April 20, 2017

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By Nicolaas Du Plessis

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Control volume intersecting surface of discontinuity of vector eld  used for deriving generalized divergence theorem. 96),we can also write as Z r 1  dx = Z @ n dS 0 Z Sk 1n0 d S (3:98) where 1n0 = 1 1 n0 and Sk is the part of the surface k = 0 enclosed in region . (See g. ) The divergence theorem is used in deriving conservation laws in uid mechanics and physics in di erential form. 97), implies that such conservation laws are valid when all the derivatives are interpreted as generaliz ed derivatives.

Lowson, M. : The Sound Field for Singularities in Motion. Aug. 1965, pp. 559{572. Proc. R. Soc. London, ser. A, vol. 286, no. 1407, 38. Morgans, W. : The Kirchho Formula Extended to a Moving Surface. Philos. , vol. 9, 1930, pp. 141{161. 39. ; and Myers, M. : Extension of Kirchho 's Formula to Radiation From Moving Surfaces. J. , vol. 123, no. 3, 1988, pp. 451{460. 40. Hawkings, D. " J. , vol. 133, no. 1, 1989, p. 189. Response by F. Farassat and M. K. Myers (same journal and page). 41. Ffowcs Williams, J.

62b) suitable for supersonic panel motion, we write pi j @ @t   @v n ~) (f ) = 0 ~ h(~f ) (f ) 0 vnh(f @t ~ 0 0 v2nh(f~ ) 0(f ) ~ 0 0 vn v  (~f )(f ) (4:63) where v is the velocity of the edge in the direction of the geodesic normal. 62b) such that h @ 0 @x i  i  pn i h(f~ ) (f ) = 0r 1 pnh(~ f ) (f ) ~ = 0ph(f~ ) 0(f )+ 2 pHf h(f~ ) (f ) ~ (4:64) where Hf is the local mean curvature of the surface f = 0. 23a{f). (See refs. 4. Identi cation of shock noise source strength. 61) is known as the quadrupole source.

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