9.2 NONLINEARITY OF OCEAN WAVES 
Wave theory has developed remarkably in recent decades with the assumption of a 
wave as a stochastic process. In most cases, however, the assumption is based on infini- 
tesimal amplitudes and is valid only when the wave height is small compared with the 
wave length, and wave length is small compared with the water depth. All the quadratic 
terms of the derivatives were assumed to be small and were neglected, and the 
fundamental equations of motion were linearized and the linearized potential function 
was derived. 
In this section, the results of only a few works on the investigation of the effect of 
this neglect will be reviewed to show that the effects are indeed small. 
92.1 Second Order Spectrum of Waves by L. J. Tick and Others 
To take account of quadratic terms, L. J. Tick®™°? expanded the potential function of 
a wave (x, z, t) around the mean position of the wave z = 0 by Taylor expansion, 
g(x, z,t) = o(x,z,t) + emote) Ze ng (9.1) 
z=0 0z z=0 
and carried it to the second term. Using this expression in the fundamental equations for 
waves, through the perturbation method he expressed this velocity potential as the sum of 
the linear part ¢ ;(t) and the quadratic term @2(t) as 
P(t) = Pi(t) + G2(¢). (9.2) 
As a result, the wave height is supposed to be composed of two parts, linear and 
quadratic, corresponding to these two potentials, as 
nit) = )(t) +4 (2), (9.3) 
and expressed 
n(t) = | eTIgE (wy). (9.4) 
Here 
EfidE!?] = s%w)dw, (9.5) 
268 
