Harmonic Analysis

circular, a warping is liable to take place and the analysis is very complex. Consider a tube of radius r and thickness Or and let 0 be the angular displacement of any section distant x from the origin. The "shear" of the material of the tube is pa Vax. The opposing elastic force per unit area is 0/ax where /A is the coefficient of rigidity of the material (p, = E/2 (o•+ I l in which E is Young's modulus and is Poisson's ratio). Since the area of section of the tube is 27rrbr the moment of this force round the axis is and the restoring force acting on the slice of thickness ox has the moment Now the moment of inertia of the slice is (p is the density of the material) whence the equation of motion is which is independent of r, and therefore equally applicable to tubes of all radii and to a solid rod. The velocity of torsional wave transmission is c= p). The ratio of this velocity to the corresponding longitudinal velocity is -V(,u/E) = 1: li[2(o-+ Taking = 0.25 for a steel rod the ratio of velocities becomes 1•58. The possible frequencies of torsional vibration of a rod will be analogous to the harmonic series for the longitudinal vibrations. Torsional vibrations are readily set up by applying tangential forces of a frictional character, e.g., by means of a resined cloth, to the free end of a rod clamped at a suitable point.

Vibrations of Membranes.—The transverse vibrations of stretched membranes are related to those of diaphragms and plates in a manner analogous to the transverse vibrations of stretched strings and elastic bars. In the former case the vibra tions are conditioned by the applied tension and are independent of elastic forces, whereas in the latter the elastic forces are all important and tension almost negligible.

By analogy with the case of wave motion in one dimension (a stretched string) it may be shown that the equation represents the motion of a stretched, two-dimensional-membrane where T is the tension in the mass per unit area (p X thickness) and the transverse displacement at a point xy. The velocity of wave motion is c= V(T/m) as in the case of strings. The complete mathematical analysis for stretched membranes of various shapes is given in Rayleigh and Lamb's treatises on Sound. In the case of a circular membrane of radius a the fundamental frequency is shown to be In the higher partials the diaphragm becomes divided into nodal rings and diameters.

Membranes approximating to the ideal type have been made from soap films, or films of thin collodion, stretched on a metal ring, the vibrations of such thin films being examined by optical methods. Sheets of parchment or of thin metal (steel) are more suitable when it is required to examine the effects of tension. Wente's condenser microphone (see fig. Oa) (see MICROPHONE) has a highly tensioned steel membrane of fundamental frequency about io,000 cycles per second. The various modes of vibration of a steel membrane are conveniently studied by means of a small electro-magnet (the magnet system of any ordinary telephone receiver will serve the purpose) and a valve oscillator with a suitable range of frequency-control. As a rule, the agreement

between theory and observation is only approximate, for the theory generally given takes no account of the serious damping and loading of the diaphragm due to the medium (air) in contact with it. The stiffness of the membrane is not always negligible as assumed in the theory. Examples of membranes as sources of sound are to be found in various forms of drums, tamborines, etc.

In

the drum, the vibrations of the membrane are reinforced by the vibrations of an air cavity in resonance with it.

Vibration of Diaphragms.

When elastic restoring forces .re called into play and tension is of secondary importance we come to the case of the diaphragm. Rayleigh has calculated the periods and motions in vacuo, of a thin circular elastic plate rigidly clamped at the edge. By means of a complex analysis he found that the fundamental frequency of vibration is given by where h is the thickness and a the radius of the diaphragm, E Young's modulus, p the denity and Poisson's ratio for the material. More recently, Lamb (Roy. Soc. Proc. A., Vol. 98, 1920) has calculated the frequency and damping of circular diaphragms in air and in water, the value of the frequency in air agreeing closely with Rayleigh's estimate. Thus for a steel diaphragm E = 2 X p = 7.8, o = 0-28, whence c= [E/P ( I = 5.27 X cms./sec., and N= 2.5 X • cycles per second. If h=o-I cm. and a= 5 cms., N will be 1,000 cycles per second in air. The addition of a load m to the centre of a diaphragm of mass M has the effect of lowering the frequency given by (27) in the ratio I/V(I-F5m/M).

Diaphragm Vibrating in Contact with Water or Other Medium.— The presence of an extensive medium, say water, in contact with the diaphragm has two effects, (I) the frequency is lowered on account of the loading due to the added mass of water vibrating with the diaphragm, and (2) the vibrations are damped owing to the energy radiated as sound waves into the water. Lamb has shown in the case of a diaphragm with one side only in water, that the inertia of the diaphragm is increased in the ratio (1 -H3) where (3= being the density of the water, p the density of the material of the diaphragm, a the radius and h the thick ness). The frequency given by (27) must therefore be divided by .V(1 +i3) in this case. When both sides of the diaphragm are immersed the value of (3 must be doubled. The persistence of the vibrations is given by N / k=o-385(1-1- where ci is the velocity of sound in the medium and c in the material of the diaphragm. Thus, in the above example where the frequency, is modified by the water, is about 55o, the range over which the energy will exceed half the maximum (indicating sharpness of resonance, see page 8) lies between 53o and 57o p.p.s. The per sistence is increased, and the resonance is sharpened, the thinner the diaphragm and the greater the load.