Geodesic
Plane waves, Batmen's solutions and sources at infinity
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 23-33 
A. S. Blagovestchensky. Plane waves, Batmen's solutions and sources at infinity. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 23-33. http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a3/
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     title = {Plane waves, {Batmen's} solutions and sources at infinity},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a3/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

For threedimensional 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 obtained by the application of Kelvin–Bateman transformation to a plane wave) is the solution to wave equation everywhere in $\mathbb R^4$.

[1] A. S. Blagoveshchensky, “On wave fields generated by the sources disposed in the infinity”, J. Inv. Ill-posed Problems, 16 (2008), 825–835 | DOI | MR | Zbl

[2] V. I. Smirnov, Kurs vysshei matematiki, v. IV, ch. 1, Nauka, M., 1981

[3] H. Bateman, “The conformal transformations in four dimensions and their applications to geometrical optics”, Proc. London Math. Soc., 7 (1909), 70–89 | DOI | MR | Zbl

[4] R. Kurant, Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[5] L. Khermander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. I, Mir, M., 1986