PHY 711 Fall 2016 Lecture 29 1 PHY 7 11 Classical Mechanics and Mathematical Methods 111150 AM MWF Olin 107 Plan for Lecture 2 9 Chapter 10 in F amp W Surface waves Nonlinear contributions and ID: 602482
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PHY 711 Fall 2016 -- Lecture 29
1
PHY
7
11 Classical Mechanics and Mathematical Methods
11-11:50 AM MWF Olin 107
Plan for Lecture
2
9:
Chapter 10 in F & W: Surface waves
-- Non-linear contributions and
soliton
solutionsSlide2
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2Slide3
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3Slide4
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p
0
h
z
x
z
y
General problem
including
non-
linearities
Surface waves in an incompressible fluidSlide5
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p
0
h
z
x
z
ySlide6
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p
0
h
z
x
z
y
z=0
Non-linear effects in surface waves:Slide7
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Detailed analysis of non-linear surface waves
[Note
that these
derivations
follow
Alexander
L. Fetter and John Dirk
Walecka
,
Theoretical
Mechanics of Particles and
Continua
(
McGraw Hill, 1980),
Chapt
. 10
.]
The surface of the fluid is described by
z=
h+z(x,t
). It is assumed that the fluid is contained in a structure (lake, river, swimming pool, etc.) with a
structureless bottom defined by the z = 0
plane and filled to an equilibrium height of z = h.Slide8
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Defining equations for
F
(
x,z,t
)
and
z
(
x,t
)
Bernoulli equation (assuming
irrotational
flow) and gravitation
potential energySlide9
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Boundary conditions on functions –
Zero velocity at bottom of tank:
Consistent vertical velocity at water surfaceSlide10
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Analysis assuming water height
z
is
small relative to
variations in the direction of wave
motion
(x)
Taylor’s expansion about
z
=
0:
Note that the zero vertical velocity at the bottom ensures that all odd derivatives
vanish
from
the
Taylor expansion . In addition, the Laplace equation allows us to convert all even derivatives with respect to
z
to derivatives with respect to
x
.Slide11
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11
Check linearized equations and their solutions:
Bernoulli equations --
Using Taylor's expansion results to lowest orderSlide12
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Analysis of non-linear equations --
keeping
the lowest order nonlinear terms and include up
to 4th
order derivatives in the linear terms.
The expressions keep the lowest order nonlinear terms and include up
to 4th
order derivatives in the linear terms
.Slide13
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Note that the wave
“speed”
c
will be consistently determinedSlide14
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Integrating and re-arranging coupled equationsSlide15
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Integrating and re-arranging coupled equations – continued --
Expressing modified surface velocity equation in terms of
h
(u):Slide16
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Solution of the famous
Korteweg
-de
Vries
equation
Modified surface
amplitude
equation in terms
of
h
Soliton
solutionSlide17
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17Slide18
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Relationship
to “standard”
form
of
Korteweg
-de
Vries
equationSlide19
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More detailsSlide20
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Soliton
solution
SummarySlide22
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Some links:
Website –
http://www.ma.hw.ac.uk/solitons/
Photo of canal
soliton
http://www.ma.hw.ac.uk/solitons/