Appendix A Mathematical appendix A.1 The fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equation One-dimensional case Let f be a function of a single variable x. The Fourier transform of f(r)is the function F(k) defined by the integral Flf())=F()=f(r)e-jkx dx Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f(x) Not every function has a outer tl ansform. The existence of(A 1)can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: /_oo If(x)ldx <oo 2. f has no infinite discontinuities 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval(a, b) While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell 22 mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse F{F(k)}=f(x)= (A.2) 0 2001 by CRC Press LLCAppendix A Mathematical appendix A.1 The Fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equations. One-dimensional case Let f be a function of a single variable x. The Fourier transform of f (x) is the function F(k) defined by the integral F{ f (x)} = F(k) = ∞ −∞ f (x)e− jkx dx. (A.1) Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f (x). Not every function has a Fourier transform. The existence of (A.1) can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: ∞ −∞ | f (x)| dx < ∞; 2. f has no infinite discontinuities; 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval (a, b). While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell [22]: a good mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse transform: F−1 {F(k)} = f (x) = 1 2π ∞ −∞ F(k) e jkx dk. (A.2)