4 ioir Gr. Greenhill on 



4. The proof of the formula in (1) § 2 is given in a 

 combination of the theory of the elastic lateral vibration of 

 the sheet of ice, propagated in the direction of the waves,, 

 with a varying upward pressure hp (gravitation), derived 

 by hydrodynamical theory from the wave-motion in the 

 water below. 



Then, if rj is the elevation of the ice at a distance x, a- the 

 superficial density in g/cm 2 , and EI the flexure rigidity, in 

 gravitation g/cm, 



«S-^S+^ a> 



'$-$->" & 



<f> denoting the velocity function of the wave-motion. 

 Writing m for — , and n for — — - , — = U, assume 



v = b sin (mar— n*), ^ = - w «7> ^ = m ^ 5 W 



and in water of depth h, take 



<£ = Bch??i(j/ + A) cos (m# — ?i£), ... (4) 

 so that just under the ice, y — 0, 

 -T- = mBshmh cos (mx—nt) 



y dv 



= — -j —nb cos (mx—nt), ?nBsh m/i = nb, , . (5) 



^ -d i. 7 • / *\ ^ 2 coth?n7i rax 



-J-z= nnchmhsm(nix--iit) = r), . . . {of 



tilt wi 



9 P 



^ = (~cothmfc-VU. . (7) 



p \m * J 



Substituting in (1), dividing out ij, 



— n 2 ar =— #EIm 4 + — p coth m7i — gp, . . (8) 

 9 -^ + Im 3 ^- 



tj»»!L^J2 £_, (9) 



com ??i/i + m - 



