physics 451/551 theoretical mechanics · moves with the fluid. if a fluid is stationary and acted...

Theoretical Mechanics Fall 2018

Physics 451/551

Theoretical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 18


Sound Waves

• Properties of Sound

– Requires medium for propagation

– Mainly longitudinal (displacement along propagation

direction)

– Wavelength much longer than interatomic spacing so

can treat medium as continuous

• Fundamental functions

– Mass density

– Velocity field

• Two fundamental equations

– Continuity equation (Conservation of mass)

– Velocity equation (Conservation of momentum)

• Newton’s Law in disguise

, , ,v x y z t

, , ,x y z t


Fundamental Functions

• Density ρ(x,y,z), mass per unit volume

• Velocity field

0 , , , lim

, , ,

V

Mx y z t

V

dM x y z t dxdydz

, , ,v x y z t

o

, , ,v x y z t

, ,x y z


Continuity Equation

• Consider mass entering differential volume element

• Mass entering box in a short time Δt

• Take limit Δt→0

2

, , , , , ,

, , , , , ,

, , , , , ,

, , , , , ,

x x

y y

z z

v

dV

v x y z t v x dx y z t dydz t

v x y z t v x y dy z t dzdx t

v x y z t v x y z dz t dxdy t

x y z t t x y z t dxdydz t

dx

dy dz , ,x y z


• By Stoke’s Theorem. Because true for all dV

• Mass current density (flux) (kg/(sec m2))

• Sometimes rendered in terms of the total time derivative

(moving along with the flow)

• Incompressible flow and ρ constant

dV dV

dV dxdydz v dxdydzt t

mJ v

0vt

0d

v v vt dt

0v


Pressure Scalar

• Displace material from a small volume dV with sides given

by dA. The pressure p is defined to the force acting on the

area element

– Pressure is normal to the area element

– Doesn’t depend on orientation of volume

• External forces (e.g., gravitational force) must be balanced

by a pressure gradient to get a stationary fluid in

equilibrium

• Pressure force (per unit volume)

dFp

dA

pr

pF

x


Hydrostatic Equilibrium

• Fluid at rest

• Fluid in motion

• As with density use total derivative (sometimes called

material derivative or convective derivative)

0app

app

f p dV

pf

net app

dv dvF p f dV m dV

dt dt

dv v

v vdt t


Fluid Dynamic Equations

• Manipulate with vector identity

• Final velocity equation

• One more thing: equation of state relating p and ρ

2

v vv v v v

app

dv v pv v f

dt t

2

app

v v v pv v f

t


Energy Conservation

• For energy in a fixed volume

ε internal energy per unit mass

• Work done (first law in co-moving frame)

• Isentropic process (s constant, no heat transfer in)

23

2tot

V

vE d x

2

2,

MpMd pdV d

ps d

2

p

t t


2 2

2 2

1 1

2 2

1 1

2 2

1

2

app

app

E

v v v v p v ft

p pv p v v

t

p p

p v v pt

v v p v f vt

j v

2 p


Bernoulli’s Theorem

• Exact first integral of velocity equation when

– Irrotational motion

– External force conservative

– Flow incompressible with fixed ρ

• Bernouli’s Theorem

• If flow compressible but isentropic

appf U

0v v

2

02

pU

t

2

02

pU

t


Kelvin’s Theorem on Circulation

• Already discussed this in the Arnold material

• To linear order

1

v

C t C

dv v v pv U

dt t t

t ds v

, , , ,

,, ,

,, ,

C s t t C s t t v C s t t

C s t tt t v C s t t t t ds

s

C s tt v C s t t ds

s


• The circulation is constant about any closed curve that

moves with the fluid. If a fluid is stationary and acted on

by a conservative force, the flow in a simply connected

region necessarily remains irrotational.

2

, , , , , ,

2

0 (the integrand is exact!)

d dv CC s t t ds v C s t t v C s t t ds

dt dt s s

vp C CU ds ds

s s


Lagrangian for Isentropic Flow

• Two independent field variables: ρ and Φ

• Lagrangian density

• Canonical momenta

2

0

02

t

pU

t

2

2U

t

L

/

0/

t

t

LP

LP


• Euler Lagrange Equations

• Hamiltonian Density

internal energy plus potential energy plus kinetic energy

2

0 0

02

t t

pU

t t

p

L LP

L LP

2

2U

t t

H P P L


Sound Waves

• Linearize about a uniform stationary state

• Continuity equation

• Velocity equation

• Isentropic equation of state

0 0, , 0 ,x t v x t v p x t p p

0

0

10 0 0v v

t t

0

1vp

t

2

0 0 0,s

pp p p s p p c


Flow Irrotational

• Take curl of velocity equation. Conclude flow irrotational

• Scalar wave equation

• Boundary conditions

0

2 2

2

0 0

1

v pv

t t t

p c

t t t

2 2

2 2

2 2 2

1c

t c t

ˆ ˆ 0 for a fixed boundary

0 free surface

n n V

t


3-D Plane Wave Solutions

• Ansatz

• Energy flux

22

0 02

0

2

0 02

0

, Rei k x t

e

kc

v ik

c ii v ik

c

2* 2

0 0 0 0 0

1 1 ˆRe2 2

Ej ik i k c kt


Helmholz Equation and Organ Pipes

• Velocity potential solves Helmholtz equation

• BCs

• Cylindrical Solutions

2 2 0r k r

0 0 0,r zv r a v z Lr z

2 22

2 2 2

1 10

, ,

cos zero possibleim

r kr r r r z

r z R r F Z z

pF e Z z z

L


Bessel Function Solutions

• Bessel Functions solve

• Eigenfunctions

• Fundamental

• Open ended

22 2 2 2

2

10 m m

d rd mr J r J r k

r dr dr r

0

2 2

2 2 2

, Re cos expmnmnp m

mnmnp mnp

pr t J r z im i t

a L

pc k c

a L

011 011ck cL

010 0102

cck

L


Green Function for Wave Equation

• Green Function in 3-D

• Apply Fourier Transforms

• Fourier transform equation to solve and integrate by parts

twice

2 2u r u r f r

3

3

3

1

2

ip r

ip r

f p d re f r

f r d pe f p

2 2p u p u p f p


Green Function Solution

• The Fourier transform of the solution is

• The solution is

• The Green function is

2 2

f pu p

p

3

3 2 2

1

2

ip xf p

u r e d pp

3 3

3 2 2

3

3 2 2

1 1

2

1 1

2

ip r ip r

ip r ip r

u r e e d pf r d rp

G r r e e d pp


• Alternate equation for Green function

• Simplify

• Yukawa potential (Green function)

2 2 3

3

1

2

ip r ip rG r r e e d p r r

cos cos3 2

3 22 2 2 2

0 0

2 22 2 2 2

0

1 1sin

2 2

1 2 sin 1 sin

42 2

ipR ipR

R

e eG R d p p dp d

p p

p pR p pR edp dp

RR p R p

4

r re

G r rr r


Helmholtz Equation

• Driven (Inhomogeneous) Wave Equation

• Time Fourier Transform

• Wave Equation Fourier Transformed

2

2 2

1, ,r t f r t

c t

1, ,

2

1, ,

2

i t

i t

r t d e r

f r t d e f r

2

2

2, ,r f r

c


Green Function

• Green function satisfies

3

22

2

3

4 22

2

3 3

4 22

2

, , ,

, ,

,1,

2

1 1, ,

2

i k r t

i k r t i k r t

r t d r dt G r r t t f r t

k k f kc

f kr t d k d e

kc

r t d r dt d k d e e f r t

kc


• Green function is

• Satisfies

• Also, with causal boundary conditions is

3

4 22

2

1,

2

i k r t i k r te e

G r r t t d k d

kc

2

2 2

1,G r r t t r r t t

c t

/

,4

i r r ce

G r rr r


Causal Boundary Conditions

• Can get causal B. C. by correct pole choice

• Gives so-called retarded Green function

• Green function evaluated

ω k plane

kc i

kc i

/i c

/

i c

3

3 22 2

/

22 2 2

,2 /

1

8 4/

ik R

ikR ikR i R c

d k eG R

k i c

e e ekdk

iR Rk i c


Method of Images

• Suppose have homogeneous boundary conditions on the x-

y half plane. The can solve the problem by making an

image source and making a combined Green function. The

rigid boundary solution has

• To satisfy the boundary condition so that the solution

vanishes on the boundary

//

, ,4 4

i r r ci r r c

x y z

e eG r r r r r r

r r r r

//

, ,4 4

i r r ci r r c

x y z

e eG r r r r r r

r r r r


Kirchhoff’s Approximation

• We all know sound waves diffract (easily pass around

corners). Standard approximation “schema”

• Zeroth solution the Image GF

• Boundary condition not correct at hole

2 2

0r k r r r

//

, ,4 4

i r r ci r r c

x y z

e eG r r r r r r

r r r r

3 2 2

3 2 2 3 2 2

lim

V A

RR H

d r dA

d r G G d r k G r r Gk r

dA G G dA G


In RHP

• Exact relation

• For short wavelengths, evaluate RHS as if screen not there!

Huygens’ Principle

2

ik r r

H H

er dA G dA

r r z

0

0

2

0 08

ik r rik r r

H H

zik e er dA G dA

r r r r r r


Babinet’s Principle

• Apply Green’s identity

0

0

0

0

0

0

,4

,4

,4

ik r r

H

ik r r

P H

ik r r

P

er dA G r r

z r r

er dA G r r

z r r

er r dA G r r

z r r

0

0

0

0

,4

4

ik r r

inc

ik r r

inc

diff diff

er r r G r r

r r

er r r

r r


Diffracted Amplitude

• Fresnel diffraction: phase shifts across the aperture

important. Full integral must be completed

• Fraunhofer diffraction

Pattern is the transverse Fourier Transform!

0

22

0

0 0 0

0

22

ˆ ˆ0

02 220

0

ˆˆ

2

ˆ2

cos exp8 ˆ

2

ik r r r rikr k k

H

r r rr r r r r

r

ikr k r

rik er dAe

ikr rr k r

r

0 0ˆ ˆ

2 2

0 08 8

ik r r ik r rikr k k ir q

H H

ik e ik er dAe dAe

r r r r

2 2/ 1 / 1ka r ka r


Two Cases

• Rectangular aperture

• Destructive interference at qxa=π

• Circular aperture

• Airy disk (angle of first zero)

22

0

sinsin yx

x y

q bq aI r I

q a q b

2

10

2 sinJ q aI r I

q a

sin 0.61a


Equation for Heat Conduction

• Field variable: temperature scalar

• Additional inputs: heat capacity (at constant pressure) cp,

thermal conductivity kth

• Thermal diffusivity

• Heat Equation

p

H th

dT c dE

j k T

th

p

k

c

2

p

T qT

t c


Boundary Conditions

• Closed boundary surface held at constant Tex

• Insulating surface

• Separate variables

• Helmholtz again

0n T

, tT r t T r e

2 2

p

qT r k T r

c


Long Rectangular Rod

• Long ends held at temperature T0

• Eigensolutions

2 2 2

2

sin 1, 2,3

cos 0,1,2,3

cos 0,1,2,3

mnp

T r X x Y y Z z

m xX x m

a

n yY y n

b

p zZ z p

c

m n pk

a b c


General Solution

• Find expansion coefficients with the orthogonality

relations

• Long term solution dominated by slowest decaying mode

, sin cos cos

, 0 sin cos cos

mnpt

mnp

mnp

mnp

mnp

m x n y p zT r t C e

a b c

m x n y p zT r t C

a b c

100

0 100, sintx

T r t T C ea


Thermal Waves

• Put periodic boundary condition on plane z = 0

• 1-D problem

00, cosT z t T t

2

2

2

2

2

1

, Re

1

2

i t

z

T T

z t

T z t T z e

d T iT

dz

T z e

i i


Penetration Depth

• Exponential falloff length (for amplitude)

• Solution for thermal wave

• On earth, 3.2 m with a one year period!

1/2 1/22 T

/

0, cosz zT z t T e t


Green Function for Heat Equation

• Fourier Transform spatial dependence

• Solve using initial condition

2

2

2,

,

, k t

TT

t

T k tk T k t

t

T k t A k e

2

2

3

3 3

3

3

3

, 0 , 0

1, , 0

2

1,

2

ik r

ik r k t ik r

ik r rk t

T r t e d r T k t A k

T r t T r t e d r e e d k

G r r t e e d k


• Complete the square

2

2 2

22

cos 2

3

1

cos 2

2 2

0

1 1/2

2 2

1, cos

2

1cos

2 2

2 2

ik r rk t

ik r r ik r r ik r rk t k t

t k ik r r t rik r rk t

G r r t e e k dkd d

r re e k dkd e e e kdk

i

r r r re e kdk e

i i

2 2 2 2

2 22

/4 /4

1

/4 /4

2 3/2

/ 2

1/ 2

2 4

r t r r t

r r t r r tl

kdk e

l t k i r r t

r r dlG e e l i r r t e

ti t

physics 451/551 theoretical mechanics · moves with the fluid. if a fluid is stationary and acted...

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