ with $\lambda (q)=2\arcsin a$ and $a=d(0,q)$. We will give a more direct proof and compute the oscillating corrections to this linear asymptotics. The proof is based on the “stationary phase” approximation.

Keywords:

• Euclidean disk
• wave front
• stationary phase
• numerical experiment

Notes

1 https://www.youtube.com/channel/UCMTvpxuhYwbYBYDErSlU0EA/