548 Edward V. Lewis and John P. Breslin 
Equations (4) and (5) represent a pair of linear coupled equations in z, and 0 of the 
form 
LZ, + L,6 - 
it 
i=) 
(6) 
zene = 0 
where L, 4 are the linear second-order operators displayed in Eqs. (4) and (5). The 
solution for either z, or 0 involves integration of single, fourth-order, linear differential 
equations of the following form: 
Z 
(oe tp} ps 0 (7) 
! IC 
where (upon omission of the primes and the subscript o on 2) 
(Zeer) DZD) raz 
Eo = 2,02 (2, + 2.) +.(Z& 2) 
3 (8) 
Lz, = M.D“ +M,D+ Mm, 
EL, =. (4; ~1,)D* + (He + H,yD +. + Mg) 
and D = d/ds. 
Thus, the equations for z and @ are in the form 
Zz 
(aD* + bD3 + cD* + dD + e) \- 0 (9) 
c 
and the solutions are of the form 
4 o.s $ o,;,s 
mh YT ne oa nes (10) 
i=1 1=1 
where h is the original depth and the o,’s are the roots of the quartic equation 
aot + bo3 + co* + do +e = 0. (11) 
For stability, or no exponential divergence, a, b, c, d, and e must each be greater than zero. 
In addition, for no oscillatory divergence 
bed - (ad? + b7e) > 0. 
