174 



Subetit'^ting for R in (8) and for v in (5) fro.n (3) and (/,), we find, since 



the deterirdnant 



sin cxi , - cos '^c 



7^0 



CCS ^^ , sin o<' 



that the speed of the bending wave with respect to the material on either 

 side of it is 



dL 



dt 





and that the shear stress behind the bending wave is 



4h = ° 



(9) 



(10) 



Substituting the value of U obtained from (l) into (6) and utilizing (8) 



and (9), we find that 

 I 

 ^RR = ^R 



(11) 



so that there is no stress iiscontinuity and no shock in tliis sense at the 

 bending vrave. In addition (3) and (4) nay be combined to give 



tan o^ = 1 



R 

 Similarly (l) and (2) yield 



U^ - 2UR - v^ = 



(12) 

 (13) 



2 9 • 



while utilization of U ♦ v'- = 2U(U - li) , obtained previously fro~i 



(1) and (2), in (?), and then substituting for (U - fi) from (1) and for 



(^) from (2), yields 



dt 



P (ft2 ♦ v'''-) - Sj^ 



(u) 



Equations (9) through (IZ*) inclusive are entirely equivalent to 



the original six independent relations described by (l) through (?) inclusive. 



Clearly vie mnv regard these equations as defining the six quantities dL, S_„, 



dt ™ 



■'RR' 



, R, and U in teri.is of v and p, and Sn. 



-11- 



