PROFESSOR K. PEARSON AND MISS A. LEE ON THE VIBRATIONS 



We have 



(t\ff <t\fr dr p* 

 " P ~dr dp~''^ 

 d fe-PiC-o 



Q = P? = 



r dp 



sin 



-ar)| 



= ~Ele~ p>(t ~ ttr) \ (p t a -- -I sin p. 2 (< ar) op, cos p, (t ar) I. 

 Now put pi = i) sin x. #z = "n cos X ! then if p = r sin 6, we have 



sn 



Q = 

 In our notation HERTZ finds* 



cos 



sin p t (t ar 

 rap t 



(Vii). 



Q = EZop, sin 2 { cos p, (t - ar) + sin Pt (t ~ ar) } . 



[ ra Pi J 



(** \ 

 vm). 



It is clear that the damped wave train, such as actually occurs with the Hertzian 

 oscillator, makes very considerable changes in the form of Q. Thus : 



(a.) As we might expect the damping factor e~ p>t is introduced, or rather a damping 

 factor e~>' 1 (t ~' r) where t,. is the time at which the disturbance reaches a distance r from 

 the centre of the oscillator. Clearly the factor Pl r will sensibly modify the form of 

 the lines of electric force obtained by tracing the curves 



Q = constant. 



(b.) The first term in the curled brackets is also sensibly modified both as to 

 amplitude and phase. 



The reader must not imagine that the difference between the formulae (vii) and 

 (viii) marks as soon as the vibrations are set up a great difference in the lines of 

 electric force. All we contend for is that it marks a sensible difference in the shape 

 after a few periods, and that this difference increases with the length of time and the 

 distance of the part of the field considered from the oscillator. In fact loops which 

 would remain at the end of each period finite according to formula (viii), have to 

 shrivel up into points and disappear from the field altogether. 



Suppose 2irp t /p., = '4, than tan x = '2/V, and we find x = 3 38/ 33 "' 3 , while 

 sec x = 1 '002,0244. Hence, the amplitude of the cosine term is increased by about 2 per 

 thousand, and the phase by 3 to 4. It is the factor e^" 1 ", however, which produces most 



sensible change. For ,a = -~^ - = -4/X, if X be the wave length. Now, 



(wave length) 



if X be 9'60 metres.t which was about its value for one of HERTZ'S oscillators 



* It seems better to write p, (t - ar) than p. t (ar - 1) with HERTZ, because ar must be less than /, or 

 Q = 0, ' Electric Waves,' p. 142. 



t HERTZ'S X is J (wave-length). Considerable confusion has arisen from this in various translations of 

 his papers. He assumes X = 4 - 8 metres, i.e., a wave-length = 9-6. 



