α

2

multiplied by (2), where it is to be observed that

2a denotes the amplitude of vibration at an antinode.

125. The foregoing reasoning applies equally to longitudinal vibration, and shows that the energy of the sonorous vibration of the air within a cylindrical pipe, when executing stationary vibration parallel to the length of the pipe, is equal to the mass of this air multiplied by

(2), where 2a still denotes the amplitude of vibration

at an antinode.

126. We shall now investigate the energy of each of the two equal travelling undulations into which stationary undulation can be resolved.

If we employ the expression for one of these undulations (equation (10), § 49)

The mean value of the square of this velocity is (27 v a)2

multiplied by the mean value of sin2 6, where ◊ denotes 2o (vt−x), and we may either make constant, and thus

λ

2

take the mean at a given moment along a wave-length, or make a constant, and take the mean at a given point for

the period of one vibration. In either case we shall have to take the mean value of sin2 6 for an entire circle, which is the same as its mean value for the first quadrant, and is, as shown in § 124. The mean value of (velocity)2

and the kinetic energy is the mass multiplied by the half of this, that is

The amount of the statical energy can be deduced from the fact that the wave now under consideration has half the amplitude of the stationary wave. The energy of the stationary wave of amplitude 2a is entirely statical at the moment of extreme displacement, and is, as we have seen,

Hence the statical energy of the travelling wave of amplitude a is

and is equal to its kinetic energy. The total energy of the travelling wave is

and is half the energy of the stationary wave, as we

should have expected from the circumstance that the stationary wave is resolvable into two travelling waves.

127. The travelling waves which combine to form stationary waves are in opposite directions. When two systems of waves travelling in the same direction, of approximately the same wave length (both longitudinal, or both in the same plane if transverse), are compounded, we have, by § 33, at any instant,

c2=a2+b2+2a b cos 0,

c denoting the resultant amplitude of any particle, a and b the amplitudes of the two components, and their difference of phase at the instant considered. Hence, as shown in § 35, the mean value of c2 is a2 + b2, whence it can be shown that the energy of the resultant system is the sum of the energies of the two components. If a and b are equal, the values of c2 will range between zero and 4a2, so that the smallest resultant waves will have no energy (being in fact evanescent), and the largest will have four times the energy of a wave of either compo

nent.

128. When waves of sound spread out uniformly in all directions from a centre, they form spheres which are continually enlarging. As each wave carries with it its original amount of energy unchanged, the same amount of vibratory energy is propagated across all the imaginary spherical boundaries which can be described about the centre. But the surfaces of these spheres are propor

tional to the squares of their radii, and hence the amounts of energy propagated across equal areas of two of the spheres are inversely as the squares of their radii. The intensity of sonorous vibration at a point, being measured by the quantity of energy which crosses unit area around the point in unit time, is therefore inversely proportional to the square of the distance of the point from the source, the source being supposed small in comparison with the distance. The amplitude of the sonorous vibrations will be inversely as the distance, since the energy of simple harmonic vibration is as the square of the amplitude.

129. When two sounds are alike, both in pitch and quality, their loudness is naturally measured by the energy of the sonorous vibrations which they excite. If they differ in pitch but agree in quality, and both lie within the usual compass of music, the sensation excited by the sound of higher pitch will in general be the more intense, if the energies are equal. This is proved by the fact that the bass pipes of an organ require a great deal more wind, and therefore more work in blowing, than the treble pipes.

Again, sounds of piercing quality strike the ear more powerfully than sounds of smooth quality containing the same amount of energy. A piercing quality is usually due to the presence of high harmonics.

CHAPTER XII.

SIMPLE AND COMPOUND TONES.

130. IT was discovered by Ohm that a simple harmonic vibratory motion produces the sensation of a simple tone, and that when several simple tones are heard together each of them is due to its own simple harmonic component, which is present in the resultant sonorous vibration. Those musical tones which we call rich are not simple. The sound produced by striking one of the keys of a pianoforte is usually composed of some four or five simple tones, due to the co-existence in the wire of so many different modes of simple harmonic vibration. The tones of a violin are still more highly compound. We have pointed out in § 112 that the periods of the several modes of simple harmonic vibration. of a string are proportional to 1,,,, &c., or, what amounts to the same thing, that the numbers of vibrations made in a given time are proportional to the series of natural numbers 1, 2, 3, 4, &c. When the partial tones which together compose a compound tone have this relation to one another, the deepest of them-that which corresponds to the number 1-is called the fundamental,