On the analyticity of waveforms in complexified relatively undistorted progressing waves

Abstract

We consider two classes of exact solutions of the wave equation, which play an important role in the linear theory of localized wave propagation. These are the complexified spherical wave, known as the “complex source wavefield”, and the complexified Bateman solution. Both involve an arbitrary analytic function of a complex argument, known as the waveform. We explore a possibility of considering non-analytical waveforms and establish that in the both cases the analyticity is unavoidable.

Keywords:

• The Bateman solution
• complex source
• localized waves
• relatively undistorted waves
• wave equation

Acknowledgements

We are indebted to M. I. Belishev and V. V. Sukhanov for helpful discussions and to the anonymous reviewer for suggestions made for improving the paper.

Notes

No potential conflict of interest was reported by the authors.

1 Analyzing the age-old discussion of non-causality of a particular Bateman-type solution, see [24], one can observe that the analyticity of the waveform played there an important role, though not explicitly underlined. The importance of analiticity of the waveform of the complexified Bateman solution was clearly recognized in [16].

2 In the pioneering papers [8,9] the time-harmonic regime where , , was considered. The relatively undistorted solution (9(9) ) was considered first (in a somewhat different but equivalent form) in [11]. For a review of further development, see, e.g. [10].

3 A different approach to construction of localized solutions of this kind, based on integral transforms, is described in [25].

Additional information

Funding

This work was supported by the Russian foundation for Basic Research [the grant 17-01-00529].