THE CARBON MICROPHONE 189 



that of hemispherical surfaces of random height fastened to a flat 

 plate, about 100 hemispheres being used. 



We see from the slopes of these curves that the model made with 

 hills of random height on a flat plate behaves most like the actual 

 contacts, the slopes of the corresponding curves being 3.2 and 3.1, 

 respectively. This arrangement is also the one which most nearly 

 represents the carbon surfaces as viewed under the microscope. Here 

 the hills have various heights and the radius of the underlying base 

 (0.015 cm.) is so much larger (1000 fold) than that of the average hill 

 that within the region of the contact area the surface of the former 

 may be regarded as plane. 



The slope of curve / is in accord with a formula derived from the 

 theory of elasticity by Hertz connecting the force F pressing together 

 two elastic spheres and the movement D between the centres of the 

 spheres: 



F = const. i)3/2 (3) 



The constant includes such factors as the elastic moduli of the contact 

 materials and the radii of the spheres and need not concern us here. 

 The case of a sphere pressed against a flat plate, as in our experiments, 

 is a particular case of this general equation, the constant only being 

 affected.® 



The slopes of curves // and III are also in accord with theory, as 

 we shall see, when one makes the simple assumption that the elastic 

 deformation is confined to such a small region near the contact in each 

 hill that the underlying base is not appreciably deformed. This 

 assumption was tested in the case of the model having the spherical 

 distribution of hemispheres by changing the stiffness of the rubber 

 used in the underlying sphere. No effect was produced on the stress- 

 strain characteristic (curve //). We may therefore consider that the 

 elastic reactions produced in each hill are independent of each other 

 and that the base is not deformed, so that with a given distribution 

 of hills it becomes a simple matter to calculate their combined effect 

 over a given compressional range. We may represent the conditions 

 essential for our calculation by the diagram, Fig. 25, in which A 

 represents the plane surface of the smooth contact element just making 

 contact with the highest hill of the rough contact element. Under 

 compression, A may be considered as moving in the direction of its 

 normal x, compressing B and, with increasing motion, coming into 



* Formula (3) is known to hold accurately for values of D not greater than about 

 1 per cent, of the radius of the sphere (J. P. Andrews, Phys. Soc. Proc, Vol. 42, No. 

 236). This condition is fulfilled in the case of curve I but D is as great as 10 per cent. 

 of the radius in the case of a few of the hills involved in the maximum compression 

 shown in curves // and /// (Fig. 24). 



