The loudness of a sound is capable of be ing variously defined. If by the loudness of a sound is meant physical energy and if the sound is a pure tone then its loudness depends on the amplitude of vibration and the pitch, being pro portional to the square of each, and on the density of the medium, to which it is directly proportional. The loudness of a sound is ordi narily defined, however, by the intensity of the sensation which it is. capable of producing. Thus defined loudness is a function not merely of the amplitude of vibration and the density of the medium, but of the pitch and the quality as well, and moreover it is a complicated func tion of each. It is an interesting fact that in man there is a definite sense of loudness which renders it possible to compare, in respect to the intensity of the sensations which they produce, sounds differing in pitch by the whole of the musical scale. Moreover, this sense of loud ness is apparently physiological and not de pendent on familiarity with the °balance° of any musical instrument, and is to a high degree of accuracy the same for different persons, in dependent of age, sex or musical training.
Production 0 f Sound.— The best example of the single impulse as a source of sound is an explosion in unconfined and therefore non resonant space. The result is an approximately single wave. When, however, the explosion oc curs in a resonant cavity the result is a note of definite pitch determined by the cavity. Or a single explosion and impulsive wave may re sult in a train of waves and therefore a sound of definite pitch, by being reflected from uni formly spaced surfaces, such for example as the pickets of a fence. The next simplest source of sound is a siren, long a laboratory instru ment, more recently made familiar by use in fog signals and steam whistles. The siren con sists of two circular discs, the one fixed, the other pivoted to revolve nearly in contact with it. Both discs are pierced by a circle or by cir cles of holes through which steam or com pressed air escapes as the holes in the two discs come opposite each other.
A straight bar of metal or wood may vibrate either transversely or longitudinally. If dis torted transversely it vibrates to and fro through its normal straight form. The simplest form of this transverse vibration is that in which the bar at points one-quarter the total length from either end remains at rest. These points of rest are called nodes and the inter mediate part of free vibration is called an anti node. When vibrating in this manner the bar emits its fundamental note, the lowest note of which it is capable if entirely free. The next simple mode of vibration is that in which there are three nodes, or points or rest, at points one sixth the total length from either end and in the middle. In this case the bar emits a note having twice the frequency of the fundamental and in pitch an octave above it. Continuing in
this way a series of simple types of motion may be determined. The notes thus produced have twice, three times, four times, etc., the vibration frequency of the fundamental. Any transverse free vibration of the bar is a combination of these forms, and the sound which it emits is a combination of these notes. In this manner the quality of the sound is determined. If the bar is clamped at one end the lowest note which it emits is an octave lower than the lowest when entirely free; and the higher tones, instead of being two, three, etc., multiples of the funda mental, skip every other one, being three, five, seven, etc., multiples of the fundamental. Touching the bar at any, point tends to pro duce a node at that point and to strengthen the corresponding partial tone, and to diminish the partial tones having antinodes at that point. The exact converse is true in regard to striking the rod. Finally, the frequency of the several notes is proportional inversely to the length, and to the square root of the density, and directly to the square root of the rigidity, other dimen sions being the same in each case.
When the rod is rubbed or stroked so as to vibrate longitudinally, either free or clamped at one end, its fundamental and overtones form the same systems as before, but all are of a different pitch, determined now by the length, density and modulus of elasticity. Thus the longitudinal vibrations of the free rod have as vibration frequencies of its overtones all inte gral multiples of the fundamental. If the same rod is rigidly clamped at one end, its fundamental is an octave lower than the funda mental of the free rod, and the even integral overtones are absent.
A stretched string or wire, so small in diameter in comparison with its length that its rigidity is insignificant in comparison with its tension, vibrates for its fundamental over its whole length with nodes at each end. The first overtone is an octave above this in pitch, the wire vibrating with a node at the centre. The second overtone (third partial) is three times the fundamental in pitch frequency, the wire vibrating with nodes a third of the whole length of the wire from either end. The third overtone (fourth partial) is four times the fundamental in pitch frequency, with nodes at the quarter and middle points. A string set in vibration by any ordinary method vibrates in a more or less complex manner, emitting a sound containing the fundamental and overtones. The overtones present and their relative intensities are determined by whether the string is plucked, struck or bowed, and also by the point of appli cation. The fundamental note emitted by a string is of a vibration frequency equal to the square root of the tension divided by the mass per centimetre of length, divided by twice the length.