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BELL SYSTEM TECHNICAL JOURNAL 



displacement, when the time consumed is infinitesimal, that is when the 

 angular velocity is infinite. It is therefore an instantaneous local angular 

 stiffness from which the instantaneous local generalized stiffness -q is derived 

 as in (19a). 



To simplify these expressions, let the direction of propagation be .v and 

 that of q be y. Then 



V X 9 = f ^ {jq) =k^l, 

 ox dx 



so l^ is in the direction of s, and represents a clockwise rotation about z. 

 (5) then becomes the scaler equation 



dq_ 

 dx 



1 



d T 



pc- \_dt \ 2 



T is also in the z direction, so 



^ + 2 



dt 



(6) 



V X 



dx 



T 



. d 

 ax 



But q is in the y direction, so 



dx 



^ dt 



(7) 



These, then, are the desired equations of motion, for the type of wave 

 under consideration. 



The Generation of Reflected Waves 



In this section we shall show that when a finite wave is proi)agated in 

 this medium each element of the medium becomes the source of auxiliary 

 waves which propagate in both directions from the source. 



To do this we shall make use of the argument by which Riemann" sliowed 

 tliat this does not occur for sound waves in an ideal gas. This will first be 

 restated in more modern language. We consider a plane wave pro])agating 

 along the .v axis. We picture the finite pressure p and the longitudinal 

 velocity u at a jwint in the medium as having been built up by the successive 

 superposition of waves of infinitesimal amplitude, each propagating relative 

 to the medium in its condition at the time of its superposition. If the first 

 increment is propagating in the positive direction, 



du 



dp 

 P'" 



^ I-amh, HvdnKlvnamics, Sixth Edition, p. 481. Rayleigh, Theory of SouiicI, Second 

 Ivlilion, Vol. II. |>.\^<S. 



