Equation (2.1) is the potential equation which originally 
comes from the equation of continuity. Equation (2.2) is 
Bernoulli's equation where an arbitrary function of time has been 
neglected. Equation (2.2) need not be considered explicitly in 
solving wave problems; it simply gives the pressure after 9 has 
been obtained. Equation (2.3) is a boundary condition equation 
and states that there is no fluid motion normal to the bottom. 
Equation (2.4) is the free surface boundary condition for pressure 
continuity where 7 is the free surface. Finally, equation (2.5) 
is the kinematic boundary condition at the free surface. It 
states the condition that a particle at the free surface must 
remain at the free surface. In this paper, partial derivatives 
will be denoted by subscripts; for example, ®, means O® At. 
These equations have never been solved completely. Partial 
solutions have been obtained only after simplifying the equations. 
Even the known solutions for waves of finite height are approxi- 
mations. The difficulty arises in equation (2.4). The term 
(p,- + Py + 95° )/28 is the cause of the difficulty. It is a 
non-linear term. Suppose, for example, that Py satisfies equations 
Wel) Ce ig) Cea) ge 4) ance (2.5) and = that P5 also satisfies 
the same equations. Then Py plus 95 will not satisfy the equations, 
and P5 plus P, and 14 plus 75 have no meaning. 
Thus the original equations for wave motion are non-linear. 
At this point, then, in the study of wave motions there are two 
possible ways to proceed. One way to proceed is to concentrate 
on the non-linear properties of the equations. The second way to 
proceed is to reduce the equations to a linear form with the smatl~ 
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