4 vibrations per second. The lowest of the series makes 256 vibrations, and the highest 512, thus completing an octave. Any note within this range can have its vibration-frequency at once determined, with great accuracy, by making it sound simultaneously with the fork next above or below it, and counting beats.

With the aid of this instrument, a piano can be tuned with certainty to any desired system of temperament, by first tuning the notes which come within the compass of the tonometer, and then proceeding by octaves.

In the ordinary methods of tuning pianos and organs, temperament is to a great extent a matter of chance; and a tuner cannot attain the same temperament in two successive attempts.

898. Pitch modified by Relative Motion. We have stated in § 889 that, in ordinary circumstances, the frequency of vibration in the source of sound, is the same as in the ear of the listener, and in the intervening medium. This identity, however, does not hold if the source of sound and the ear of the listener are approaching or receding from each other. Approach of either to the other produces increased frequency of the pulses on the ear, and consequent elevation of pitch in the sound as heard; while recession has an opposite effect. Let ʼn be the number of vibrations performed in a second by the source of the sound, v the velocity of sound in the medium, and a the relative velocity of approach. Then the number of waves which reach the ear of the listener in a second, will be n plus the number of waves which cover a length a, that is (since n waves cover a length v), will be n+an or "+an.

α

v

The following investigation is more rigorous. Let the source make n vibrations per second. Let the observer move towards the source with velocity a. Let the source move away from the observer with velocity a'. Let the medium move from the observer towards the source with velocity m, and let the velocity of sound in the medium be v.

Then the velocity of the observer relative to the medium is a - m towards the source, and the velocity of the source relative to the medium is a'm away from the observer. The velocity of the sound relative to the source will be different in different directions, its greatest amount being v+a'-m towards the observer, and its least being v-a'+m away from the observer. The length of a wave will vary with direction, being of the velocity of the sound

PITCH CHANGED BY APPROACH OR RECESS.

907

relative to the source. The length of those waves which meet the observer will be

v+a-m
n

and the velocity of these waves relative to

the observer will be v+a-m; hence the number of waves that meet

Careful observation of the sound of a railway whistle, as an express train dashes past a station, has confirmed the fact that the sound as heard by a person standing at the station is higher while the train is approaching than when it is receding. A speed of about 40 miles an hour will sharpen the note by a semitone in approaching, and flatten it by the same amount in receding, the natural pitch being heard at the instant of passing.1

1 The best observations of this kind were those of Buys Ballot, in which trumpeters, with their instruments previously tuned to unison, were stationed, one on the locomotive, and others at three stations beside the line of railway. Each trumpeter was accompanied by musicians, charged with the duty of estimating the difference of pitch between the note of his trumpet and those of the others, as heard before and after passing.

CHAPTER LXIV.

MODES OF VIBRATION.

899. Longitudinal and Transverse Vibrations of Solids.-Sonorous vibrations are manifestations of elasticity. When the particles of a solid body are displaced from their natural positions relative to one another by the application of external force, they tend to return, in virtue of the elasticity of the body. When the external force is removed, they spring back to their natural position, pass it in virtue of the velocity acquired in the return, and execute isochronous vibrations about it until they gradually come to rest. The isochronism of the vibrations is proved by the constancy of pitch of the sound emitted; and from the isochronism we can infer, by the aid of mathematical reasoning, that the restoring force increases directly as the displacement of the parts of the body from their natural relative position (§ 111).

The same body is, in general, susceptible of many different modes of vibration, which may be excited by applying forces to it in different ways. The most important of these are comprehended under the two heads of longitudinal and transverse vibrations.

In the former the particles of the body move to and fro in the direction along which the pulses travel, which is always regarded as the longitudinal direction, and the deformations produced consist in alternate compressions and extensions. In the latter the particles move to and fro in directions transverse to that in which the pulses travel, and the deformation consists in bending. To produce longitudinal vibrations, we must apply force in the longitudinal direction. To produce transverse vibration, we must apply force transversely.

900. Transverse Vibrations of Strings.-To the transverse vibrations of strings, instrumental music is indebted for some of its most

VIBRATIONS OF STRINGS.

909

precious resources. In the violin, violoncello, &c., the strings are set in vibration by drawing a bow across them. The part of the bow which acts on the strings consists of hairs tightly stretched and rubbed with rosin. The bow adheres to the string, and draws it aside till the reaction becomes too great for the adhesion to overcome. As the bow continues to be drawn on, slipping takes place, and the mere fact of slipping diminishes the adhesion. The string accordingly springs back suddenly through a finite distance. It is then again caught by the bow, and the same action is repeated. In the harp and guitar, the strings are plucked with the finger, and then left to vibrate freely. In the piano the wires are struck with little hammers faced with leather. The pitch of the sound emitted in these various cases depends only on the string itself, and is the same whichever mode of excitation be employed.

901. Laws of the Transverse Vibrations of £trings. It can be shown by an investigation closely analogous to that which gives the velocity of sound in air, that the velocity with which transverse vibrations travel along a perfectly flexible string is given by the formula

t denoting the tension of the string, and m the mass of unit length of it. If m be expressed in grammes per centimetre of length, t should be expressed in dynes (§ 87), and the value obtained for v will be in centimetres per second. The sudden disturbance of any point in the string, causes two pulses to start from this point, and run along the string in opposite directions. Each of these, on arriving at the end of the free portion of the string, is reflected from the solid support to which the string is attached, and at the same time undergoes reversal as to side. It runs back, thus reversed, to the other end of the free portion, and there again undergoes reflection and reversal. When it next arrives at the origin of the disturbance it has travelled over just twice the length of the string; and as this is true of both the pulses, they must both arrive at this point together. At the instant of their meeting, things are in the same condition as when the pulses were originated, and the movements just described will again take place. The period of a complete vibration of the string is therefore the time required for a pulse to travel over twice its length; that is,

I denoting the length of the string between its points of attachment, and n the number of vibrations per second.

This formula involves the following laws:

1. When the length of the vibrating portion of the string is altered, without change of tension, the frequency of vibration varies inversely as the length.

2. If the tension be altered, without change of length in the vibrating portion, the frequency of vibration varies as the square root of the tension.

3. Strings of the same length, stretched with the same forces, have frequencies of vibration which are inversely as the square roots of their masses (or weights).

4. Strings of the same length and density, but of different thicknesses, will vibrate in the same time, if they are stretched with forces proportional to their sectional areas.

All these laws are illustrated (qualitatively, if not quantitatively) by the strings of a violin.

The first is illustrated by the fingering, the pitch being raised as the portion of string between the finger and the bridge is shortened. The second is illustrated by the mode of tuning, which consists in tightening the string if its pitch is to be raised, or slackening the string if it is to be lowered.

The third law is illustrated by the construction of the bass string, which is wrapped round with metal wire, for the purpose of adding to its mass, and thus attaining slow vibration without undue slackness. The tension of this string is in fact greater than that of the string next it, though the latter vibrates more rapidly in the ratio of 3 to 2.

The fourth law is indirectly illustrated by the sizes of the first three strings. The treble string is the smallest, and is nevertheless stretched with much greater force than any of the others. The third string is the thickest, and is stretched with less force than any of the others. The increased thickness is necessary in order to give sufficient power in spite of the slackness of the string.

902. Experimental Illustration: Sonometer.-For the quantitative illustration of these laws, the instrument called the sonometer, represented in Fig. 618, is commonly employed. It consists essen