the string will have the same periodic time (and therefore the same number of vibrations per second) as the string itself, but will have a different wave-length unless the velocity of propagation v along the string happens to be equal to the velocity of sound in air.

113. In giving its first or fundamental tone, the string passes backwards and forwards between the two positions shown by the continuous and dotted lines in Fig. 43.

In giving its second tone, its two extreme positions are those indicated by the continuous and the dotted line

in Fig. 44. There is here a node in the centre besides the two fixed points at the ends.

In like manner, for its third tone, it divides into three equal parts separated by two nodes as in Fig. 45; and higher tones are in like manner obtained by carrying the division further.

If the string is forcibly started in any one of these modes of vibration and then left to itself, it will continue to vibrate in the same manner, the nodes remaining at the same places, but the amplitudes gradually becoming smaller. Hence these are called modes of free vibration of a string. A mode of free vibration for any body is a

mode of vibration which the body can maintain of itself when once started.

Modes of vibration resembling those here described can be started in a string by lightly touching it at one of the points where a node is required, while a fiddle-bow is drawn across it at a place where a node is not required. As many as eight or twelve successive tones can thus be elicited from an ordinary fiddle-string or piece of pianoforte wire of suitable length. A piece of wire stretched upon a sounding-box is sold, under the name of a sonometer or monochord, by makers of acoustic apparatus.

114. All stringed instruments have a sounding-box or board, for the purpose of communicating the vibrations of the strings to the surrounding air. A string stretched between two massive and firmly fixed blocks would give but a very faint sound; for the very small surface of the string itself is too small to enable it to produce powerful undulations in the air. In the violin and piano the string or wire is stretched over a bridge supported by a board (called in the piano the sounding-board); and it is the vibration of this board, with its large surface, that has the principal share in communicating disturbance to the air. In the violin the belly, on which the bridge rests, transmits its vibrations to the back with the help of the sound-post, and thus both the belly and the back act as sounding-boards. The agitation of the strings by the bow rocks the bridge from side to side, throwing pressure on its two feet alternately, and causing the two soundingboards to vibrate normally.

CHAPTER X.

DYNAMICAL INVESTIGATION.

115. WE shall now show that the elastic force of a stretched string is competent to produce such motion as we have been describing..

This motion is specified by the equation

which is obviously the same as equation (15) of § 62, the amplitude of the stationary vibration at the points of maximum amplitude being now denoted by a instead of by 2a.

and our present business is to show that the forces of elasticity will produce this acceleration, if the tension F of

the string, and its mass per unit length m, satisfy the condition

the inclination of the string to the axis of x being supposed to be everywhere so small that its square is negligible.

Since cos is 1-102+higher terms, and sin ✪ is 0 – 1 03 +higher terms, we may write cos 61, sin 6-0, and tan 6

The component tension parallel to the axis of x at any point is F cos 0=F, and the component tension parallel dy; since in any

to the axis of y is F sin = F tan = F 6 0

dx

of the inclination of

the curve to the axis of x.

The tension F being supposed the same at all points, the component parallel to the axis of x will therefore be the same, and in computing the resultant force acting on an element dx of the string, we may leave the components in this direction out of account. The component normal to the axis of x at the point x of the string is

and the corresponding component at the point x+dx is greater than this by the amount

This is the resultant force upon the element dx, and the acceleration will be found by dividing by the mass. m dx. The acceleration will therefore be

2

F

Substituting for its value (+), this expression becomes

2

m

y, which was to be proved.

(2) pi

116. Very similar reasoning applies to the stationary undulation of a cylindrical column of air. Let the motion of the particles of air be specified by the equation

y denoting the longitudinal displacement of the particle whose undisturbed position was x, so that x+y is its actual dy d2y

position at time t. Then and will still denote re

dť

dt2

spectively the velocity and acceleration of the particle,

and we have to show that the value of

from equation (18), namely

dt2'

as deduced