77. Importance of the connexion of musical strings with a sounding-board.

It has been seen in Article 48 that in a divergent oscillating wave of air, such as we may suppose to be caused by the vibrations of a string, the motion of the particles is of the order of R, whose first term varies as

1

the distance raised to the power. Moreover, the smallness of dimension of a wire makes it impossible that it can communicate great motion even to the air which it touches. Hence, it is impossible that a wire can, by immediate action on the air, produce a sound easily audible to a considerable or convenient distance. To make it audible, the wire must be connected with an intermediate substance whose vibrations can produce a stronger effect on the air, and those vibrations must be excited by the vibrations of the wire. The intermediate substance used for this purpose is the soundingboard.

In the violin, the wires pass over a bridge which rests by two feet upon the upper board; and under that board, at the place where one foot of the bridge presses, is a little post (known by the name of the "sound-post" or the "soul") connecting the upper board with the lower board. Every tremulous motion of a wire of the violin acts directly upon the bridge and upon the upper and lower boards; and the tremors

of these produce effective vibrations of the air, and diffuse the sound. We know that every thing depends on the elastic properties of these boards; but we know nothing of their precise laws of vibration.

In the piano-forte, the general construction is simpler, but the sounding board is so connected with the supports of the wires that it is made to vibrate by the vibrations of the wires.

Though we cannot give a theory which shall apply accurately to the motions of the sounding-board, we can give one founded on motions which have a certain degree of analogy with the motions of the wire and sounding-board. Consider the wire as a pendulum, whose length is L and weight W, and the soundingboard also as a pendulum, whose length is land weight w; and suppose these two pendulums to be connected. When both are displaced through the same space z, the pressure-force tending to bring them back is

the mass to be moved is W+w; the equation of motion is therefore

and the time of a complete vibration, in which sin (ct+B)

goes through all its changes, is

The time of complete vibration of the first pendulum,

if not connected with the second, would be 2π

Hence the time of vibration is altered, by the connection,

in the proportion of 1 to

WI+wl
Wl+wL*

If T be the

time of complete vibration of the first pendulum, and ↑ that of the second, supposed to be unconnected, the time of vibration of the first pendulum is altered by

the connection in the proportion of 1 to

If, in one combination, the first pendulum vibrates in

T, the connection-alteration is as 1 to

T

if, in another combination, it vibrates in

the con

2'

'4WT2 + 4WT*
4W+*+wT

Hence

nection-alteration is as 1 to

it appears possible that the frequency of vibration may be altered in different proportions for the fundamental note and for its harmonics; and the series of sounds which reach the ear may not possess the character of

N

harmonics. The pleasurable character of the complex sound will therefore depend entirely upon laws of vibration of the sounding-board, which we are unable to investigate mathematically, but which perhaps in some cases are rudely mastered in practice.

We may add to this subject that, in the common tuning-fork, the separation and approximation of the two branches appear to produce a longitudinal retraction and extrusion of the central fibres of the stalk; and, when the stalk is planted downwards upon a table, the sound of the fork is very much increased; the table acting as a sounding-board. In this manner it is commonly used by the tuners of musical instruments.

78. Theory of the vibrations of air in an organ-pipe stopped at both ends.

In Article 23 we found as the general expression for the disturbance of air in a tube (omitting, for convenience, the factors n and 0),

from which

dX

dt

X=$(at −x) + (at + x);

=a.4′ (at − x) + a. f' (at + x).

The further treatment of these formulæ requires us to distinguish three cases: (1), that of a tube stopped at both ends; (2), that of a tube open at both ends;

(3), that of a tube stopped at one end and open at the other.

When a tube is stopped at both ends (it being supposed that adequate means are provided for putting the air into a state of vibration) the condition required is, that the disturbance at both ends is 0. Let the length bel, and let a be measured from one end; then, exactly as in Article 73,

(at) + (at) = 0,

$(atl) + (at + 1) = 0 ;

the terminal equations in this case; from which it appears that, and that & is a periodic function going through all its changes while at increases by 21,

or while t increases by 21. The number of complete

α

vibrations per second will be second will be Now on comparing

21

the expression in Article 23 with the discussion in Article 24, it appears that nea, for which (with a convenient abbreviation) we have here used a, is the space described by external sound in one second of time. Hence the number of complete vibrations per second of the air in the closed organ-pipe is

velocity of sound in feet per second

2 x length of pipe in feet

In ordinary temperatures the velocity of sound is about 1090 feet per second (Article 65); a pipe 103 foot long