The space-time ℝ4 is compactified by adding the manifold of infinitely distant points. The problem of constructing a solution of the wave equation with the right-hand side (the source of waves), which is a generalized function with support on the manifold of infinitely distant points, is posed and solved. Strict necessary and sufficient conditions that the source must satisfy are stated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Blagoveshchensky, “On wave fields generated by sources disposed in the infinity,” J. Inv. Ill-posed Problems, 16, 1–11 (2008).
A. S. Blagoveshchensky, “Plane waves, Bateman solutions, and sources at infinity,” Zap. Nauchn. Semin. POMI, 426 (2014).
R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964).
A. S. Blagoveshchensky, “On some new problems for the wave equation,” in: Proceedings of the 5th All-Union Symposium on Diffraction and Wave Propagation, Nauka, Leningrad (1971).
R. Penrose and W. Rindler, Spinors and Space-Time. Spinor and Twistor Methods in Space-Time Geometry [Russian translation], Mir, Moscow (1988).
A. S. Blagoveshchensky, “Generalized d′Alembert operator on a compactified pseudo-Euclidean space,” Mat. Zam., 85, No. 5 (2009).
H. Bateman, “The conformal transformations in four dimensions and their applications to geometrical optics,” Proc. London Math. Soc., 7, 70–89 (1909).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 471, 2018, pp. 59–75.
Rights and permissions
About this article
Cite this article
Blagoveshchensky, A.S. On Waves Generated by Sources Localized at Infinity. J Math Sci 243, 671–681 (2019). https://doi.org/10.1007/s10958-019-04568-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04568-4