306 Prof. K. Pearson, On plane waves of the [May 25, 



and the equation for normal vibrations may be written 



/. « \ d* w o / , Ci _ c dw\dw d l w , d . „ , 2N 



(x + m jj, + 3 ^ + 2/^) ^ gj, + . 5 K- + o 



7 , „ „ N d 2 w 7 dw d , „ , 2X c? 2 w 



+ A W + <) 35, tljj « + », ! ) - p le ■ 



Cv (tXJO ' 



The coefficient therefore of the term -=- (w/ + vj) is e 4- h -j- , 



dz ' dz 



and unless this vanishes there will be normal vibrations. There are 



two important cases however in which it can be made to vanish. 



First, if o- = — y a constant quantity. The expression for W will 



r 



then be of the form fx (a 2 + /3 2 ) + =- (a 2 + /3 2 ) 2 , where however the 



constants /j,' and v will not be equal to the previous fi and v. 

 This result seems noteworthy. It would appear that : if once a 

 certain definite strain be given to the medium parallel to some 

 straight line, then it is possible to send a wave of pure transverse 

 vibrations in the direction of this strain. 



For in this case the equation for w is satisfied and we have 

 similar equations to those of Art. 5 (b) for u and v. 



This result suggests various inquiries : (a) as to whether a 

 wave motion could be started which would produce or be accom- 

 panied by such a strain ; (6) as to whether such states of strain 

 permitting of transverse waves unaccompanied by normal waves 

 may not exist for one or more directions in certain bodies. 



The other case in which there would be no normal wave what- 

 ever is in a medium for which e = 0. This involves 



X + 2fi + 2c = 0. 



Hence we see that if the expression for the work contains no 

 terms of the third order (e = and e = 0) — a by no means impro- 

 bable supposition — then there will be to a high degree of approxi- 

 mation no normal wave. 



a "A -4- ^ a 



Should e = 0, our expression for v reduces to -j — — and 



thus involves only the hitherto unconsidered elastic constant g. 

 If g might be neglected as small in any case, we have for v the 

 following physical meaning — it is \ of the squared velocity of 

 propagation of waves of normal displacement in the medium. 



The above equation for w would doubtless give not uninterest- 

 ing results for the normal wave which must accompany one of 

 transverse vibration. 



