A TRANSITION FUNCTION IN PROBLEMS HYDROE LASTIC INTERACTION OF V. D. Kubenko UDC 533.6.013.42 In [2,3] the problem of the diffraction of a nonstationary hydroacoustic wave by a rigid shell of rota- tion was investigated. For this case the potentiai of the reflected wave r is determined by the potential of the incident wave r using the equation t (' F ( / - - ~) 0tp~ d~. (1) . J O~, ~=o 0 Here t is the time; Y is a coordinate measured along the outward normal to the surface of the body (the sys- tem of orthogonal eurvilinear coordinates a, fi, y is such that the coordinate lines a and fl coincide with the lines of the principal curvature); F is a transition function having the form t F (t) = e-~a -- lw I exp (-- kc V t ~ -- u ~) J~ (kcu) du, (2) 0 where J1 is a Bessel function; c is the wave-propagation velocity; k = 1/2 (k i + k2), ki and k 2 are the prin- cipal curvatures; the function F {t) is obtained under the assumption that the potential r varies significantly only along y. Since the second term in Eq. (2) has an unwieldy form and is an oscillating, damped function, in [2, 3] consideration is restricted to only. the first term, or the more exact expression F (t) = e -kct COS kCt, (3) is used; this equation is constructed on the basis of the asymptotic equivalence of the function and its ap- proximation. The aim of the present article is to obtain analytically simple expressions that approximate the quan- tity F (t) with the required accuracy. In the space of Laplace-transform functions [5], the transform of the function F (t) has the form L iF (t)] 1 , (4) 1 +1/ 1+s 2 where "L [F (t)l = ~ e-'th"(t)dt; o s is the transformation parameter; "t = kct is the dimensionless time; the bar over the symbol will be dropped. To transform Eq. (4), taking into account that s ~ 0, we expand the fraction in the series I 1 1 L iF (t)l = V 1 + s ------~ 1 + s 2 +'" ' + (- 1)~ ~+~ + . . . (5) (1 + s~) T Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya" Mekhanika, Vol. 7, No. 10, pp. 131-134, October, 1971. Original article submitted February 25, 1971. 9 I974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g'est ITth Street, New York, N. Y. 1001i. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by an)" means, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. ,t electronic, I copy of this article is available from the publisher for $15.00. 1171