CHAPTER SEVENTH.

ACOUSTICAL.

SECTION L.-MUSICAL SOUND.

ART. 233.-Period and Frequency. It is a property of a vibrating body that it vibrates always in the same amount of time, whether the amplitude of its vibration is large or small, provided that the amplitude does not exceed certain limits, which differ for different bodies. For instance, if an iron bar be held in a vice, and the upper end be displaced from the perpendicular, the bar, when let go, will vibrate on either side of the perpendicular, each point in the bar performing a simple harmonic motion (Art. 123). Each succeeding vibration has a less amplitude than its predecessor, but the time occupied in making the vibration remains the same. This constant time is called the period of the body, and is expressed in terms of

T per vibration.

The vibration may be defined in one or other of two ways; either as the movement from one side to the other, or the movement from one side to the other and back again. The latter is the more appropriate definition; for distinction it is sometimes called a complete vibration.

The reciprocal idea is the frequency; it is expressed in terms of vibrations per T.

ART. 234.-Wave-length. A disturbance initiated at any part

of a material medium is propagated outwards in all directions and with a constant velocity, provided the medium is uniform.

Let the velocity of propagation be

This gives us the idea of wave-length. The wave-length is the distance-period, that is the uniform distance from one point of greatest condensation to the next point of greatest condensation. The reciprocal is

n/v vibrations per L.

ART. 235.-Pitch. The pitch of a sound depends on the number of vibrations received by the ear per unit of time. It is the same as the frequency, when the spectator and the source of sound are at rest relatively to one another.

If the spectator and the vibrating body move towards one another with a velocity v L per T, the velocity with which the vibrations will arrive will be v + v1 L per T ; and as there are n/v vibrations per L, he will receive n(v + v1)/v vibrations per T.

If they move from one another with a velocity v L per T, the velocity with which the vibrations will arrive will be v v1 L per T, and the spectator will receive n(vv)/v vibrations per T.

ART. 236. Intensity. By the objective intensity of a source of sound is meant the amount of energy transformed per unit of time. It is expressed in the form

μ W per T.

Its amount at any time is proportional to the square of the amplitude of the vibrations.

By the intensity of the sound at a given position in the medium is meant the amount of energy received per unit of time per unit of cross-section; it is expressed in terms of

W per T per S cross-section.

Suppose concentric spheres drawn round the source of sound. If there be no absorption, the same amount will pass across

4 S spherical surface = (L radius),

hence the intensity is given by

i.e.,

or

μ W per T = 4π S per (L radius)2,

μ W per TS cross-section per (L radius)?,

4π

The last form shows that the current through a constant crosssection is proportional to the solid angle subtended by the cross-section.

The intensity perceived by the ear is the objective intensity per unit of cross-section, modified by difference in sensitiveness to sounds of different wave-lengths.

EXAMPLES.

Ex. 1. Find the wave-length in air of a note making 50 vibrations per second, taking the velocity of propagation in air at 1100 feet per second.

Ex. 2. An express train rushes past a station at the speed of 40 miles an hour, and blowing a whistle, the frequency of which is 1000 vibrations per second. What will be the difference in pitch of the notes heard by a spectator as the train comes up and

T

goes away? The velocity of sound in air is then 1090 feet per second.

1. Find the wave-length of a note making 1,000 vibrations per second, both in air and in water; the velocity of sound in air being 1,100 ft. per sec., and in water 4,900 ft. per sec.

2. A tuning-fork makes 256 vibrations per second, and the velocity of sound is 340 metres per second; what is the value of the wave-length of the note produced?

3. It is observed that 6 seconds elapse between the flash and report of a lightning discharge. At what distance did the discharge take place?

4. A stone is dropped down a well, and is heard to strike the bottom after an interval of 3 seconds; determine the depth, the velocity of sound being 1,140 ft. per sec.

5. Three observers are stationed, the first at a mile, the second at two miles, and the third at three miles from a gun. At 12 o'clock precisely the gun is fired; state the times at which the explosion will be heard at the several stations.

6. Taking 1,120 ft. per sec. as the velocity of sound in air, find the number of vibrations which a middle C tuning-fork (which vibrates 264 times per second) must make before its sound is audible at a distance of 154 feet.

7. A shot is fired at 500 yards, and a man standing at a distance of 100 yards from the target hears the reports of the firing and of the impact at the same time. The time of flight of the shot is 1 sec.; find the distance of the man from the place of firing.

8. A locomotive moving at 100 ft. per sec. carries a steam-whistle which produces 1,000 vibrations per second. What will be the pitch of the note heard by a person standing close to the rails before and after the locomotive has passed? Velocity of sound as in question 1.

SECTION LI.-VELOCITY OF SOUND.

ART. 237.-Extensibility of a Solid. Suppose that a bar of a substance of uniform cross-section is subjected to an equal pull at either end. The effect will be an alteration of the length of the bar; and so long as the stretching force is not sufficiently great to strain the bar beyond the limit of elasticity, the extension is proportional to the force. The bar, in its strained state, exerts a force equal and opposite to the stretching force. The relation between the effect and the cause is expressed in the form L increment per L original = F per S.

This is called the extensibility of the solid. The reciprocal idea is 1/ F per SL increment per L original;

it is called Young's modulus, or the modulus of elasticity, or the resilience due to longitudinal extension.*

The rate of longitudinal compression, when the bar is subjected to an equal compressing force at either end, is expressed in terms of

L decrement per L original=F per S.

Its value is the same as that of the extensibility.

ART. 238.-Compressibility of a Solid or Liquid. When a solid or liquid is subjected to an equal increment of pressure all over the surface, the change of volume produced per unit of original volume is proportional to the additional pressure. The relation between the effect and the cause is expressed in the form

cV decrement per V original = (F per S) increment. This is called the compressibility of the substance. The reciprocal is 1/c (F per S) increment = V decrement per V original; it is called the modulus of compressibility, or (by Everett) the resilience due to hydrostatic pressure.

The expansibility of a solid or liquid is expressed in terms of V increment per V original = F per S.

Its value is the same as that of the compressibility.

*Everett, Units and Physical Constants, p. 47.