Method of handling the divergences in the radiation theory of sources that move faster than their waves

Journal of Mathematical Physics

Volumn 40, Issue 9, 1999, Pages 4331-4336

  Ardavan, H ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033240761     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532970     Document Type: Article

References (12)

1. B. M. Bolotovskii and V. L. Ginzburg, Sov. Phys. Usp. 15, 184 (1972).
2. H. Ardavan, Phys. Rev. E 58, 6659 (1998).
3. J. H. Hannay, Proc. R. Soc. London, Sci. A 452, 2351 (1996).
4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part 1 (McGraw-Hill, New York, 1953).
5. J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale Univ. Press, New Haven, 1923).
6. R. F. Hoskins, Generalised Functions (Horwood, London, 1979).
7. A. P. Dowling and J. E. Ffowcs Williams, Sound and Sources of Sound (Horwood, London, 1983).
8. R. Courant and D. Hubert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962).
9. H. Ardavan, Proc. R. Soc. London, Ser. A 424, 113 (1989).
10. H. Ardavan, Phys. Rev. D 29, 207 (1984).
11. 11 The analysis in Ref. 3 is based on the assumption that the domain of integration in Eq. (7) extends over all space regardless of what the expression for the source density may be. That this is not so may be seen by writing out the explicit form of the integral in Eq. (7) for any source distribution - such as the one considered here - whose density vanishes outside a finite volume. For such source distributions, the integration in (7) does not extend over all space unless we introduce step functions that would incorporate the finiteness of their support into the mathematical expressions for their densities. The contributions of the required step functions toward the derivatives of the integral in (7), on the other hand, are the same as those of the original limits of integration.
12. R. Courant, Differential and Integral Calculus, Vol. 2 (Blackie, New York, 1967).