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The Quarterly Journal of Mechanics and Applied Mathematics 1967 20(1):107-125; doi:10.1093/qjmam/20.1.107
© 1967 by Oxford University Press
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RADIATION OF SHORT WAVES BY A CONVEX CYLINDER

F. G. LEPPINGTON

( Department of Mathematics, Imperial College London )

The problem under consideration is that of scalar radiation of sound waves by a smooth convex cylinder pulsating with prescribed normal velocity V (s)exp (–i{omega}t), where s denotes arc length of the curve C bounding the cylinder cross -section, and {omega} is the radian frequency. For {omega} large, a formal expansion for the velocity potential ø exp (–i{omega}t) can be constructed by the methods of ray theory. The present work is to verify that the leading term of this expansion is the leading term of tho asymptotic expansion for ø. The potential at a field point P outside C is given by the formula


where is the Hankel function of the first kind and zero order, n denotes the outward normal from C, r is the distance from P, k = {omega}/c and c is the wave speed. Ursell has calculated the leading term for the potential ø(s; k) on C, and the functions V and are known.

Kelvin's priniciple of stationery phase is slightly modified to evaluate ø(P; k) asympototically for large k, and is valid provided ø(s; k) is sufficiently smooth: sufficient conditions are established, the leading term of the formal expansion.


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