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1 - Green's Functions

from 1 - Time-Harmonic Waves

Published online by Cambridge University Press:  14 October 2009

N. Kuznetsov
Affiliation:
Russian Academy of Sciences, Moscow
V. Maz'ya
Affiliation:
Linköpings Universitet, Sweden
B. Vainberg
Affiliation:
University of North Carolina, Charlotte
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Summary

The simplest “obstacle” to be placed into water is a point source. The corresponding velocity potential (up to a time-periodic factor) is usually referred to as the Green's function. This notion is crucial for the theory we are going to present in this book, since a wide class of time-harmonic velocity potentials (in particular, solutions to the water-wave problem) admit representations based on Green's function (see Section 1.3).

Potentials constructed by using Green's functions form the basis for such different topics as proving solvability theorems (see Chapters 2 and 3) and constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems given in Chapter 4).

The plan of this chapter is as follows. Beginning with Green's functions of point sources in water of infinite (Subsection 1.1.1) and finite (Subsection 1.1.2) depths, we proceed with straight line sources and ring sources (Section 1.2) arising in two-dimensional problems and problems with axial symmetry, respectively. Green's representation of velocity potentials and related questions are given in Section 1.3. Bibliographical notes (Section 1.4) contain references to original papers treating the material of this chapter as well as other related works.

Three-Dimensional Problems of Point Sources

Point Source in Deep Water

In the present subsection, we consider in detail Green's function describing the point source in deep water. In Subsection 1.1.1.1, we define it as a solution to the water-wave problem having Dirac's measure as the right-hand-side term in the equation.

Type
Chapter
Information
Linear Water Waves
A Mathematical Approach
, pp. 21 - 49
Publisher: Cambridge University Press
Print publication year: 2002

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  • Green's Functions
  • N. Kuznetsov, Russian Academy of Sciences, Moscow, V. Maz'ya, Linköpings Universitet, Sweden, B. Vainberg, University of North Carolina, Charlotte
  • Book: Linear Water Waves
  • Online publication: 14 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546778.003
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  • Green's Functions
  • N. Kuznetsov, Russian Academy of Sciences, Moscow, V. Maz'ya, Linköpings Universitet, Sweden, B. Vainberg, University of North Carolina, Charlotte
  • Book: Linear Water Waves
  • Online publication: 14 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546778.003
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.

  • Green's Functions
  • N. Kuznetsov, Russian Academy of Sciences, Moscow, V. Maz'ya, Linköpings Universitet, Sweden, B. Vainberg, University of North Carolina, Charlotte
  • Book: Linear Water Waves
  • Online publication: 14 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546778.003
Available formats
×