65.8 EXPLORATION GEOPHYSICS 



Velocities of Elastic Waves and Elastic Constants. — The elastic 

 wave velocities of any medium depend on the elastic constants and density 

 of that medium.* In the case of a homogeneous isotropic solid, the elastic 

 properties are described by two moduli of elasticity. The two moduli most 

 frequently used are: the compressibility k and rigidity n, or Lame's con- 

 stants A and /*, or Young's modulus E and Poisson's ratio a. t These sets 

 of moduli are not independent of each other. Thus 



r, .a. 2 E , , 2 aE 



« — A + -Z- /A = -ZJZ TT^ , \= k — n = 



n = fx 



3 '^ 3(1 -2a)' 3 (1+ a) (1-2(7) 



jj_ fi(3\ + 2fi) _ 9kn 



X + [i 3k + n ^ ^ 



X. 3k — 2n 



2(1+ a) 2{X + fi) 6k + 2n 



A comprehensive investigation of the elastic constants was the research program 

 sponsored by the Harvard University Committee for Geophysical Research4 The 

 investigations included static and dynamic laboratory measurements of elastic constants 

 and field measurements of the wave velocities in rock bodies from which the laboratory 

 specimens were taken.§ 



The velocities of longitudinal and transverse waves in an isotropic 

 homogeneous solid are related to the elastic constants and the density 

 by the equations :f f 



V - -v/ E(l-a) J A + 2/x _ J k + 4/3n 



^-^d(l + CT){l-2a)-^ d -^ d ^^^ 



-Vf[5(,V^] = Vf=Vf (9a) 



where d denotes the density, V the velocity of the longitudinal wave, and 

 V the velocity of the transverse wave. 



The elastic constant E varies over a wide range while ct is very nearly 



* The expression elastic constants of a medium refers to the strains produced 

 in the medium by the application of stresses. 



t Compare A. B. Broughton Edge and T. H. Laby, Geophysical Prospecting, pp. 330-331 

 (Cambr. Univ. Press, 1931). 



t L. Don Leet, Practical Seismology and Seismic Prospecting (Appleton Century, 1938) p. 94. 

 § W. A. Zisman, "Young's Modulus and Poisson's Ratio with Reference to Geophysical Appli- 

 cations," Proc. Nat. Acad. Sci., 19, pp. 653-665, 1933; "Compressibility and Anistropy of Rocks 

 at and near the Earth's Surface," Proc. Nat. Acad. Sci., 19, pp. 666-679, 1933; "An Improved 

 Apparatus for the Measurement of Poisson's Ratio," Rev. of Sci. Inst., 4, pp. 342-344, 1933; 

 "Comparison of the Statically and Seismologically Determined Elastic Constants of Rocks," Proc. 

 Nat. Acad. Sci., 19, pp. 680-686, 1933. 



John M. Ide, "Some Dynamic Methods for the Determination of Young's Modulus," Rev. 

 of Sci. Inst., 6, pp. 296-298, 1935; "The Elastic Properties of Rocks: A Correlation of Theory and 

 Experiment," Proc. Nat. Acad. Sci., 22, pp. 482-496, 1936; "Comparison of Statically and Dynami- 

 cally Determined Young's Modulus of Rocks," Proc. Nat. Acad. Sci., 22, pp. 81-92, 1936. 



L. D. Leet, "Velocity of Elastic Waves in Granite and Norite," Physics, 4, pp. 375-385, 1933. 

 F. Birch and R. R. Law, "The Measurement of Compressibility at High Temperature and 

 High Pressures," Bull. Geol. Soc. America, 46, pp. 1219-1250, 1935. 



F. Birch and R. B. Dow, "Compressibility of Glasses and Rocks at High Temperatures and 

 Pressures," Bull. Geol. Soc. America, 47, pp. 1235-1255, 1936. 

 tt Broughton Edge and Laby, loc. cit. 



