### 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 language | English (US) |
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Title of host publication | 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) |

State | Published - Dec 1 2007 |

Event | 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) - Rome, Italy Duration: May 21 2007 → May 23 2007 |

### Publication series

Name | 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) |
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### Other

Other | 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) |
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Country | Italy |

City | Rome |

Period | 5/21/07 → 5/23/07 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Aerospace Engineering
- Mechanical Engineering

### Cite this

*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).*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 S.; Dunn, Mark H.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Working with the wave equation in aeroacoustics- The pleasures of generalized functions

AU - Farassat, F.

AU - Brentner, Kenneth S.

AU - Dunn, Mark H.

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84884798889&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884798889&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84884798889

SN - 9781624100031

T3 - 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)

BT - 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference)

ER -