Strings excited by Plucking and by Resonance. 579 
is plucked. Complete the parallelogram of which these parts 
are adjacent sides. The opposite sides constitute the other 
position of rest of the string. To get the intermediate forms, 
suppose two points to start from the point of pluck and to 
move uniformly in opposite directions round the parallelo- 
gram, passing each other at the points of discontinuity of the 
extreme forms. The shape of the string at any instant is 
made up of the line joining these two travelling points together 
with portions of the sides of the parallelogram. 
It is easy to see that each point of the string will remain at 
rest for certain intervals at each extreme position, and will 
move from one extreme to the other at a uniform speed. So 
the vibration-curve will be made up of straight lines alternately 
horizontal and sloped (see the first vibrations on the accom- 
panying photographs), the relative durations of the different 
stages will depend on the positions of the point plucked and 
the point whose motion is observed. The appropriate formule 
are easily deduced, and will be found discussed very fully in 
the first paper of Krigar-Menzel and Raps. 
Motion of a Resonating String. 
Photographs 2 & 3 (Pl. XVI.) show the initial stages of the 
motion in our earlier experiments with two strings, stretched 
onasimple wooden frame without a resonance-cavity. It will 
be seen that the disturbance of the secondary string begins 
atonece. The pr imary vibrations rapidly lose their symmetry, 
and their behaviour is copied by the secondary vibrations, the 
downward motion being more gradual than the upward. 
Further, it is to be noted that only the lower harmonics: 
the series included in the primary vibrations are transferrea 
to the secondary string. Indeed, the secondary forms may 
be built up of the fundamental and octave only. This shows 
that, although the higher harmonics rapidly disappear, as 
we shall see, from the motion of the plucked string, they 
are not taken up by the frame. We may take the frame 
to have a much lower natural period than the string, and so 
explain the greater readiness with which it takes the lower 
notes. 
The only remaining point to be noticed concerns the pbase- 
relation between the two vibrations, which is closely connected 
with the transfer of energy taking place between the strings. 
The simple mode of steady vibration with motionless nodes, 
the same phase at all points between two nodes, and a sudden 
change to the opposite phase in passing a node, is incom- 
2 patible with an energy transfer in either direction. The work 
done by any portion of the string as its ends move outward 
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