SOURCES OF SOUND Vibrating Systems.—A fundamental advance in the theory of sound was made in 1843 when Ohm proved that the simplest and most fundamental type of sound sensation is that which cor responds to a simple harmonic motion, i.e., to the simplest mathe matical form of periodic function. (See HARMONIC MOTION.) Such motions may vary in period and amplitude but in no other manner; they are consequently ideal for the production of "sim ple" or "pure" tones. Another important feature of this form of motion is the possibility of transmission from one medium to another without change of form. Again, it has been proved by Fourier that the most complex form of periodic motion can be analysed (or synthesised) into a series of simple harmonic mo tions having frequencies which are multiples of that of the com plex motion. The vibrations of a tuning fork may approximate closely to a simple harmonic motion, the resulting sensation being described as a " pure tone." Simple Harmonic Motion (see WAVE MOTION) is typified by the oscillations of a particle attracted towards a fixed point 0 with a force varying as the distance x from 0. If s be the force at unit distance from 0, then at x it will be —sx, the sign of the force being always opposite to that of the displacement. Thus if m is the mass of the particle ma'xiar = —sx or, writing s/m, (I) of which the solution is x= a cos (nth +) (2) the constants a and E are arbitrary. The motion is therefore periodic, the values of the displacement x, and the velocity ax/at of the particle recurring whenever nt increases by 27. The periodic time of the oscillation is therefore T = 27/n= 271" V(//t/S) (3) a quantity independent of a. The type of vibration indicated by equation (2) is of fundamental importance. The equation shows that the particle oscillates between two points at a distance a on opposite sides of 0. This distance a is called the Amplitude of the vibration. The quantity E represents the initial Phase of the vibration when t=o. By simple differentiation of equation (2)
we obtain the velocity of the particle ax/ at= —ansin(nt+e) and the acceleration at' = as postulated in equation The meaning of equation (2) for S.H.M. is often expressed graphically as the projection on a diameter of the motion of a point moving uniformly, with constant angular velocity n on a circular path of radius a, the periodic time (for one revolution) being T =27/ n. The reciprocal of the Period T, viz., n/2r, is termed the Frequency. The quantity n is sometimes called the pulsatance.
Practical demonstration of the relation between a simple harmonic motion and the corresponding circular motion, may be seen in any type of reciprocating engine; the motion of the piston in the cylinder is approximately simple harmonic whilst that of the flywheel is circular with uniform velocity. A simple pendulum, a weight suspended on a spring, a rapidly vibrating wire, tuning fork, or diaphragm all illustrate, approximately, simple har monic motion; any differences in their motion arising from dif ferences in frequency and amplitude only.
The relationship (3) for the periodic time of vibration is of very wide application in the theory of sound. It indicates the general principle that the period T increases with the "inertia or mass factor m and decreases with the "stiffness" or "elastic factor" s (s being the force required to produce unit displacement). Thus in the case of a simple pendulum of length 1, s = mg/l, whence the periodic time of an oscillation is T = 21r V(//g). The period of vibration of a mass m on a helical spring of strength s is expressed directly by (3), viz. T=271,1(m/s). We also obtain from (2), for the energy of the vibrating particle Mean kinetic energy = Mean potential energy = ma2n2/4 (4) Maximum energy = 2.
The energy is a maximum at the mid-point and at the two turning points of the vibration. In the former case it is all kinetic and in the latter all potential energy.