For the three-dimensional wave equation, two equivalent statements are proved: (1) plane waves are not generated by a source at infinity; (2) Bateman’s solution (the solution that is obtained by application of the Kelvin–Bateman transformation to a plane wave) is a solution of the wave equation everywhere in ℝ4 . Bibliography: 5 titles.
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References
A. Blagoveshchensky, “On wave fields of sources disposed at infinity,” J. Inv. Ill-Posed Problems, 16, 1–11 (2008).
V. I. Smirnov, A Course in Higher Mathematics [in Russian], Vol. IV, Part 1, Nauka, Moscow (1981).
H. Bateman, “The conformal transformations in four dimensions and their applications to geometrical optics,” Proc. London Math. Soc., 7, 70–89 (1909).
R. Courant, Partial Differential Equations, New York (1962).
L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. I, Springer-Verlag, Berlin (1983).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 426, 2014, pp. 23–33.
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Blagoveshchensky, A.S. Plane Waves, Bateman’s Solutions, and Sources at Infinity. J Math Sci 214, 260–267 (2016). https://doi.org/10.1007/s10958-016-2775-4
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DOI: https://doi.org/10.1007/s10958-016-2775-4