6 Nonlinear waves
Note: this material will not be examined
6.1 Quadratic corrections
So far we have restricted ourselves to the case of waves of small amplitude, which means
we only took terms linear in the wave amplitude into account.
Now we will consider
quadratic terms in the boundary conditions. Comparing their size to that of the leading
linear calculation will permit us to give a more precise meaning to what we shall call
“small amplitude”.
As an example, consider the kinematic condition (53), where the velocities
u
and
w
are to be evaluated at the free surface
z
=
ζ
. As discussed before, this can be reduced to
quantities evaluated at a fixed position
z
= 0 by performing a Taylor expansion:
∂φ
∂x
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=
ζ
=
∂φ
∂x
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=0
+
ζ
∂
2
φ
∂x∂z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=0
+
...,
∂φ
∂z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=
ζ
=
∂φ
∂z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=0
+
ζ
∂
2
φ
∂z
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
=0
+
....
Thus (53), including terms up to second order, becomes
∂ζ
∂t
+
∂φ
∂x
∂ζ
∂x
=
∂φ
∂z
+
ζ
∂
2
φ
∂z
2
,
on
z
= 0
,
(60)
which means there are two new quadratic terms.
Now we insert the solution found in section 5.3, and compare the relative size of linear
and quadratic terms. For example, we find
∂φ
∂z
=
Z
′
(0) cos Φ =
−
gHk
ω
tanh
kh
cos Φ =
−
Hω
cos Φ
,
where we wrote Φ =
kx
−
ωt
for the phase factor for brevity.
On the other hand, the
nonlinear term on the left of (60) is
∂φ
∂x
∂ζ
∂x
=
−
k
2
HZ
(0) sin Φ cos Φ =
gH
2
k
2
2
ω
sin 2Φ
.
Thus the condition for the amplitude of the nonlinear term to be small relative to the
linear one, assuming formally cos Φ and sin 2Φ assume their maximum values, is
gH
2
k
2
2
ω
2
H
=
Hk
2 tanh
kh
≪
1
.
(61)
In the case of infinite depth
h
→ ∞
this simplifies to
Hk/
2
≪
1. In other words, the
small parameter that measures the size of the nonlinearity is
Hk
, staying with the case
of infinite depth for the rest of this chapter.
6.2 Perturbation theory
Our aim is to include terms of higher order in the amplitude in a systematic fashion, as
was done first by Stokes in 1847. The resulting finiteamplitude waves are therefore often
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called
Stokes’ waves
. From now on, we will consider the case of infinite depth and the
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 Fall '11
 Eggers
 Fluid Dynamics, Cos, Boundary value problem, Nonlinear system

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