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or using the Bernoulli theorem for irrotational motion, (10.2.30), for a fluid of
constant density and a gravitational potential ! = gz ,

!" #" 2 p
+ + g$ + a = 0 (10.4.7)
!t 2 %

We will only consider the relatively easy problem of small amplitude motions

when the wave amplitudes are small enough so that the nonlinear terms is the

equations are negligible compared to the linear terms. When will that be so? Let’s
suppose that the characteristic magnitude of the velocity of the fluid elements in the

wave is characterized by a scale U. Suppose the period of the wave is measured by a
scale T and the wavelength of the wave is of order L. Then !" = O(U) which

implies that ! = O(UL)so the condition,

#"
| !" | <<
2
#t
(10.4.8 a, b)
L
2
$U << U
T
or,

L
U << = c (10.4.9)
T

so that linearization is possible only if the velocity of fluid elements is small
compared with the phase speed (the ratio of wavelength to period) of the wave. So,

if we ignore terms that are quadratic in the amplitude of the wave, the boundary
conditions become, at z = η,

!" !#
= ,
!z !t
(10.4.10 a, b)
!" p
+ g# + a = 0.
!t $

Chapter 10 22