Working with the wave equation in aeroacoustics- The pleasures of generalized functions

F. Farassat, Kenneth Steven Brentner, Mark H. Dunn

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The theme of this paper is the applications of generalized function (GF) theory to the wave equation in aeroacoustics. We start with a tutorial on GFs with particular emphasis on viewing functions as continuous linear functionals. We next define operations on GFs. The operation of interest to us in this paper is generalized differentiation. We give many applications of generalized differentiation, particularly for the wave equation. We discuss the use of GFs in finding Green's function and some subtleties that only GF theory can clarify without ambiguities. We show how the knowledge of the Green's function of an operator L in a given domain D can allow us to solve a whole range of problems with operator L for domains situated within D by the imbedding method. We will show how we can use the imbedding method to find the Kirchhoff formulas for stationary and moving surfaces with ease and elegance without the use of the four-dimensional Green's theorem, which is commonly done. Other subjects covered are why the derivatives in conservation laws should be viewed as generalized derivatives and what are the consequences of doing this. In particular we show how we can imbed a problem in a larger domain for the identical differential equation for which the Green's function is known. The primary purpose of this paper is to convince the readers that GF theory is absolutely essential in aeroacoustics because of its powerful operational properties. Furthermore, learning the subject and using it can be fun.

Original languageEnglish (US)
Title of host publication13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)
StatePublished - Dec 1 2007
Event13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) - Rome, Italy
Duration: May 21 2007May 23 2007

Publication series

Name13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)

Other

Other13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)
CountryItaly
CityRome
Period5/21/075/23/07

Fingerprint

Aeroacoustics
Wave equations
Green's function
Derivatives
Mathematical operators
Conservation
Differential equations

All Science Journal Classification (ASJC) codes

  • Aerospace Engineering
  • Mechanical Engineering

Cite this

Farassat, F., Brentner, K. S., & Dunn, M. H. (2007). Working with the wave equation in aeroacoustics- The pleasures of generalized functions. In 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) (13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)).
Farassat, F. ; Brentner, Kenneth Steven ; Dunn, Mark H. / Working with the wave equation in aeroacoustics- The pleasures of generalized functions. 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference). 2007. (13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)).
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Farassat, F, Brentner, KS & Dunn, MH 2007, Working with the wave equation in aeroacoustics- The pleasures of generalized functions. in 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference). 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), Rome, Italy, 5/21/07.

Working with the wave equation in aeroacoustics- The pleasures of generalized functions. / Farassat, F.; Brentner, Kenneth Steven; Dunn, Mark H.

13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference). 2007. (13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Farassat F, Brentner KS, Dunn MH. Working with the wave equation in aeroacoustics- The pleasures of generalized functions. In 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference). 2007. (13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)).