By F. Farassat

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Extra info for Introduction to generalized functions with applications in aeroacoustics

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Assume that  satis es the wave equation in the exterior of the surface f = 0, which is the region de ned by f > 0. Now extend  to the entire unbounded space as follows: ( (f > 0) ~(x; t) = (x; t)  (4:56) 0 (f < 0) Clearly, ~ satis es the wave equation in the unbounded space. However, ~ has discontinuities across f = 0 that appear as source terms of the wave equation. Note that the jumps in ~ and its derivatives depend on corresponding values for  because 1~ = (f = 0+) and  ~ t = @ = @ t (f = 0+).

The sphere with center at the observer x and radius c (t 0  ) with the element of the surface denoted by d ). 23a) are known as the retarded time and the collapsing sphere forms of the solution, respectively. 23b) can also be written in several forms. (See refs. ) We give two forms here. For a rigid surface f (x; t) = 0, let Mr = M 1 r^ be the local Mach number in the radiation direction. Then   Z Q(y ;  ) 4 8(x ; t) = dS (4:28) r j1 0 Mr j ret f =0 4 8(x ; t) = Note that to get this equation, the formal Green's function solution, which is Z 1 Q(y;  ) (f )(g) dy d 48(x; t ) = r (4:29) is integrated as follows.

1965. 14. Synowiec, John: Distributions: The Evolution of a Mathematical Theory. Historia Mathematica, vol. 10, 1983, pp. 149{183. 15. Lutzen, Jesper: The Prehistory of the Theory of Distributions. Springer-Verlag, 1982. 16. Dieudonne, Jean: History of Functional Analysis . Elsevier Sci. Publ. , 1981. 17. Antosik, Piotr; Mikusi nski, Jan; and Sikorski, Roman: Theory of Distributions|The Sequential Approach. Elsevier Sci. Publ. , 1973. 18. Mikusi nski, Jan: Operational Calculus|Volume I. , 1983.

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