Skip to main content

Research Repository

Advanced Search

Introducing pre-impact air-cushioning effects into the Wagner model of impact theory

Moore, Matthew R.



In this analysis, we consider the effects of non-quiescent initial conditions driven by pre-impact air–water interactions on the classical Wagner model of impact theory. We consider the problem of a rigid, solid impactor moving vertically towards a liquid pool. Prior to impact, viscous forces in the air act to deform the liquid free surface, inducing a flow in the pool. These interactions are then incorporated as initial conditions in the post-impact analysis. We derive expressions for the size of the effective contact set, the leading-order pressure and force on the impactor, and the speed and thickness of the jet at its base. In all cases, we show that the effect of the pre-impact behaviour is to cushion the impactor, reducing the size of the effective contact set and, hence, the force on the impactor. Small- and large-time asymptotic solutions are derived for general power-law impactors, and we show that the effects of the air die away as the impact progresses, so that we approach the classical Wagner solution.


Moore, M. R. (2021). Introducing pre-impact air-cushioning effects into the Wagner model of impact theory. Journal of engineering mathematics, 129(1),

Journal Article Type Article
Acceptance Date Apr 24, 2021
Online Publication Date Jun 22, 2021
Publication Date 2021-08
Deposit Date Nov 17, 2021
Publicly Available Date Nov 22, 2021
Journal Journal of Engineering Mathematics
Print ISSN 0022-0833
Electronic ISSN 1573-2703
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 129
Issue 1
Article Number 6
Keywords Air cushioning; Ship slamming; Water entry
Public URL


Published article (761 Kb)

Publisher Licence URL

Copyright Statement
© The Author(s) 2021.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

You might also like

Downloadable Citations