Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T02:07:20.022Z Has data issue: false hasContentIssue false

10 - Conformal Mapping and Complex Topographies

Published online by Cambridge University Press:  05 February 2016

By
André Nachbin
Edited by
Thomas J. Bridges ,
Mark D. Groves  and
David P. Nicholls
André Nachbin
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada
Thomas J. Bridges
Affiliation:
University of Surrey
Mark D. Groves
Affiliation:
Universität des Saarlandes, Saarbrücken, Germany
David P. Nicholls
Affiliation:
University of Illinois, Chicago
Get access

Summary

Abstract

Many interesting research problems consider long water waves interacting with highly disordered bottom topographies. The topography profile can be of large amplitude, not smooth and rapidly varying. Disordered topographies do not have a well defined structure and therefore can be modelled as a random coefficient in the wave equations. A summary of results that arise through random modelling is provided. The main goal of this article is to present the Schwarz-Christoffel conformal mapping as a tool for dealing with these problems. Both from the theoretical point of view as well as regarding computational aspects that make use of the Schwarz-Christoffel Toolbox, developed by T. Driscoll.

Introduction and Mathematical Motivation

About 25 years ago we became interested in studying the effects of small scale features of a topography on long waves propagating on the water surface for large distances, such as a tsunami. Our goal was to study solitary waves over rapidly varying disordered topographies. At the time the theory for linear acoustic waves over rapidly varying (one dimensional-1D) layered media was maturing and becoming quite sophisticated [1, 2]. The theory developed by George Papanicolaou and collaborators considered linear hyperbolic systems with disordered rapidly varying coefficients as a model to understand linear pulse-shaped waves travelling in a medium with a random propagation speed. This setup is very useful for understanding the effect of uncertainty on a travelling wave, in direct and inverse problems related to the Earth's subsurface. Mathematically it called for new technology showing that random modelling produced some universal results of interest. These included effective wave propagation properties that were independent of a specific realisation. To study linear travelling waves in the presence of random multiple scattering, Papanicolaou and collaborators developed an asymptotic theory for stochastic ordinary differential equations (SODEs) regarding the transmitted and reflected signals. Limit theorems for randomly forced oscillators characterised asymptotically the expected value for the transmitted and reflected signals after the wave had propagated for large distances. Three scales are involved in this analysis: the medium's microscale ε2, the pulse's characteristic width ε (the mesoscale) and the large propagation distance O(1) (the macroscale).

Type
Chapter
Information
Lectures on the Theory of Water Waves , pp. 203 - 225
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] R., Burridge, G., Papanicolaou and B., White, Statistics for pulse reflection from a randomly layered medium, SIAM J. Appl. Math., 47 (1987), 146-168.Google Scholar
[2] R., Burridge, G., Papanicolaou, P., Sheng and B., White, Probing a random medium with a pulse, SIAM J. Appl. Math., 49 (1989), 582-607.Google Scholar
[3] J. P., Fouque, J., Garnier, G. C., Papanicolaou and K., Sølna, Wave Propagation and Time Reversal in Randomly Layered Media (Springer Verlag, 2007).Google Scholar
[4] J. P., Fouque, J., Garnier and A., Nachbin, Time reversal for dispersive waves in random media, SIAM J. Appl. Math., 64 (2004), 1810-1838.Google Scholar
[5] J. P., Fouque, J., Garnier and A., Nachbin, Shock structure due to stochastic forcing and the time reversal of nonlinear waves, Physica D, 195 (2004), 324–346.Google Scholar
[6] J. C., Muñoz Grajales and A., Nachbin, Dispersive wave attenuation due to orographic forcing, SIAM J. Appl. Math., 64 (2004), 977-1001.Google Scholar
[7] R., Burridge and L., Berlyand, The accuracy of the O'Doherty-Anstey approximation for wave propagating in highly disordered stratified media, Wave Motion, 21 (1995), 357-373.Google Scholar
[8] A., Nachbin and K., Sølna, Apparent diffusion due to topographic micro structure in shallow waters, Phys. Fluids, 15 (2003), 66-77.Google Scholar
[9] G.B., Whitham, Linear and Nonlinear Waves (John Wiley, 1974).Google Scholar
[10] J. P., Fouque and A., Nachbin, Time reversed refocusing of surface water waves, Multiscale Modell. Simul., 1, (2003) 609-629Google Scholar
[11] J.P., Fouque, J., Garnier, J.C., Muñoz Grajales and A., Nachbin, Time reversing solitary waves, Phys. Rev. Lett. 92 (2004), 094502.Google Scholar
[12] J., Garnier, J. C., Muñoz Grajales and A., Nachbin, Effective behavior of solitary waves over random topography, Multiscale Modell. Simul., 6 (2007), 995-1025.Google Scholar
[13] J., Hamilton, Differential equations for long-period gravity waves on a fluid of rapidly varying depth, J. Fluid Mech., vol. 83 (1977), 289-310.Google Scholar
[14] J. C., Muñoz Grajales and A., Nachbin, Improved Boussinesq-type equations for highly-variable depths, IMA J. Appl. Math., 71 (2006), 600-633.Google Scholar
[15] O., Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterway, Port, Coastal and Ocean Engineering, 119 (1993), 618-638.Google Scholar
[16] R.A., Kraenkel, J., Garnier and A., Nachbin, Optimal Boussinesq model for shallow-water waves interacting with a micro structure, Phys. Rev. E., vol. 76 (2007), 046311-1.Google Scholar
[17] A., Nachbin, A terrain-following Boussinesq system, SIAM J. Appl. Math., 63 (2003), 905-922.Google Scholar
[18] R.S., Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, 1997).Google Scholar
[19] M., Belzons, E., Guazzelli and O., Parodi, Gravity waves on a rough bottom: experimental evidence of one-dimensional localization, J. Fluid Mech., 186 (1988), 539-558.Google Scholar
[20] P., Devillard, F., Dunlop and B., Souillard, Localization of gravity waves on a channel with a random bottom, J. Fluid Mech., 186 (1988), 521-538.Google Scholar
[21] A., Nachbin and G.C., Papanicolaou, Water waves in shallow channels of rapidly varying depth, J. Fluid Mech., 241 (1992), 311-332.Google Scholar
[22] A., Nachbin, The localization length of randomly scattered water waves, J. Fluid Mech., 296 (1995), 353-372.Google Scholar
[23] A., Nachbin, Modelling of Water Waves in Shallow Channels (Computational Mechanics Publications, Southampton, U.K., 1993).Google Scholar
[24] R.R., Rosales and G.C., Papanicolaou, Gravity waves in a channel with a rough bottom, Studies in Appl. Math., 68 (1983), 89-102.Google Scholar
[25] T.A., Driscoll, T.A. and L., Trefethen, Schwarz-Christoffel Mapping (Cambridge University Press, Cambridge, 2002 UK).Google Scholar
[26] V., Rey, M., Belzons and E., Guazzelli, Propagation of surface gravity waves over a rectangular submerged bar, J. Fluid Mech., 235, (1992), 453-479.Google Scholar
[27] A.S., Fokas and A., Nachbin, Water waves over a variable bottom: a non-local formulation and conformal mappingsJ. Fluid Mech., 695 (2012), 288-309.Google Scholar
[28] W., Artiles and A., Nachbin, Nonlinear evolution of surface gravity waves over highly variable depth, Phys. Rev. Lett., 93, (2004), 234501.Google Scholar
[29] J. C., Muñoz Grajales and A., Nachbin, Stiff microscale forcing and solitary wave refocusing, SIAM Multiscale Model. Simul., 3 (2005), 680-705.Google Scholar
[30] J., Garnier and A., Nachbin, Eddy viscosity for gravity waves propagating over turbulent surfaces, Phys. Fluids, 18 (2006), 055101.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×