DETERMINATION OF WAVE FOI;\J 



tion, represented by 0', rapidly diminishes to /<-m after tin- 

 circuit is closed. 



The particular integral to which 0' must be added to obtain 

 the complete integral is 



8 = c " sin 



^-v-,, <> 



This expression for should be compared with the corresponding 

 one for the flow of current in a circuit containing resistance, 

 inductance and capacity after the steady state has been estab- 

 lished. In both cases there are "impedance" and phase-dis- 

 placement terms. 



Inspection of (3) shows that after the transient term has 

 disappeared, 



1. The various harmonics do not have the same proportional 

 effect on the deflection. 



2. The harmonics suffer different phase displacements. 



In consequence, the oscillograph can never give a mathematic- 

 ally correct picture of a wave form. This being so, it is neces- 

 sary to find the conditions which will make the instrument 

 sufficiently correct for practical purposes. 



Obviously, if the moment of inertia and the damping were both 

 zero the wave would be followed exactly. Therefore, the mass 

 of the moving parts must be reduced to a minimum and an ar- 

 rangement adopted which will make the moment of inertia as 

 small as possible. At the same time the directive moment on 



the movable system must be increased so that the ^ 2 and the 



^T terms are small in comparison with it; hence the usual state- 



ment that the rate of free vibration must be high. 



The instrument must closely follow sudden changes of cur- 

 rent, therefore, the damping should be near the critical vulu<>, 

 or mathematically, 



P = 4rP. 

 Then 



sn 



-ft.- tan- - (4) 



