General scaling method for electromagnetic fields with application to a matching problem
1.For general principle see, e.g., L. D. Landau and E. M. Lifshitz, Fluid Dynamics (Pergamon, New York, 1959), Secs. 19, 53, 118, 119;
1.also, Mechanics (Pergamon, New York, 1960), Sec. 10;
1.for an earlier example of such transforms, see E. J. Routh, Proc. Lond. Math. Soc. 12, 73 (1881).
2.For the theory of conformal mapped waveguides see, e.g., F. E. Borgnis and C. H. Papas, “Electromagnetic Waveguides and Resonators,” in Handbuch der Physik (Springer‐Verlag, Berlin, 1958), 16, 358;
2.for some examples see F. J. Tischer, Proc. IEEE 51, 1050 (1963);
2.F. J. Tischer, 53, 168 (1965); , Proc. IEEE
2.also J. A. Stratton, Electromagnetic Theory (McGraw‐Hill, New York, 1941), p. 217;
2.also P. Krasnooshkin, J. Phys. USSR 10, 434 (1946);
2.for an approach using invariance groups in differential forms, see B. K. Harrison and F. B. Estabrook, J. Math. Phys. 12, 653 (1971);
2.for a frequency scaling of reflection, see J. H. Davis and J. R. Cogdell, IEEE Trans. Antennas Propag. 19, 58 (1971);
2.for a scaling for reducing constantly moving uniform simple media, see R. J. Pogorzelski, IEEE Trans. Antennas Propag. 19, 455 (1971).
3.For most recent treatments on scaling see C. E. Baum, EMP Sensor and Simulation Notes DASA 32, 1800 (1967),
3.and Ph.D. thesis, Caltech Antenna Lab. Report 47, California Institute of Technology (1968).
3.For an example of tapering the dielectric to suit propagation, see P. L. Uslenghi, IEEE Trans. Antennas Propag. 17, 644 (1969).
4.See any text on relativistic electrodynamics, e.g., V. Fock, The Theory of Space, Time and Gravitation (Pergamon, New York, 1964), Sec. 24.
5.Notice that the signature (+−−−) is used therefore for special relativity flat space‐time Also geometrized MKS unit with for vacuum is used for simple formalisms. For retrieval to MKS see, e.g., attached table in T. C. Mo, Radio Sci. 6, 673 (1971). For any applications in special relativistic EM theory, only insert appropriate powers of in the final answer to fix dimensions right.
6.T. C. Mo, J. Math. Phys. 11, 2589 (1970), Sec. 4. Also for 3‐vectors, etc.
7.See Ref. 6, Sec. 2; and any textbook with tensor calculus, e.g., J. L. Synge, General Relativity (Interscience, New York, 1960), Sec. 3.
8.Notice that the of this scaled current‐density is the contravariant 4‐vector component in the frame of (1).
9.For both P and to be really physical, care must be taken into account to make have dimensions of length such that are dimensionless pure numbers.
10.Here we use in the diagonal instead of the conventional notation just to avoid confusion with the scaled magnetic field h.
11.Ref. 3, Antenna Report 47, p. 76.
12.Notice that for geometry (46), is either a sphere or a plane; see Ref. 11, p. 83;
12.and L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Dover, New York, 1960), p. 449.
13.The ρ in cylindrical () coordinates from here on should not cause confusion with the previous charge density ρ.
14.P. Moon and D. E. Spencer, Field Theory Handbook (Springer‐Verlag, Berlin, 1961), p. 112.
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