Geodesic
Simple solutions of the wave equation, singular at a~ranning point, based on the complexified Bateman solution
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 45, Tome 438 (2015), pp. 73-82

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We suggest simple solutions of the homogeneous wave equation with constant propagation speed having a power-like singularity in a moving spatial point. The construction is are based on the complexified Bateman-type solution. Example of such a solution showing exponential decay with distance from the singular point is presented. 
@article{ZNSL_2015_438_a4,
     author = {A. S. Blagovestchenskii and A. P. Kiselev and A. M. Tagirdzhanov},
     title = {Simple solutions of the wave equation, singular at a~ranning point, based on the complexified {Bateman} solution},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {73--82},
     publisher = {mathdoc},
     volume = {438},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a4/}
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A. S. Blagovestchenskii; A. P. Kiselev; A. M. Tagirdzhanov. Simple solutions of the wave equation, singular at a~ranning point, based on the complexified Bateman solution. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 45, Tome 438 (2015), pp. 73-82. http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a4/