The Bell System Technical Journal 



October, 1928 



The Practical Application of the Fourier Integral ^ 



By GEORGE A. CAMPBELL 



Abstract: The growing practical importance of transients and other 

 non-periodic phenomena makes it desirable to simplify the application of 

 the Fourier integral in particular problems of this kind and to extend 

 the range of problems which can be solved in closed form by this method. 

 Unless the physicist or technician is in a position to evaluate the definite 

 integrals which occur, by mechanical means, he is usually entirely de- 

 pendent upon the results obtained by the professional mathematician. 

 To facilitate the use of the known closed form evaluations of Fourier 

 integrals many of them have been compiled and tabulated in Table I. They 

 are presented, however, not as definite integrals but as paired functions, one 

 function being the coefficient for the cisoidal oscillation (or complex expo- 

 nential) and the other function the reciprocally related coefficient for the 

 unit impulse. This arrangement simplifies the table and promises to be 

 most convenient in practical applications, since it is the coefficients of which 

 immediate use is made, just as in the case of the Fourier series. Applica- 

 tions of the tabulated coefficient pairs to 85 transient problems are given, 

 together with all necessary details, in Table II. 



Introduction 



THE Fourier integral and the Fourier series are alternative expres- 

 sions of the Fourier theorem, the series being a limiting case of 

 the integral and vice versa. Usually the theorem is approached 

 from the side of the series, but there are also advantages in the approach 

 from the integral side, which is the method followed in this paper. 

 The generality and importance of the theorem is well expressed by 

 Kelvin and Tait who said: "... Fourier's Theorem, which is not 

 only one of the most beautiful results of modern analysis, but may 

 be said to furnish an indispensable instrument in the treatment of 

 nearly every recondite question in modern physics. To mention only 

 sonorous vibrations, the propagation of electric signals along a tele- 

 graph wire, and the conduction of heat by the earth's crust, as subjects 

 in their generality intractable without it, is to give but a feeble idea 

 of its importance." For any real understanding of the theorem it is 

 necessary to appreciate why it is one of the most beautiful mathe- 

 matical results and why it furnishes an indispensable instrument in 

 physics. 



The Fourier integral is a most beautiful mathematical result because 

 of the economy of means employed in obtaining a most general result. 



1 Presented September 13, 1927, in preliminary form at the International Congress 

 of Telegraphy and Telephony in Commemoration of Volta. 

 41 639 



