Emendations to a proof in the general three-dimensional theory of oscillating sources of waves
Michael James Lighthill · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · 1990
Abstract 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. A careful analysis in §4 of why they cancel shows that the original intermediate result regains validity if S+, besides including that part of the real wavenumber surface S on which the inequality ∂ω/∂k1 > 0 is satisfied (where ω is frequency and k1 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 ∂ω/∂k1 = 0, in the negative pure-imaginary k1-direction. This change is required to ensure the proper application of Cauchy’s theorem. Furthermore, the removal of any discontinuity at C prevents the appearance of an additional asymptotic term that would be unavoidably associated with such a singularity. I am grateful to Professor V. A. Borovikov for stimulating me to make these necessary clarifications.