The string may vibrate with any of the frequencies given by equation (r g) at the same time, i.e., a note may be produced which is made up of the fundamental and a number of harmonics. The laws of a vibrating string indicated by equation (DO may easily be verified by means of a monochord or sonometer, which consists essentially of a thin metallic wire (e.g., steel piano wire) stretched over two bridges by means of a weight hanging over a pulley, or by a spring tensioning device. A movable bridge provides a convenient means of varying the vibrating length of the wire. The monochord is a very useful means of comparing frequencies— the frequency of the string being inversely proportional to the vibrating length. Exact tuning is indicated by the " beats " between the monochord note and the note compared with it. The various overtones of a string may also be very simply demonstrated by means of the monochord, the string being lightly damped at any point corresponding to a node and plucked or bowed at a point corresponding to an antinode of the overtone required to be excited. The positions of the antinodes are easily determined by means of little paper riders.
Stiffness of Strings and Yielding of Supports.—When the thickness of the string becomes appreciable in relation to the length of a loop, the stiffness may have a perceptible effect on the frequency of vibration—this effect becoming more and more important the higher the overtone excited (i.e., the greater the number of loops). Where great accuracy is required a modifica tion of equation (19) is necessary, viz.
where r is the radius of the circular section of the wire and E Young's modulus of elasticity for the material. Yielding of the " bridges " supporting the wire may have the effect of increasing or decreasing the frequency according as the supports have (a) very large mass M but small spring factor or (b) very large spring factorµ and negligible mass. The effects of (a) and (b) are equivalent to a change of length of the string in the ratios — 2 T11 M 70) and I :(1-1-- 2 T/p./) respectively.
Methods of Producing Vibration in Strings. Quality.--A stretched string may be set in vibration by numerous methods. Plucking, bowing and striking are the more familiar; exemplified in the harp, the violin, and the piano, respectively. A string may also be set in vibration by forced oscillation of a point of support, e.g., if one end is attached to the prong of a vibrating tuning fork (Melde's experiment). Electromagnetic methods may also be used to excite a metallic string. In one of these methods a light metal wire is attached at right angles to the vibrating wire and arranged to dip in a small cup of mercury at each downward movement. A current passing through this intermittent contact actuates a small electromagnet which maintains the wire in vibration in the same manner as an electric bell is operated. In another method the vibrating wire itself carries an alternating electric current and lies in a permanent magnetic field. When the frequency of the current and position of the magnet are suitably chosen one of the numerous possible overtones of the wire will be readily excited. It will be appreciated that the method of excitation has a very important influence on the form of the wave which travels along the string. The quality of the note is of
course dependent on this wave form, i.e., on the relative amplitudes of the various overtones present in the vibration. It is just this addition of overtones to the fundamental which makes it possible for the ear to distinguish between the sounds of a piano, a violin, and a harp, emitting the same fundamental note.
Strings as a Source of Sound: The Sounding Board.—A vibrat ing wire rigidly supported would radiate extremely little sound energy to the surrounding air on account of the local reciprocating flow of air between opposite sides of the vibrating wire. It is necessary in an efficient stringed sound source that the bridges should yield and communicate the vibrations to a surface of large area, i.e., to a sound board, in contact with the medium. This sound board is therefore a vital part of all stringed musical instruments—it is important, however, in a good instrument, that the sounding board should have no predominant resonance fre quencies of its own, otherwise these would reinforce dispropor tionately the corresponding frequencies of the strings.
Transverse Vibrations of Elastic Rods.—The vibrations of a stretched wire are controlled by the tension whilst the stiffness of the wire may generally be disregarded. In the case of a rel atively thick wire or rod the stiffness may become all-important and tension may ultimately be disregarded. Even when sim plified as far as possible the theory of transverse vibrations of elastic rods is very complex in comparison with the theory of strings. In the case of strings, harmonic waves travel with a velocity independent of wave-length but in the case of rods or bars this is not so. It is shown in textbooks of sound (see Bib liography) that the velocity of a transverse elastic wave in a rod is proportional to t. J(E/ p)./X where t is the thickness in the direc tion of displacement, E elasticity, p density, and X the wave length. The velocity is thus dependent on the wavelength, a fact which makes the theory much more complex. It may be shown that the possible frequencies of transverse vibration of a bar are given by (20) where k is the radius of gyration of the section of the bar about the neutral plane, and 1 is the length of the bar = P/I 2 for a rectangular bar of thickness t). The value of the constant C depends on the method of supporting or clamping the bar and on the overtone to be excited. The frequency is therefore propor tional to the velocity of longitudinal waves MEI p)] in the material of the bar. It also varies as the thickness (or radius) and inversely as the square of the length of the bar. As in the case of strings, stationary waves are set up in rods by the combined effects of the direct and reflected waves. The possible forms of these stationary waves depend on the method of supporting the rod. Some of the modes of vibration of a free-free rod, i.e., entirely free or supported at two nodes, are shown in fig. 4a; whilst fig. 4b illustrates those for a clamped-free bar, i.e., a bar fixed 1 frequencies of the first three tones relative to the fundamental are consequently 1, 6.25 and 17.6 approximately. The relation Na k. NI(E/p)IP has been experimentally verified by a number of physicists, both in the case of the free-free and the clamped-free bar.