Fluid Dynamics 3
2011/12
Sheet 10
Homework to be handed in Friday 20th January: questions 2,5.
1. Consider small amplitude twodimensional oscillations on the freesurface of an unbounded
fluid of
infinite
depth. At large depths, the fluid is a rest. The elevation of the freesurface
is given by
z
=
ζ
(
x, t
), and the fluid occupies
z < ζ
.
(i) Assuming incompressibility and an irrotational flow, write down the linearised equations
of motion in terms of
ζ
and a velocity potential
φ
(
x, z, t
).
(ii) Assume a timeharmonic dependence with angular frequency
ω
and wavenumber
k
of
the form
ζ
(
x, t
) =
H
sin(
kx

ωt
)
and hence show that the corresponding velocity potential is
φ
(
x, z, t
) =

ω
k
H
e
kz
cos(
kx

ωt
)
In doing so, find the dispersion relation for deep water,
ω
2
=
gk.
(iii) Assume in that
H
1, so that small amplitude waves are supposed to travel in
the positive
x
direction. Now consider the paths of fluid particles which are initially
located at
x
= (
x
0
, z
0
). Denoting their subsequent position by
x
= (
x
0
+
X
(
t
)
, z
0
+
Z
(
t
))
and on the assumption that the displacements from the initial position are small (i.e

X/x
0

1,

Z/z
0

1), show that the paths taken approximately satisfy
X
2
(
t
) +
Z
2
(
t
) =
H
2
e
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 Fall '11
 Eggers
 Fluid Dynamics, group velocity, dispersion relation, velocity potential

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