Silo 
x(a,t)=a- Da. sin(k.a- w.t + €.) 
i i i i 
a.a. w.(w.+w.) 
r a) : 2 f, a : 2 
ea 5 378, sin((k; k,)a (o; ote. €)+ Za, w kit 
(29) 
z(a,t)= Za. cos(k.a-w.t + €.) 
i i i i 
aa. 5 
tes pe eaicy: cos((k,-k.)a-(w.-w.)t + €.-€.) 
oil d i jin rd i} re ia 
The parametric equations, x = x(a,t) and z = z(a,t), imply 
an inverse for x = x(a,t) such that a = a(x,t), and so 
(30) Ze ez (oxsub) eit) 
is the equation for the free surface correct to second order as obtained by 
this solution of the Lagrangian equations of motion. 
Comparison of the Lagrangian and Eulerian models 
Results obtained in the Lagrangian system of equations are 
difficult to compare with results obtained in the Eulerian system of 
equations. In the Eulerian system the fluid velocity at a fixed point below 
the surface will be Gaussian in the linear Gaussian model. The velocity 
at a fixed point in the linear Gaussian model in the Lagrangian system has 
not been found but it involves finding 
w= Gee BX, FARE ON Ore5 F45 1)n 15)) 5 
(31) ai—salcazet)ee and 
OF = 10l(ecsezeat) 
from equations (21), even in the long crested case, and this velocity is 
not Gaussian unless the waves are so low that a = x and 6 =z are suf- 
ficiently accurate inverses of x = x(a,6,t) and z = z(a,6,t). 
However, a higher order model in the Eulerian system of equations 
may well yield velocities comparable to those given by a model to a lower 
order inthe Lagrangian equations. 
The first and second order models in the Lagrangian equations 
appear to have advantages over the comparable first and second order 
models in the Eulerian equations, as will be shown later. Since the 
