Harmonic Analysis

For the various methods of generating alternating currents, (e.g., by valve oscillators, interrupters, alternators, etc.) the reader should consult the article on WIRELESS TELEGRAPHY.

Motional Impedance.

On account of the extensive use of electrical forms of sound generators and receivers it is very important in design to have a knowledge of their characteristics. In any form of machine which converts electrical energy into motion the moving mechanism reacts on the electrical circuit. Thus the revolving armature of an ordinary electric motor generates what is called a "back e.m.f." in opposition to the applied voltage. This back e.m.f. may alternatively be regarded as an increased resistance to current flow in the armature, the efficiency of the motor being measured by the ratio of this resistance R relative to the total resistance (Rd-r) in the circuit; the net power used in the motor being a maximum when R=r, in which case the efficiency is 5o per cent. In a similar manner the mechanical vibrating element of an electrical sound generator or receiver reacts on the electrical circuit, the back e.m.f. due to the vibration appearing as a change of impedance of the circuit. The change of impedance due to this cause is termed motional impedance and, relative to the total impedance, is a measure of the efficiency of the sound generator.

Transverse Vibrations of Strings.

When any point of a thin flexible wire stretched between two fixed clamps or bridges is displaced transversely and released, the wire commences to vibrate. This vibration results from transverse motions travel ling in opposite directions along the string and successive reflec tions of these motions from the fixed ends. In order to visualise such reflection of a transverse motion it is well to make a few simple experiments on waves travelling along a stretched rope, one end of which is held in the hand whilst the other end is fixed. It will be observed that the movement of each particle of the rope when forming part of the reflected wave is in the opposite direction to its motion in the original wave. The rope assumes the form of a sine wave if the end is moved up and down harmonically. Before we can deduce the modes of vibration of a stretched string it is necessary to know the velocity with which a wave of displacement travels along the string and also the manner in which the direct and reflected waves affect each other to produce what are called stationary waves.

Velocity of a Transverse Wave Along a String.—The following method is due to Tait. The string is supposed to be drawn through an imaginary smooth tube with velocity c. The tube

is straight except for an isolated curved portion which represents the wave on the string. If R is the radius of curvature at any point of the tube, the force acting in the direction of the normal to an element Ss is TbsIR where T is the tension in the string. Now the centrifugal force of the element as, of mass m per unit length, and velocity c will be mos-0/R, and this must balance the force Tasl R if there is to be no reaction on the tube. Thus if then c=-V(T1m) and there is no reaction on the tube, i.e., the tube may be regarded as absent and the wave travelling along the string with the velocity c =11(T/ m).

Reflection. Formation of Stationary Waves. (See WAVE MO TION.) —If both ends of the string of length I are fixed, the wave is reflected successively from end to end, and the resultant motion is determined by the superposition of the direct, or incident wave, and the reflected wave. The resultant displacement y at the instant of a point distant x from one end will be y=f(ct—x)—gct-Fx) where f indicates "a function of." Since y = o when x=1 we have also f(ct—l) = f (ct +l), or f(z) = f ( z+ 2/) where z = (ct —/), which indicates that z is a periodic function repeating at intervals of 2/. Consequently the displacement at any point of the string is periodic, the period T =211c being the time taken for a wave to travel along the full length of the string and back again. The frequency of this form of vibration is consequently N= c/2/ where c= /(T/m), i.e., FIG. 3 to a sine law. The result is a series of loops on the string of amplitude o to 2a. The condition for the formation of loops on a string of finite length 1 is clearly that in which 1 is a whole mul tiple of the length of a loop, i.e., provided the number of loops is I, 2, 3, • • • etc. The fundamental frequency n of the string is given by (r7), the various possible overtones are simple mul tiples of n, i.e., they form a harmonic series. The modes of vibra tion of a string vibrating with r, 2, 3 loops is shown in fig. 3. The points marked n which are permanently at rest during the vibrations, are called nodes, whilst the points marked a where the amplitude is a maximum are called anti-nodes or loops. It will be evident that the string could be clamped at the nodes a without affecting the motion of the remainder of the string. Denoting the wave-length of the vibration by X we must have N'X=c= -V(T/m) whence the frequency s being the number of loops, i.e., half wave-lengths, into which the string of length 1 is divided.