﻿980 Composition of Harmonic Motions. 



adopting this longitudinal mode, for not only does the cir- 

 cular motion tend not to occur, but the fact that the number 

 of loops in a given length of string under the same tension 

 is now halved gives longer loops and larger amplitudes. 

 The two vibrations seem always to set themselves at right 

 angles, though there is no very apparent reason for such 

 behaviour. It will be found advantageous not to attach the 

 string at the tip of the prongs, but rather lower, as the effect 

 of varying tension seems to hinder the two vibrations being 

 maintained together if the amplitudes of the forks are too 

 large. If the adjustments have been properly made, every 

 point on the string will execute a Lissajous* figure. At 

 any pla'ce which is a node for one vibration but not for the 

 other the motion is in a straight line, and in passing through 

 the node along the string a change of phase difference 

 between the two vibrations of it occurs. 



The two photographs PI. XVI. figs. 3 and 4 exhibit ex- 

 ceedingly well the surface traced out by a brightly 

 illuminated string executing such vibrations, and everyone 

 will recognise two phases in the composition of vibrations 

 in the ratio 1 : 3. The figure which stands out so distinctly 

 is traced out by a point rendered specially brilliant by a 

 piece of wax which was attached to the string. It can easilv 

 be seen that every little hair is carrying out a similar 

 motion. The size of the pattern photographed was about 

 5 cm. in diameter, and it is possible to obtain them much 

 larger. The phase difference in this case caused the figure 

 to go quickl} r through all the forms usually figured in text- 

 books. The effect of the relative amplitudes of the two 

 vibrations at different points in the string was easily observed; 

 where a node occurred for one vibration but not for the 

 other, the motion was in a straight line ; the change in phase 

 in passing through a node turned the figure the opposite way. 



It is obvious that any of Lissajous' figures may be obtained 

 in this manner. 



A case of some interest is that in which two vibrations with 

 frequencies in the ratio 1 : 2 may be compounded by the use 

 of a single fork. In the well-known experiments of Melde 

 two positions of the fork only are considered : (1) The trans- 

 verse mode, in which the period of the string is equal to that 

 of the fork ; and (2) the longitudinal mode, in which the 

 period of the string is twice that of the fork. If the fork be 

 rotated, so that the direction of its motion is inclined to the 

 direction of the string, it is clear that two transverse vibra- 

 tions whose frequencies are in the ratio 1 : 2 will be pro- 

 duced in it, and sometimes these set themselves at right 



