Strings excited by Plucking and by Resonance. o¢7 
Integrating through a _ period we find the energy 
transfer from the string at «=0 
le 
= 77 147 (0) LOR 
rm A9(%) 
energy loss 
(amplitude)2” 
-, rate of change of phase a 
- Wesee that ifa string is losing energy by its ends, as is the 
case with the plucked string in our experiments, Snecse is a 
lag of phase from the centre outwards. When the ends are 
outside the nodal points, this lag is greater than : but less 
than 7. This is, as Lord Rayleigh has shown, the state of 
the case when the ends are yielding and have a natural 
period longer than. that of the string, the pitch of the note 
being then higher than for a string of the same length with 
= °2) © fo) 
fixed ends. Conversely, for the secondary string, which is 
y] 
absorbing energy by its ends, we have the central parts 
lagging behind the ends. Using “crest ” and “trough” for 
the extreme positions, as shown on the vibration-curves, we 
should expect the crest of the end-motion to fall before the 
trough of the primary vibration, and the crest of the secondary 
vibration to come after the crest of the end-motion. The 
result is that the crest of the secondary is found, as is shown 
on the photographs, after the trough of the primary vibration. 
Deviations from the ideal forms in the motion of a 
Plucked String. Theoretical. 
In the experiments of Krigar-Menzel and Raps the de- 
viations from the ideal shown by the vibration-forms consisted 
ina slow forward inclination of the horizontal parts and a very 
slight curvature of the sloping parts. ‘The tops and bottoms 
remained straight, or showed very small wrinkles, and the 
corners remained sharp. We get this result when we employ 
a fine steel wire (photo. 1, PI. aa ie 
Working with strings of other thicknesses and materials, 
we have found in addition, first, marked wrinkles generally 
unsymmetrical in position with rounding of the corners; and 
second, in certain cases a backward sloping of the tops and 
bottoms of the vibration-curves. 
If we regard the form of the vibration-curve as built up of 
a Fourier series, we have two ways in which the shape can 
be altered ; viz. (1) by a change in the relative amplitudes, 
and (2) by a change in the relative phases of the simple 
