MEK4350
Karsten Trulsen
1
Nonlinear resonance conditions
We inquire if three or four waves can resonantly interact such that the conditions
k1 + k2 = k3
and
ω1 + ω2 = ω3
(1)
k1 + k2 = k3 + k4
and
ω1 + ω2 = ω3 + ω4
(2)
or
are satisfied. Here ωn and kn are related according to the dispersion relation
γ
ω 2 = (gk + k 3 ) tanh(kh).
ρ
1.1
(3)
Three wave resonance of long waves
We notice that for non-dispersive waves, ω = αk with α a constant, both of the
resonance conditions (1) and (2) are trivially satisfied for co-linear wave vectors.
The dispersion relation (3) is approximately non-dispersive for long waves. This
is seen by Taylor expansion of (3) around k = 0
√ p
gh 3γ
2
(4)
− h k 3 + O(k 5 ),
ω = ghk +
6
ρg
to second order in k equation (4) is non-dispersive.
It is even possible to achieve non-dispersive gravity–capillary
waves on finite
q
3γ
depth, accurate to fourth order in k, simply set h =
. With typical values
ρg
g = 9.81 m/s2 , ρ = 998 kg/m3 and γ = 0.0728 N/m (values for 20◦ C), we get the
target depth h = 4.7 mm, which is not very interesting for ocean waves, but might
be quite interesting for waves that occur on paved roads on a rainy day.
1.2
Three wave resonance of one long and two short waves
In equation (1) let k1 ≈ k3 and thus k2 be small. In this case there can be three-wave
resonance if the phase speed of the long wave (wave 2) is equal to the component of
the group velocity of the short waves (e.g. wave 1) in the direction of the long wave.
This is seen by Taylor-expanding ω3 around k1
∂ω ω3 = ω1 + k2 ·
+ O(k22 )
(5)
∂k
k1
thus we have to leading order
c2 ≡
ω2
k2
=
· cg1 .
k2
k2
(6)
It is interesting to show that the dispersion relation (3) allows this condition to
be satisfied!
1
5
ky/k1
4
3
2
1
0
0
0.5
1
1.5 2
kx/k1
2.5
3
3.5
Figure 1: Resonant triad of gravity–capillary waves on infinite depth, normalized
against k1 with unit length along horizontal axis: Γ = 0.5 (green); 1 (blue); 2 (red).
1.3
Three wave resonance of deep-water capillary–gravity
waves
γk2
Consider infinite depth and set Γ = ρg1 . In figure 1 wave vector k1 is oriented along
the first axis and has been normalized to unit length. The green, blue and red curves
show the location where wave vectors k2 and k3 meet for Γ having values 0.5, 1 and
2.
Resonant triads of gravity–capillary waves were investigated by McGoldrick (1965).
1.3.1
Three wave resonance of two unidirectional capillary–gravity waves
(Wilton’s ripples)
Consider infinite depth and let k1 = k2 . In q
this case it can be shown that the
ρg
resonance condition (1) is satisfied when k = 2γ
. With typical values g = 9.81
m/s2 , ρ = 998 kg/m3 and γ = 0.0728 N/m (values for 20◦ C), we get k1 = 259 m−1
= 2.4 cm and short wavelength
which corresponds to the long wavelength λ1 = 2π
k1
2π
λ3 = k3 = 1.2 cm.
These waves are commonly called Wilton’s ripples, after Wilton (1915), although
they were previously described by Harrison (1909).
1.4
No three-wave resonances of deep-water gravity waves
Assume the depth is infinite and capillarity can be neglected, then the dispersion
relation is ω 2 = gk. It can be shown that there are no resonant triads gravity waves.
One way to show this is to let the angle between wave vectors k1 and k2 be θ,
eliminate k3 from equations (1) by the expression k32 = k12 + k22 + 2k1 k2 cos θ, and
derive the following expression for the angle
cos θ =
2(k1 + k2 )
√
+ 3 ≥ 3.
k1 k2
This is clearly impossible.
2
(7)
ky/k1
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.5
1
1.5
kx/k1
2
2.5
Figure 2: Resonant quartet of gravity waves on infinite depth.
The nonexistence of resonant triads of gravity waves on deep water was shown
by Phillips (1960).
1.5
Quartet resonance of gravity waves
Let us simplify the problem by letting k1 = k2 = (k, 0), let k3 = k(1 + x, y) and let
k4 = k(1 − x, −y). In the case of infinite depth the resonance condition (2) requires
that
1 1
(1 + x)2 + y 2 4 + (1 − x)2 + y 2 4 = 2.
(8)
For y = 0 we have the three solutions x = 0 and x = ± 54 . The full solution for
infinite depth is shown in figure 2 where the first axis is x and the second axis is y.
This figure is known as the “figure 8” of Phillips after Phillips (1960).
It is interesting to show how equation (8) and figure 2 are modified as the depth
decreases from infinite to small!
References
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite
amplitude. Proc. Lond. Math. Soc. 7, 107–121.
McGoldrick, L. F. 1965 Resonant interactions among capillary–gravity waves.
J. Fluid Mech. 21, 305–331.
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688–700.
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