USSC3002 Oscillations and Waves Lecture 11 Continuous Systems

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sghttp://www.math.nus/~matwml

Tel (65) 6874-2749

1

POTENTIAL ENERGY OF A TAUT STRING

2

Consider a string without boundary that moves in the x-y plane. If the string displacement is given by a function y = f(t,x) with support [-b(t),b(t)] and small then its length increases by

enT

dxdxtbt x

ftb

tb xf 2

21

)(

)(

21)(2)(

Therefore, if the string is taut and has tensionthen its elastic potential energy increases by

dxTftV xf

en

2

21),(

xf

KINETIC ENERGY OF A STRING

3

The kinetic energy of the string equals

where

dxftT t

ftf 2

21),,(

is its linear density (mass per length).

Therefore its Lagrangian equals

dxt,f,ftL t

fxf

tf ),(),,( L

where the Lagrangian density is defined by

221

2

21),( x

fent

ftf

xf Tt,f,

L

CONFIGURATION AND STATE VARIABLES

4

We recall that the Euler-Lagrange equations for a physical system with conservative forces and a finite dimensional configuration variable q and

0

qL

qL

dtd

Lagrangian ),,( qqtL can be expressed as

where we interpret the partial derivatives to bevector valued Frechet derivatives. This suggests that for the vibrating string we treat the Lagrangian to bea function of infinite dimensional configurationvariable f and velocity variable .t

f

Question 1. What should the EL equations be ?

EULER-LAGRANGE EOM FOR A STRING

5

are where is the Frechet

dxggdxgdxg

t

fLdtd

tf

tf

tf 2

2

)( L

derivative of L with respect to the velocity

.02

2

2

2

x

fent

f T

0

f

LLdtd

tf

tf

L

tf

hence

it is a linear functional whose value at a function g is

0

),,(),,(

0|),,(lim)(

st

fdsd

s

gsftLgsftL

s

L gsftLg tf

tf

tf

and

dxggsftLg dx

dgfst

fdsd

fL

xf

LL0|),,()(

dxgTdxg

x

fendx

df

xf 2

2LL so the EOM

is the wave equation

VARIATIONAL PROBLEMS FOR MULTIPLE INTEGRALS

6

Green’s theorem

If dxdyfyxFL yf

xf ),,,,(

dxdygg ygF

xgF

fF

fL

yf

xf

)(

gdxgdydxdygxf

xf

yf

xf

FFFy

Fxf

F

where is a bounded planar region and is its boundary oriented counterclockwise (+) direction. If g vanishes on the boundary then the Frechet derivative of L is represented by a density function

.yf

xf

Fy

Fxf

FfF

(note the new notation)

EL EOM FOR A VIBRATING MEMBRANE

7

dxdyf,yt,xL tf

yf

xf

),,,,(L

.2

2

2

2

y

f

x

ff

whose density is

having small

and with Dirichlet boundary conditions

A vibrating membrane with constant area density

RRf :

has Lagrangian

2

21

2

21

2

21),,,,( y

fenx

fent

ftf

yf

xf TTf,yt,x

L

0| fyf

xf

,

and EL EOM is fTent

ffdt

d

tf

2

2

LL

where the Laplacian

with vertical displacement

,enTand tension

FUNCTION SPACES

8

),(),(),( 2 ggffgf

Consider a bounded planar region 2R

Question 2. Show that the scalar product

and define the following space of functions

satisfies the Schwartz inequality

dydxyxfRfL ),( with :)( 22

dydxyxgyxfgfLgf ),(),(),(),(, 2

and with equality holding iff either g = 0 or.0),( agfagfRa

We define the norm

),(|||| fff

and orthogonality .0),( gfgf

FUNCTION SPACES

9

and its subspace

Define the Sobolev space (there are many others)

then

dydxLfH y

fxf 2221 : )()(

Question 3. Show that the scalar product

exists and satisfies the Schwartz inequality.

.0| : )()( 110 fHfH

dydxgfHgf y

gyf

xg

xf

11 ),(),(,

Question 4. Show that if )(),(, 10

2 HgLff

).,(),( 1 gfgf

EIGENFUNCTIONS OF THE LAPLACIAN

10

1and since

Problem 1. Find

1),( ff

The solution must satisfy

subject to the constraint

is a Lagrange multiplier. Question 3 implies

.1),( ff

),(2)(),(

gfgf

ff

where the partials denote Frechet derivatives and

),(2),(2)(),(

11 gfgfg

f

ff

it follows that if

f

ff

f

ff

),(),(1

1

1f

the minimization problem then

solves

.111

)(10 Hf that minimizes

that

EIGENFUNCTIONS OF THE LAPLACIAN

11

2 and

Problem 2. Find

1),( ff

The solution

with constraints

are Lagrange multipliers.

1),( ff)(1

0 Hf

Question 4

2

that minimizes

where 1222

and .0),( 1 f

must satisfy

),(),( 121122 Clearly

.0),(),( 12112 hence

Continuing we construct an orthonormal basis

)(2 L ,,,, 4321 for consisting of

eigenfunctions of . Also each .)(10 Hi

EXAMPLES

12

Example 1. ],0[],0[ BA

Example 2.

is a rectangle

22222 seigenvalue BnAm

Nnmyx Bn

Am ,:)sin()sin(ionseigenfunct

}1:),({ 22 yxyx is a unit disc.

1,0:)()sin(),()cos( ,, nmrkJmrkJm nmmnmm

nmnm kk ,2

, where)(seigenvalue

eigenfunctions

mJfunction Besselth -m of roots theare

NORMAL MODES

13

for zero boundary conditions on a bounded domain

Wave Equation ft

f

2

2

),()cos(),,(1

yxtyxtf ii iii

Heat Equation ftf

),(),,(1

yxeyxtf ii

ti

i

Question 5. How can i and i be determined ?

FOURIER MODES

14

can be use to expand solutions in )( 22 RL

Wave Equation ft

f

2

2

dudveevucyxtf vyuxi

Rvu

vuit )(

),( 2

22

),(Real),,(

Heat Equation ftf

Question 6. How can ),( vuc be determined ?

dudveevucyxtf vyuxi

Rvu

vut )(

),(

)(

2

22

),(Real),,(

REFLECTION AT A CHANGE OF DENSITY

15

Consider a solution of the wave equation ffor transverse displacements

2

2

2

2

x

fent

f T

on an infinite string,but whose linear densityenTwith constant tension

1)( x for 0x and 2)( x for ,0x

that has the form dtransmittereflectedinciden ffff t

Question 8. What is the physical significance ?

,0),(cos),( 1ii xxktAxtf ,0),(cos),( 1rr xxktAxtf .0),(cos),( 2tt xxktAxtf

REFLECTION AT A CHANGE OF DENSITY

16

?)0,()0,()0,( tri tftftf

?, 2,1 jjenj Tk

These two boundary conditions give

Question 9. Why does

Question 10. Why does

Question 11. Why does ?)0,()0,()0,( tri ttt xf

xf

xf

2it

2ir )(,)( AATAAR

We define coefficients of reflection & transmission21

1

i

t

21

21

i

r 2, kkk

A

A

kkkk

A

A

Question 12. What is their physical meaning ?

LONGITUDINAL WAVES IN BARS

17

In a longitudinal wave the displacement u

is stretched or compressed by the factor

is in thesame direction as the wave as shown below

),( xtT xu

en

so by

hence a small length dx of the bar between x and x+dx

x

Hook’s law results in tension

dxx

ux dxux xu )1(

xu1

at the point x where enT is the constant tension. The

net force on a length ],[ xxx (with mass

2

2

2

2

2

2

),(),(x

u

t

u

x

uxu

xu xxtxxt

x ) is

WAVES IN ELASTIC SOLIDS

18

The displacements are described by a vector function),,( 321 uuuu of a coordinate vector

For an isotropic material with Lame constants

Tension is described by the stress tensor

3,2,1,,, jijiT

that is linearly related to the strain tensor

and the wave equations are

).,,( 321 xxxx

i

j

j

i

x

u

xu

i

j

j

i

k

k

x

u

xu

k xu

ijijT

3

1

,

3,2,1,)(3

1 2

22

2

2

i

j x

uxx

u

t

u

j

i

ji

ji

TUTORIAL 11.

19

Problem 1. Fix an angle

to the functionRRf 2:

define the rotation operator

R that maps a function

RRfR 2: defined by )cossin,sincos(),( yxyxfyxfR

Show that if f is twice differentiable then

fRfR

where 2

2

211

r

g

rrg

rr rf

Problem 2. Show that if f is twice differentiable then)sin,cos(),( rrfrg

Problem 3. Use this polar coordinate expression for

to derive the following differential equations

.0,01 2

21 mJr mrm

rdJd

rdd

rm

and the properties of the Bessel functions on vufoil 12

TUTORIAL 11.

20

Problem 4. Let

be a bounded region withRg :

that minimizes

boundary

2Rbe continuous.and let

)(1 Hfsatisfies1),( ff

Prove that the functiongf |subject to the constraint

0 f (ie it is harmonic)on the interior of and satisfies the Dirichlet boundary conditions. Then prove the solution of this Laplace problem is unique.Problem 5. Derive the solution for the reflection problem on vufoil 15 if the incident wave has theform .0),(),( 1i xxktgxtf Problem 6. Use equations in vufoil 18 to computespeeds of 221 ,, uuu if u depends only on t and 1x