A Family of Solutions to the Two-Dimensional Eikonal Equation


The propagation of a wave from a point source when the velocity \(v\) in the medium is expressed as \(v = \frac {1}{\sqrt y}\) is considered. Exact solutions to the corresponding eikonal equation are obtained and numerically verified. It is shown that the solution of this equation when the point source is at the origin of coordinates is not unique.

This is a preview of subscription content, log in to check access.


  1. 1

    Smirnov, V.I., A Course of Higher Mathematics, vols. 1–5, New York: Pergamon Press, 1964.

  2. 2

    Moskalensky, E.D., A New Class of Exact Solutions to the Two-Dimensional Eikonal Equation where the Velocity in the Medium Depends on One of the Spatial Coordinates Alone, Sib. Zh. Vych. Mat., 2018, vol. 21, no. 3, pp. 259–271.

  3. 3

    Borovskikh, A.V., Two-Dimensional Eikonal Equation, Sib. Mat. Zh., 2006, vol. 47, no. 5, pp. 993–1018.

  4. 4

    Marchuk, An.G., Minimizing Errors in Numerical Calculations of Wave Rays and Tsunami Fronts, Vestnik NGU, Ser. Inform. Tekhnol., 2013, vol. 11, no. 3, pp. 27–36.

  5. 5

    Stocker, J.J., Water Waves: The Mathematical Theory with Applications, New York: Interscience, 1957.

Download references

Author information



Corresponding authors

Correspondence to An. G. Marchuk or E. D. Moskalensky.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Marchuk, A.G., Moskalensky, E.D. A Family of Solutions to the Two-Dimensional Eikonal Equation. Numer. Analys. Appl. 13, 127–135 (2020). https://doi.org/10.1134/S1995423920020044

Download citation


  • wave propagation
  • wave front
  • eikonal equation
  • point source