during the passage of a wave, then the compression and expansion will be adiabatic (§ 258).

Now in the case of an adiabatic compression or expansion the volume and pressure are connected by the relation

PV constant,

where is the ratio of the specific heat at constant pressure to the specific heat at constant volume.

Hence, making the assumption that the changes of pressure and volume caused by a sound-wave are adiabatic, and not isothermal, as we previously assumed, we have, using the same notation as before,

PV'k=(P+p){V′ – z)k.

Expanding (1-7) by the binomial theorem, and, since is small, neglecting all terms which involve v2 or higher powers of ʊ, we get

PV=(P+p)(V ́ k − k V’k−1v)

=VPk-k.PVk 1q+p V k - k Vk-up,

or, neglecting the term involving the product of the small quantities p and ',

kPVk 1y—pVk,

and hence the equation for the velocity of sound becomes

In the case of air k = 1.41, and hence the calculated value of the velocity is 28026× √1.41=33240 cm. per second, which agrees fairly well with the observed value.

The above is Laplace's formula for the velocity of sound in a gas, and the fact that the calculated value of the velocity agrees with the observed value is a proof of the accuracy of the assumptions on which it is based. We shall, indeed, see further on that the most accurate method of determining the ratio of the specific heats for a gas is to measure the velocity of sound in the gas, and to calculate the value of k from Laplace's equation for the velocity.

In the case of liquids, the compressibility is so small that the thermal changes which take place on this account have no appreciable effect on the elasticity, so that in this case Newton's equation is applicable. For water we have at 4° C. p = 1, and an increase of pressure of one atmosphere causes unit volume to decrease by .000049 units of volume. Hence in the c.g.s. system the elasticity is given by

a result which agrees fairly well with the observed value.

286. Effect of Temperature on the Velocity of Sound.-We have seen in § 197 that in a case of a gas

PV
I + at

is a constant.

Hence

if P, is the standard pressure, and V, is the volume of unit mass of a gas at this pressure and at a temperature of o°, we have

But if I' is the volume of unit mass, we have, since the density ℗ is the mass of unit volume, the relation V=1]p; and hence

Hence, if v is the velocity of sound at a temperature 1, we have

ย

For if v, is the velocity of sound at o°, and under standard pressure,

Now if we expand (1+a) by the binomial theorem, and, since a is small, neglect terms in a2 and in higher powers of a, we get

Changes of pressure unaccompanied by changes of temperature, such as a change in the barometric pressure, will not affect the velocity of sound in a gas, for by Boyle's law

Hence the change in pressure effects the elasticity and the density in the same ratio, so that the velocity of sound is unaffected.

CHAPTER III

PITCH-MUSICAL SCALE

287. Quality of Sounds.-Sounds which affect our ear are divided into two classes. One of these consists of short sounds which last only for a short time, or, if they last for any length of time, are continually changing their character, and are called noises. The other class consists of sounds the character of which is that the vibrations by which they are caused are periodic; these are called musical notes.

Musical notes differ from one another in three important particulars : (1) They may be of different intensity or loudness. Thus when we move away from a sounding body the intensity of the note given by the body decreases, but does not otherwise alter. (2) The pitch of two notes may be different. We shall see that the pitch or highness of a note depends on the frequency of the vibrations of the sounding body. (3) The notes given out by two different instruments, such as the cornet and the piano, although they may be of the same pitch and intensity, are clearly distinguishable by the ear. This quality of a musical note is called its timbre.

288. Pitch of a Note.-That the pitch of a sound depends on the frequency, or the number of vibrations per second made by the sounding body, can be shown by the instrument called a syren. The usual form of this instrument is shown in Fig. 233. It consists of a circular disc BB', mounted on a vertical spindle D, so that it turns freely. This disc is pierced by a number of holes at equal distances apart. The disc is pivoted so that it just clears the upper surface AA' of a small wind-box, H, which is connected with a bellows, by means of which a continuous current of air can be forced into the instrument. The plate A is pierced by the same number of holes as the movable disc. The holes in the fixed and movable discs are not drilled at right angles to the surface of the plates, but those in A and B are inclined in opposite directions, as shown at a and b. Hence

FIG. 233.

the air, when forced out through a, strikes against the side of the hole ¿, and causes the disc to rotate in the direction of the arrow. Each time the holes in the upper disc come opposite the holes in the lower plate, a puff of air escapes, and the disc receives an impulse. If the disc is rotating sufficiently rapidly, these puffs will produce a musical note, and as the velocity of rotation of the disc, and therefore also the frequency of the puffs increases, the pitch of the note rises.

The syren also permits of the determination of the frequency of a musical note, for if there are r holes in the upper and lower plates, then, during a complete revolution of the upper plate, a hole in the upper plate will coincide with a hole in the lower plate r times. For the angular distance between two holes in the lower plate is 360, and hence, when the upper plate has turned through this angle, each hole on the upper plate will have just moved on one, and will coincide with the next hole in the lower plate. The number of coincidences during one turn, or 360°, is therefore

360°÷360/x or x.

If the movable plate makes n revolutions per second, the number of puffs per second, or the frequency of the sound, will be nx.

In performing the experiment the pressure of the wind is increased, thus causing the movable plate to rotate faster and faster, till the pitch of the note emitted is the same as that of the note whose frequency has to be measured. The wind pressure is then kept constant, and the number of revolutions made by the movable plate in a given time is obtained by means of the toothed wheels R and S, which can be put into gear with the endless screw, V, attached to the spindle by pressing on the knob E, at the commencement of the interval, and put out of gear at the end of the interval by pressing on the knob F. The wheel R moves on one tooth for each revolution of the disc, and has 100 teeth. The wheel s is moved on one tooth every time R completes a revolution. Hence the number of turns and hundreds of turns can be read off on two dials by means of pointers attached to R and S.

In performing such an experiment, there is considerable difficulty in keeping the speed of rotation constant, and such that the note given by the syren is in unison with the note whose frequency is being measured. For this reason, the more modern forms of syren are driven by a small electric motor, and not by the pressure of the escaping air in the inclined holes. The speed of the motor is kept constant by means of an electric regulator.

Other methods of measuring the pitch of the musical note given out by a sounding body will be considered in subsequent sections.

289. The Musical Scale. We have in the previous section referred to a note as being higher or lower than another, and this statement has probably conveyed the required impression to all readers. We