Electrically maintained forks of this nature are used to control electrical "multi-vibrator" circuits which serve as frequency standards for all frequencies which are multiples of that of the fork, e.g., in steps of ',coo cycles, up to I.5 million cycles per second. The tuning fork thus functions as a standard of radio frequencies. It is possible to maintain an electrically driven fork by means of the "seconds" contacts of a standard pendulum clock. Such clock controlled forks are not subject to temperature variations, etc., and remain accurately in step with the standard clock over relatively long periods of Longitudinal Vibrations of Rods.—When a rod is set into longitudinal vibration its axis remains undisplaced whilst cross sections vibrate to and fro in the direction of the axis of the rod. The quantities involved are the density of the material and its elastic properties. Consider a rod of cross section A and an elementary slice of it bounded by two planes at x and x+hx at right angles to the axis x of the rod. If the plane at x is displaced at the time t to the plane at x+hx will be displaced to x-i-ox-FO-dVdx•ox, that is, the actual elongation of the slice is db/dx•8x and its fractional elongation d/dx. This change in thickness of the slice implies a difference in the forces acting on its faces. The total force acting on the face at x will be EAd/dx where E is Young's Modulus of elasticity. Similarly at the op posite face of the slice the force will be in the opposite sense and equal to EA Consequently the resultant force acting on the slice will be EA • ox• d'E/ . Now the mass of the slice is equal to pAbx, where p is the density of the material, and the acceleration is We have therefore an equation of the same form as that obtained for the transverse motion of a string, indicating a wave travelling along the rod with velocity c= Al(E/ p). The above treatment is approximate only, since it ignores lateral bulging or contracting of the sides of the rod. The result is, however, sufficiently true provided the wave length X is great compared with the width or thickness of the rod. As in the case of vibrating strings the direct and end reflected waves in a rod of finite length form stationary waves with nodes and antinodes in accordance with the relation c=yx= J(E/p) or N= •V(Elp). In the fundamental mode of vibration of a free bar X = 2/ and the various partials require sly = 2/. Consequently the expression for the frequencies of the partial tones is which is similar to equation (DO for strings, the tension T being now replaced by the elasticity E. When the bar is clamped at the mid point, an important practical case, all partials requiring an antinode at that point are suppressed, consequently only the odd partials are present. It will be seen that the partials given by (23), form a harmonic series—provided X/2 is always large com pared with the diameter or thickness of the rod and that lateral bulging and shrinking may be neglected.
The frequency of longitudinal vibrations in bars is usually very high compared with that of the transverse modes, the ratio increasing rapidly as the diameter or thickness diminishes relative to the length. The nodes of a bar vibrating longitudi nally are points of maximum stress, whereas the antinodes are points of zero stress. Biot and Tyndall demonstrated this fact to a large audience by means of passing polarised light through the node of a vibrating strip of plate glass. The analyser was set at extinction before the rod was set in vibration so that no light was seen on the screen. On stroking the rod with a resined cloth vigorous vibrations were set up and intermittent light was passed on to the screen. No such effects were observ able at the antinodes where there is no stress.
Methods of Excitation.—Rods of metal, wood, or glass, clamped at the midpoint or at one end, are readily set in vibration by the steady frictional drag of a resined cloth drawn, with a moderate pressure, along the rod towards an antinode. Another method, more suitable for relatively short stiff rods, is to strike the end a sharp blow with a hammer. This method usually results in a complex sound due to both transverse and longitudinal vibra tions—the one or the other may be rapidly damped out by clamping at a suitable point.
Electrical Methods of Excitation.—Powerful longitudinal vibra tions may be set up in bars of magnetic material by acting on them with alternating magnetic fields of the resonating fre quency. Using a small alternating current magnet (with iron
wire core) mounted close to the end of a steel bar, the funda mental and harmonics can readily be excited as almost pure tones. The resonance is extremely sharp (indicating very small internal damping) and careful tuning is required. Vibrations have been obtained in non-magnetic rods by electrostatic meth ods (J. H. Vincent, Nature. Dec. 31, 1927) the end of the vibrat ing rod forming one plate of a condenser supplied with high frequency alternating current from a valve oscillator'. (See THERMIONIC VALVE.) Longitudinal Vibrations of High Frequency in Piezo Electric Crystals.—Certain crystals, notably quartz and rochelle salt, have the property of changing their linear dimensions when subjected to electrostatic stress, and conversely they develop electrical charges on their faces when mechanically strained. The phenomenon is known as piezo-electricity (discovered by P. Curie, 188o). The best effects are obtained when slices or rods of the crystal are cut in certain specified directions relative to the optic axis. Voltage applied to the faces at the ends of the electric axis produces a change of thickness and length of the crystal. If alternating voltage is applied, the thickness and length fluc tuate accordingly, the maximum effects being produced when the frequency of the electrical alternations coincides with the natural frequency of longitudinal elastic vibration. W. G. Cady (Phys. Rev., 19, p. I, 1922) has made use of this property of a quartz crystal in the construction of standards of high frequency, with particular application to radio-frequency standardisation. A special electrical circuit which we need not consider here was employed to detect the resonance in the quartz. Thus Cady found that a quartz resonator 3ox4x1 -4 m.ms. vibrating longitudi nally in the direction of its length had a fundamental frequency of 89,87o cycles per second. The tuning was so sharp that a change of frequency of 1 cycle per second was measurable. The overtones were also excited and found to approximate to har monics of the fundamental. G W. Pierce (Proc. Amer. Acad., Vol. 59, Oct. 1923 and Vol. 6o, Oct. 1925) has succeeded in con trolling the frequency of an oscillating valve by means of such a crystal. In this manner a quartz oscillator may be used as a frequency stabiliser for radio purposes.
Quartz plates vibrating longitudinally in the direction of their thickness (the electric axis) have been employed by Langevin, Boyle, and others as a source of super-sonic vibrations, more par ticularly for use under water. The same apparatus is used in the converse process of reception, for the alternating pressure of the arriving sound waves produces corresponding electrical effects which, when suitably amplified and heterodyned, can be heard in telephones. Langevin and Boyle have employed this apparatus for the detection by an echo method of submerged objects (wrecks, icebergs, etc.), the sound emitted from a large disc of quartz being sharply directional and therefore suitable for such purposes. (See numerous papers by R. W. Boyle in Proc. Roy. Soc. of Canada 1922-28.) R. W. Wood and A. L. Loomis, using a piezoelectric quartz crystal vibrating at fre quencies of the order cycles per second have obtained striking effects of a physical and biological nature (Phil. Mag., Sept. 1927). With the crystal vibrating under oil, they estimated the pressure of the sound radiation to be equal to I so grams weight—the free surface of the oil being raised into a mound 7 cm. high! The uses of Rochelle salt as a sound generator and receiver have been demonstrated by Nicholson (Amer. Inst. Elect. Eng. Proc., Nov. 1919) who in one application used the crystal to replace a gramophone sound-box, the amplified e.m.f.s produced in the crystal being sufficient to operate loud-speakers or telephones. The elasticity of Rochelle salt is very small, E=3 X as com pared with 8 X ion for quartz and 2 X PO" for steel. (An excellent bibliography on piezo-electricity and its applications is given by W. G. Cady in the Proc. Inst. Radio Engineers, April 1928.) Torsional Vibrations of Rods.—A solid rod or tube of circular section may be twisted in such a way that each trans verse section remains in its own plane. If the section is not 'A. W. Pierce and J. H. Vincent have excited rods of nickel (and alloys of nickel-iron) into resonant vibration by magnetostriction.