Brennen and Whitney 



of surface tension the free surface condition reduces to 



^tt "*" S^a ~ ^ (g = 1 ii^ the dimensionless variables) (30) 



only when the additional assumption that "Hi^ « g is made. In this 

 way harmonic solutions can be obtained for some simple problems. 



In passing, it may be of interest to compare Lagrajigian 

 linearization with the more common Eulerian type, at least in some 

 simple cases. For travelling waves on an infinite ocean the first 

 order Lagrangian terms are precisely those of Gerstner's waves. 

 The Eulerian solution must be taken to the third order to achieve this 

 waveform. On the other hand, while the Eulerian solution is always 

 irrotational the Lagrangian only approaches it. Thus the comparitive 

 accuracy of the two methods depends upon what particular feature of 

 the flow is under scrutiny. A comparison of the works of Zen'kovich 

 [ 1947] and Penney and Price [1952] for standing waves on an infinite 

 ocean demonstrates the same features. 



B. Example One, Figs. 4(a), 5, 6, 7, 8, 9, and 10 



In the example of Fig. 4(a), the liquid is initially at rest in 

 Ihe rectangular vessel BCDA; between t = and t = T the side EC 

 moves inward according to 



Xgj^t) = M sin^ TTt/ZT for < t < T 



= M for t > T 



With a suitable choice of M and T this creates a wave which 

 travels along the box, builds up on and is reflected by the opposite 

 wall, AD. The linearized solution (which requires a Fourier 

 analysis of the free surface boundary condition) is 



CD 



Z - c = Xedlt) [l - ^] + ^ R^B^(t) sin (^) (31) 



k=l 



where 



2 2 



R^ = MAk(^-i)cosh(i^) 



B^(t) = cos V|^t - cos ^ , < t < T 



= cos Vj^t + cos v^{t - T), t > T 



= r^ tanh ^1 



134 



