An integral equation approach to the problem of wave propagation over an irregular surface

Author:
George A. Hufford

Journal:
Quart. Appl. Math. **9** (1952), 391-404

MSC:
Primary 78.0X

DOI:
https://doi.org/10.1090/qam/44350

MathSciNet review:
44350

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**[1]**S. A. Schelkunoff,*Electromagnetic waves*, D. van Nostrand Company, New York, Chapter 12, (1943).**[2]**M. Leontovič,*On a method of solving the problem of propagation of electromagnetic waves near the surface of the earth*, Bull. Acad. Sci. URSS. Sér. Phys. [Izvestia Akad. Nauk SSSR]**8**(1944), 16–22 (Russian). MR**0011041****[3]**M. Leontovich and V. Fock,*Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equation*, Acad. Sci. USSR. J. Phys.**10**(1946), 13–24. MR**0017662****[4]**E. Feinberg,*On the propagation of radio waves along an imperfect surface*, Acad. Sci. USSR. J. Phys.**8**(1944), 317–330. MR**0013037****[5]**G. Hufford,*On the propagation of horizontally polarized waves over irregular terrain*, Master's thesis, University of Washington, 1948.**[6]**L. Brillouin, Perturbation d'un problème de valeurs propres par déformation de la frontière, C. R. Paris**204**, 1863-1865 (1937).**[7]**Herman Feshbach,*On the perturbation of boundary conditions*, Phys. Rev. (2)**65**(1944), 307–318. MR**0010231****[8]**Nicolas Cabrera,*Perturbation par changement des conditions aux limites*, Cahiers de Physique**1948**(1948), no. nos. 31-32, 24–62 (French). MR**0024012****[9]**See J. A. Stratton,*Electromagnetic theory*, McGraw-Hill Book Company, New York, 1941, p. 165. It may be noticed that in this formulation of Green's theorem the sign on the right hand side has been reversed from that usually used. This is because we have thought it more natural here to think of the normal derivative as directed into the volume V rather than in the conventional outward direction. In this way the normal derivative is directed away from the earth, and this corresponds to the direction of the normal derivative in Eq. (2).**[10]**See J. A. Stratton,*loc. cit*., p. 486.**[11]**This process is described with more detail and with more rigour by O. D. Kellogg,*Foundations of potential theory*, Julius Springer, Berlin, 1929, pp. 160-172.**[12]**J. S. McPetrie and L. H. Ford,*Some experiments on the propagation over land of radiation of*9.2-*cm wavelength, especially on the effect of obstacles*, JIEE**93**, III A, 531-538, (1946); see especially Fig. 3.**[13]**J. C. Shelleng, C. R. Burrows, and E. B. Ferrell,*Ultra-short wave propagation*, Proc. I.R.E**21**, 427-463 (1933).**[14]**Joseph B. Keller and Herbert B. Keller,*Determination of reflected and transmitted fields by geometrical optics*, Research Rep. No. EM-13, New York University, Washington Square College, Mathematics Research Group, 1949. MR**0034232****[15]**Aurel Wintner,*Remarks on the method of stationary phases*, J. Math. Phys. Mass. Inst. Tech.**24**(1945), 127–130. MR**0014148**, https://doi.org/10.1002/sapm1945241127**[16]**Gustav Doetsch,*Theorie und Anwendung der Laplace-Transformation*, Dover Publication, N. Y., 1943 (German). MR**0009225****[17]**Karl Willy Wagner,*Operatorenrechnung nebst Anwendungen in Physik und Technik*, J. W. Edwards, Ann Arbor, Michigan, 1944 (German). MR**0012172****[18]**K. A. Norton,*The propagation of radio waves over the surface of the earth and in the upper atmosphere*, Proc. I.R.E., Part 2,**25**, 1203-1236 (1937).**[19]**See W. B. Ford,*The asymptotic developments of functions defined by Maclaurin series*, University of Michigan Press, Ann Arbor, 1936, and H. K. Hughes,*On the asymptotic expansion of entire functions defined by Maclaurin series*, Bull. Amer. Math. Soc.**50**, 425-430 (1944).**[20]**G. Doetsch,*loc. cit*., p. 142.**[21]**B. van der Pol and H. Bremmer,*The propagation of radio waves over a finitely conducting spherical earth*, Phil. Mag. (7)**25**, 817-834 (1938). The equation we refer to is their Eq. (206) which appears on p. 829.**[22]**V. Fock,*Diffraction of radio waves around the earth’s surface*, Acad. Sci. USSR. J. Phys.**9**(1945), 255–266. MR**0014332**

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DOI:
https://doi.org/10.1090/qam/44350

Article copyright:
© Copyright 1952
American Mathematical Society