## Summary

The problem of scattering of an arbitrarily directed scalar incident plane wave by a fundamental surfacE, of orthogonality with mixed linear boundary conditions is solved for arbitrary separable coordinate system. From the Fredholm integral equation of the second kind, derived from the Helmholtz solution, a simple linear relationship is found between the unknown coefficients in the expansions of the wave potential function and its normal derivative on the fundamental surface of orthogonality, in terms of the eigenfunctions of the separable coordinate system. By applying the mixed linear boundary conditions on the fundamental surface of orthogonality another set of simple linear relationships between the unknown coefficients is found. From the two sets of linear equations, the unknown coefficients may be determined and the scattered field calculated. The above method may be applied to both Laplace and Helmholtz equations. Since we require for the solution only the free space Green’s function expansion, the problem of finding explicit representations of the various specialized Green’s functions is avoided.

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

## References

- 1)
Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, p. 678, McGrawHill, New York 1953.

- 2)
Sommerfeld, A., Partial Differential Equations in Physics, p. 187, Academic Press, 1949.

- 3)
Bergman, S. and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, 1953.

- 4)
Morse, P. M. and H. Feshbach, loco cit., pp. 804–806.

- 5)
Baker, B. B. and E. T. Copson, The Mathematical Theory of Huygens’ Principle, pp. 23–28, Oxford, 1950.

- 6)
Strutt, J. W., (Lord Rayleigh), Theory of Sound, Vol. II pp. 143–148, Dover, 1945.

- 7)
Morse, P. M. and H. Feshbach, loc. cit., p. 706.

- 8)
Sommerfeld, A., loco cit., pp. 188–193.

- 9)
Kellogg, O. D., Potential Theory, pp. 228–249, F. Unger, 1929.

- 10)
Borgnis, F. E. and C. D. Papas, Randwertprobleme der Mikrowellenphysik pp. 17–19, Springer-Verlag, Berlin, 1955.

- 11)
Franz, W., Z. Naturforsch.

**9J**(1954) 705. - 12)
Felsen, L. B., J. Appl. Phys.

**26**(1955) 138. - 13)
Flammer, C., Spheroidal Wave Functions, p. 46, Stanford, 1957.

- 14)
Bouwkamp, C. J., Diffraction Theory, Physical Society, Vol. XVII, 1954.

- 15)
Schwinger, J., Phys. Rev. (1947) 742.

- 16)
Sollfrey, W., The Variational Solution of Scattering Problems, Research Report E.M. 11, New York University, 1949.

- 17)
Mentzer, J., Scattering and Diffraction of Radio Waves, p. 45, Pergamon Press, 1955.

- 18)
Sleator, F. B., J. Math. Phys.

**39**(1960) 105. - 19)
Jones, D. S., A Critique of the Variational Method of Scattering Problems, I.R.E. Trans. PGAP, July 1956 issue: Symposium on Electromagnetic Wave Theory, pp. 297–301.

- 20)
Bergman, S. and M. Schiffer, loc. cit., pp. 258 ff.

- 21)
Morse, P. M. and H. Feshbach, loc. cit. pp. 914–916, 949–960.

- 22)
Morse, P. M. and H. Feshbach, loc. cit., pp. 828–832.

- 23)
Morse, P. M. and H. Feshbach, loc. cit., pp. 655–666.

- 24)
Courant, R. and D. Hilbert, Methods of Mathematical Physics, p. 277, Interscience Publishers, 1953.

- 25)
Magnus, W. and F. Oberhettinger, Special Functions of Mathematical Physics, pp. 144–145, Chelsea, 1949.

- 26)
Jeffreys, H. and B. S. Jeffreys, Methods of Mathematical Physics, pp. 492–493, Cambridge, 1956.

## Author information

### Affiliations

### Corresponding author

Correspondence to H. Unz.

## Additional information

The research reported in this paper was supported in part by Grant Number AF-AFOSR-62-265 from the Air Force Office of Scientific Research and in part by the University of Puerto Rico, Rio Piedras, Puerto Rico.

## Rights and permissions

## About this article

### Cite this article

Unz, H., Sleator, F.B. Scattering of a plane scalar wave by fundamental surfaces with mixed boundary conditions.
*Appl. sci. Res.* **10, **344 (1963). https://doi.org/10.1007/BF03177940

Received:

### Keywords

- Helmholtz Equation
- Unknown Coefficient
- Scattered Field
- Fredholm Integral Equation
- Orthogonality Property