110 DYNAMICAL THEOEY OF SOUND 



In general the states of strain and stress in a body are not 

 uniform, or " homogeneous," but vary continuously from point 

 to point; but the above notions are still applicable to the 

 infinitely small elements into which the body may be conceived 

 to be divided. 



42. Elastic Constants. Potential Energy of Defor- 

 mation. 



The theory of strains is a matter of pure geometry ; that of 

 stresses one of pure statics. When we come to connect the two 

 we require some physical assumption. The usual hypothesis, 

 known as " Hooke's law," * is that the stresses are linear 

 functions of the strains. This law ceases to hold, even approxi- 

 mately, when the strains exceed certain values called the 

 "elastic limits"; but for the purposes of acoustics it may be 

 adopted without hesitation, on account of the excessive minute- 

 ness of the varying strains with which we are concerne^f. 



In an "isotropic" substance, i.e. one in which there is no 

 distinction of properties between one direction and another, the 

 principal axes of strain must evidently coincide with those of 

 stress. Moreover the principal stress p^ must involve the 

 principal strains e 2 , e 3 symmetrically, and so on. The most 

 general assumption consistent with this requirement, and with 

 Hooke's law, is of the form 



(1) 



where X, /j, are constants depending on the nature of the 

 material]:. It will be noticed that e l5 e a , e 3 are pure ratios and 

 that the dimensions of \, p are therefore those of stress, or force 



* Robert Hooke (1635 1703), professor of geometry at Gresham College 

 16651703. 



t If Hooke's law were sensibly departed from, the frequencies of the normal 

 modes of a vibrating bar would no longer be independent of the amplitude. 

 Since the ear is very sensitive to variation of pitch, this would easily be detected. 

 This remark is due to Stokes. 



J There is a great diversity of notation as regards these constants. The 

 above symbols are those introduced by G. Lame" (1795 1870), professor of 

 physics at the Ecole poly technique 1832 44. 



