The Natural Period of Linear Conductors 



By C. R. ENGLUND 



Synopsis: This paper describes the experimental determination of the 

 frequency of free electrical oscillation of straight rods and circular loops. 

 The results agree more closely with the formula of Abraham than with that 

 of MacDonald. For three rods whose lengths were 300 cm., 250 cm. and 

 227.1 cm., the ratio of wave length at resonance to rod length had the values 

 2.11, 2.13 and 2.13, respectively. Measurements taken upon 250 cm. rods 

 bent into circular arcs of different radii gave values of the ratio of resonant 

 wave length to arc length which passed through a minimum value and 

 were virtually independent of the radius of the arc over a wide range, 

 deviating markedly only at the extreme value of minimum radius possible 

 and infinite radius. The extreme measured range of the ratio was 2.05 to 

 2.166. The wave lengths were measured upon a pair of Lecher wires and a 

 very satisfactory meter for the rapid comparison of waves of short length 

 was found to be a quarter wave length Lecher frame. This frame showed 

 a constant end correction so that X = 4:{d + 3.1), d being the length of the 

 parallel rods. 



IN 1898 Abraham ^ calculated the free period of an extended but 

 relatively narrow metallic ellipsoid of revolution when excited by 

 an electrical impulse. To a good approximation the fundamental 

 natural period found was related to the major axis length by the 

 expression X// = 2. Obviously a rectilinear conductor of circular 

 cross-section cannot differ markedly from such an ellipsoid and 

 Abraham concluded that the equation X// — 2 was also valid for this. 



In 1902 Macdonald ^ arrived, by a theoretical deduction quite 

 different from that of Abraham, at the expression X// = 2.53 for the 

 fundamental free period of a linear conductor. Moreover Macdonald 

 assigned the same value to the linear conductor when bent into a 

 nearly closed circle. In the next twelve years a variety of papers 

 were published ' giving results which were aimed at clearing up this 

 discrepancy without however definitely settling the matter one way 

 or the other. The subject has in recent years become of interest 

 again following the development of the short wave vacuum tube 

 oscillator and the conjoint use of rectilinear conductors as radiators 

 (or "reflectors") — particularly in grids of parabolic form. 



Since the universal method of measuring wave length is that of 

 determining the nodal distances for standing waves on parallel con- 



1 Abraham, Ann. der Phy., 66, 435, 1898. 



2 Macdonald, "Electric Waves," pp. 111-112. 

 * See Bibliography at end. 



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