Skating on Thin Ice. 9 



With a solution rj = £""*', the auxiliary equation (A.E.) of 

 (2) becomes 



~ke 3 m^-cem 2 + l = 0, (4) 



JLZ 



(i-! c *) 2= H c2 -H' • • (5) 



requiring 



c>\ke, Y >f>v /(lke)>p^/ e ^ > . . (6) 



for the ice to become unstable, and the surface to buckle up 

 into waves. 



The vibration equation in (8) § 4 can be adjusted so as to 

 introduce the term due to P : it makes 



n 2 

 — n 2 a=—gl$Im* + c/Y em 2 + —p coth mh--gp, . . (7) 



1 P ^_ E T 3 



9 gm coth mh + m~ 



P 



cem + ^ ke*m z 1 — cem 2 4- m* kim 



m 12 12 



(8) 



coth mh + me m coth mh + em 2 ' 



but the extra complication does not repay investigation. 



10. For the deflexion of ice at rest, bent into a cylindrical 

 surface of straight waves, by a single line-load over 0, such 

 as a log, equation (1) §4 becomes 



ia = _i2 , jUp-i, i -(£*)*.. (2) 



17 da 4, ke* * ' 12 4^ 4 5 \3 / 



suppose; so that q is the reciprocal of a length, and for 

 positive x and rj finite the solution is 



97 = £-2* (A cosqx + B sin qx) (3) 



To make -^ = at a* = 0, A=B; and measuring rj 

 ax 

 upward, 



rj = —ae~* x (cosqx-t-smqx), .... (4) 

 ~ = 2aqe~ qx sin gar, -r^ = 2aq 2 e~ qx ( cos $*— sin £^)> 



d 3 ?; 



t4 = — 4a<? 3 e- ?x cos qx, [ r\dx — - e~ gx cos <?# . 



(5) 



