578 Messrs. Morton and Vinycomb on Vibrations of 
harmonic components. The latter change will account for 
the sloping of the horizontal parts of the curve; the former 
for symmetrical wrinkling and rounding of corners; the two 
combined for the unsymmetrical wrinkling. 
(1) The alteration in the relative amplitudes is due to the 
fact that the higher frequencies are much more rapidly 
damped out by the internal viscosity of the material than are 
the lower. This was first pointed out by Lord Kelvin for the 
case of torsional oscillations. The effect in the present case 
is to continuously reduce the number of components present 
until finally, in photographs of advanced stages of the motion, 
there is only a simple sine-curve left. This preferential 
damping of the higher harmonics is specially marked with 
gut strings. 
A comparison of the results got by putting the same string 
on different frames, shows that the viscosity concerned is chiefly 
in the string itself. As already mentioned, the frame takes 
up the lowe tones more readily. 
(2) The progressive alteration in the relative phases of the 
components means that the overtones of the string do not 
form an accurately harmonic series. This may arise from 
three causes :— 
(a) The yielding of the supports.—lf M is the effective mass 
of the end-support, its elasticity being neglected ; T, /, the 
tension and length of the string, and a the velocity of sound 
along it ; then ‘the sth partial tone is raised in pitch through 
the interval * 
sa te al 
Ma2a?” 5 
Thus the higher partials are too flat in comparison with the 
lower. Krigar-Menzel and Raps have shown that this will 
cause the horizontal parts of the curve to slope forwards. 
(b) The rigidity of the wire—The effect of this is to raise 
the sth partial through the interval + 
ie hae 
1+ SPT °°? 
where 7 is the radius of the wire and q¢ is Young’s modulus 
for its material. The higher overtones are raised too much : 
so the effect of rigidity is opposite to that of end-motion, and 
will make the horizontal parts slope backwards. 
(c) Inequality in the sectional area of the string.—A small 
* Rayleigh, ‘Sound,’ vol. 1. p. 204. T 
+ Rayleigh, ‘Sound,’ vol.i. p. 298; where “T” is used for our —., 
7 
