442 



SEISMIC METHODS 



[Chap. 9 



(1) volume changes, that is, compression or dilation; (2) shearing strains. 

 These two strains propagate in a homogeneous isotropic medium with con- 

 stant but different velocities. It is true that geologic formations en- 

 countered near the surface and in the interior of the earth are far from 

 homogeneous and isotropic. However, this can be allowed for by assuming 

 continuous or discontinuous variations of the physical properties and by 

 appljdng the theory to small elements of an elastic substance. The deriva- 

 tions given below presuppose, furthermore, a perfectly elastic body, that 

 is, a body in which the stress is proportional to the strain (one to which 

 Hooke's law may be applied), and one in which there is no elastic hysteresis. 



B. Elements of Theory of Elastic Deformation and Wave 



Propagation^ 



1. Relation of strain and stress; elastic properties; volume and shear de- 

 formations (static problem). In considering an element of volume of an 



Fig. 9-1. Normal and tangential stress components in elementary p>arallelopiped. 



elastic body subjected to stress (Fig. 9-1), its orientation can be so chosen 

 that the three stress components X^ , Y^ , and Z^ , are at right angles to 

 its surfaces, Si , Sy , and S^ . These stress components are known as 

 "normal" stresses. Two stress components exist in each surface at right 

 angles to each of the three normal stresses. These components are known 

 as "tangential" stresses and are designated Z^ , Xj ; Y^ , X„ ; and Zj, , Y^ . 



* This section is not intended to go into the detail^ of the theory of elasticity; 

 and it has, of necessity, been limited to a discussion of the definition and relation 

 of elastic moduli and wave velocities. The literature on theoretical physics contains 

 numerous valuable treatises on the subject, for instance, L. Page, Theoretical Physics, 

 pp. 132-141, Van Nostrand (1928); A. E. H. Love, A Treatise on the Mathematical 

 Theory of Elasticity (2 volumes, Cambridge, 1892-1893); W. Thomson and P. G. 

 Tait, Treatise on Natural Philosophy, vol. II, chap, vii (Cambridge, 1883). 



