Expansion of the term on the right as a power series in uuP 

 and separating real and imaginary parts gives 



(a) h=-\ l-f(^) 2 -.. 

 a 8 



(42) 



(b) a ^ = 77l 1 -8 (Lu3) 



where c is the wave speed J\ Q /p . If uuB is much less than 

 unity, then the above relations can be approximated by 



U) _ uu : P _ feu(3 , v 



k = — and civ - — — — (4 3/ 



c ^ 2,0 2 



These approximations apply as long as the viscous compo- 

 nents of the stress tensor (11) are small compared with the 

 elastic part. Thus, to this order of approximation there 

 is no influence of viscosity on the wave number-frequency 

 relation, while the spatial attenuation coefficient is directly 

 proportional to the viscosity. The above attenuation factor 

 differs from that given in the energy considerations in two 

 respects. First, the factor a enters since a^ is an attenua- 

 tion coefficient per unit length along the wave ray. Second, 

 the factor -| enters since a^- pertains to the amplitude atten- 

 uation coefficient rather than the energy attenuation. Bear- 

 ing this in mind, the result (43) is entirely consistent with 

 the earlier deductions. 



If one takes as a lower limit, \ - -2|_i / 3, then it is 

 readily shown that 



2 w ^ 



3 p„c 2 



which is the result obtained by Stokes. However, for some 

 fluids the numerical factor in (44) may be significantly 

 greater than f, in view of conditions (12). 



(44) 



43 



