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11/09/2016

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

solutions

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2

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4

p

0

h

z

x

z

y

General problem

including

non-

linearities

Surface waves in an incompressible fluid

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5

p

0

h

z

x

z

y

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6

p

0

h

z

x

z

y

z=0

Non-linear effects in surface waves:

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7

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.

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8

Defining equations for F(x,z,t) and z(x,t)

Bernoulli equation (assuming

irrotational

flow) and gravitation

potential energy

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9

Boundary conditions on functions –Zero velocity at bottom of tank:

Consistent vertical velocity at water surface

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10

Analysis assuming water height

z is small relative tovariations 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 zto derivatives with respect to x.

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11

Check linearized equations and their solutions: Bernoulli equations --

Using Taylor's expansion results to lowest order

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12

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

.

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13

Note that the wave

“speed”

c

will be consistently determined

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14

Integrating and re-arranging coupled equations

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15

Integrating and re-arranging coupled equations – continued --Expressing modified surface velocity equation in terms of h(u):

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16

Solution of the famous Korteweg-de Vries equationModified surface amplitude equation in terms of h

Soliton

solution

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18

Relationship to “standard” form of Korteweg-de Vries equation

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19

More details

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21

Soliton solution

Summary

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22

Some links: Website – http://www.ma.hw.ac.uk/solitons/

Photo of canal

soliton

http://www.ma.hw.ac.uk/solitons/

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