# Problems in Two Dimensions

• James B. Seaborn
Part of the Texts in Applied Mathematics book series (TAM, volume 8)

## Abstract

Transmission of electromagnetic energy may be accomplished by means of hollow metallic cylinders called wave guides. The electromagnetic waves propagate in the nonconducting region enclosed by the metal cylinder. For the fields in this region, Maxwell’s equations reduce to1
$$\nabla \cdot E = 0,$$
(4.1a)
$$\nabla \times E + \mu \frac{{\partial H}}{{\partial t}} = 0,$$
(4.1c)
$$\nabla \cdot H = 0,$$
(4.1c)
$$\nabla \times H - \varepsilon \frac{{\partial E}}{{\partial t}} = 0.$$
(4.1d)
.

## Keywords

Bessel Function Transverse Magnetic Recursion Formula Wave Guide Transverse Magnetic Mode
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
In this book we use SI units. See, for example, P. Lorrain and D. Corson, Electromagnetic Fields and Waves,W.H. Freeman and Company, San Francisco, 1970.Google Scholar
2. 3.
Here and throughout the chapter, i denotes the imaginary number. For a discussion of complex numbers see Chapter 7.Google Scholar
3. 4.
For example, see P. Lorrain and D. Corson, op. cit.,Chapter 13.Google Scholar
4. 6.
In Chapter 9 as an exercise in contour integration, we show the equivalence of the two representations of J„(x) given by Eqs. (4.17) and (4.19).Google Scholar
5. 10.
A rationale for this definition may be seen by comparing the asymptotic forms of J,,(x) and N„(x) for large values of x. See Exercises 10.8 and 10.9. For a more complete discussion of the history of these functions, see G.N. Watson, op. cit.,p. 63 if.Google Scholar
6. 11.
Another name that is often used for this function is the Neumann function. 12An example occurs in Exercise 10.12.Google Scholar
7. 17.
For a derivation, see A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua, McGraw-Hill, New York, 1980, p. 273.Google Scholar
8. 21.
Solutions to this equation are useful in constructing the connection formulas of the WKB approximation. See, for example, L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955, p. 187.Google Scholar