English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Solutions to the wave equation with constant coefficients in $\mathbb{R}^d$can be represented explicitly in Fourier space. We investigate a reconstructionformula, which provides an approximation of solutions $u(.,t)$ to initial data$u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given$u_0$, initial data for a profile equation are extracted. 2) A profileevolution equation determines the shape of the profile at time $\tau =\varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile toa function on $\mathbb{R}^d$. The sketched construction simplifies the waveequation, since only a one-dimensional problem in an $O(1)$ time span has to besolved. We prove that the construction provides a good approximation to thewave evolution operator for times $t$ of order $\varepsilon^{-2}$.

Representation of solutions to wave equations with profile functions