Emendations to a proof in the general three - dimensional theory of oscilla , ting sources of waves

l'rlnted ln (:reat Bratam Emendations to a proof in the general three-dimensional theory of oscilla,ting sources of waves Asymptotic forms far from the source region of waves generated by oscillating sources in a linear homogeneous anisotropic system were derived by a method of proof that requires emendation although the final conclusions remain unchanged. An intermediate asymptotic result. in the form of an integral over that part S+of the whole real wavenumber surface S on which a certain inequality (related to the radiation condition) is satisfied, needs modification as described in $ 2 ; but, as shown in $3, it is the modified form that is correctly estimated as in the final conclusions. Thus the proof is given two necessary emendations that cancel out. *4 careful analysis in $ 4 of why they cancel shows that the original intermediate result regains validity if 8+,besides including that part of the real wavenumber surface S on which the inequality c?w/c?kl > 0 is satisfied (where w is frequency and k, the component of wavenumber in the direction chosen for wave estimation), is considered as being continued on the complex wavenumber surface S, beyond the curve C on which c?w/ak, = 0, in the negative pure-imaginary kl-direction. This change is required to ensure the proper application of Cauchy's theorem. Furthermore, the removal of any discontinuity a t C prevents the appearance of an additional asymptotic term that would be unavoidably associated with such a singularity. I a m grateful to Professor V. A. Borovikov for stimulating me to rnake these necessary clarifications. The general theory of oscillating sources of waves for two-or three-dimensional systems governed by linear partial differential equations with constant coefficients is set out on pp. 361-369 of U7aveCs i7zfluids (Lighthill 1978), where the proof of the results follows a method proposed earlier by Lighthill (1960). In the present paper, although the correctness of the results themselves is confirmed, the proof is given certain necessary emendations. The need for these emendations was brought to my attention when Professor V. A. Borovikov of the C.S.S.R. Academy of Sciences lnstitute for Problems in Mechanics (Rloscow) wrote to me and showed by examples (see Appendix for one of these) how a certain intermediate result derived in the course of the proof is incorrect This intermediate result is equation (284) on p. 365 of U7aves influids, where it is used as an approximation to the Fourier …