11/11/2013phy 711 fall 2013 -- lecture 291 phy 711 classical mechanics and mathematical methods...
TRANSCRIPT
![Page 1: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/1.jpg)
PHY 711 Fall 2013 -- Lecture 29 111/11/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 29 --
Chap. 9: Wave equation for sound
1. Standing waves
2. Green’s function for wave equation; wave scattering
![Page 2: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/2.jpg)
PHY 711 Fall 2013 -- Lecture 29 211/11/2013
![Page 3: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/3.jpg)
PHY 711 Fall 2013 -- Lecture 29 311/11/2013
![Page 4: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/4.jpg)
PHY 711 Fall 2013 -- Lecture 29 411/11/2013
![Page 5: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/5.jpg)
PHY 711 Fall 2013 -- Lecture 29 511/11/2013
Comments about exam
Note that this figure is misleading; a>b for physical case
12/cos2
2/sinsin
1sin
sin
sinsin
22
2
202
2
21
an
naab
nE
U
vv
m
bd
dd
d
db
d
dbb
d
d
T
0
2
2
sin2
1
sin
![Page 6: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/6.jpg)
PHY 711 Fall 2013 -- Lecture 29 611/11/2013
Comment about exam – continued
mgkT
pss
dt
dp
s
gkT
ms
p
dt
sd
dt
dp
sgkTqVp
ms
pH
s
s
s
2
0
2
2
2
2
2
2
2
for 0 : thatNote
ln22
120
2
102
10
2
21
2
10 : thatSuppose
sms
p
ss
gkT
mss
p
dt
sd
tssts
![Page 7: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/7.jpg)
PHY 711 Fall 2013 -- Lecture 29 711/11/2013
Comment about exam – continued
![Page 8: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/8.jpg)
PHY 711 Fall 2013 -- Lecture 29 811/11/2013
Linearization of the fluid dynamics relations:
0 :equation Continuity
:motion ofequation Euler -Newton
v
fvvv
t
p
t applied
0
0
:mequilibriuNear
0
0
applied
ppp
f
vv
![Page 9: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/9.jpg)
PHY 711 Fall 2013 -- Lecture 29 911/11/2013
0
:onperturbatiin order lowest toEquations
0
0
v
v
t
p
t
v :potentialVelocity
2
,, 00
:density theof in terms Pressure
cp
pps
0222
2
ct
![Page 10: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/10.jpg)
PHY 711 Fall 2013 -- Lecture 29 1011/11/2013
v0
02
222
2
Here,
0
:airfor equation Wave
pp
c
ct
s
0 0
:surface Free
ˆ ˆ ˆ
:y at velocit moving ˆ normal with surface leImpenetrab
:aluesBoundary v
0
t
Φp
nvnVn
Vn
![Page 11: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/11.jpg)
PHY 711 Fall 2013 -- Lecture 29 1111/11/2013
Time harmonic standing waves in a pipe
L
a
0222
2
ct
0 :surface freeAt
0ˆ :surface fixedAt
:aluesBoundary v
t
Φ
n
![Page 12: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/12.jpg)
PHY 711 Fall 2013 -- Lecture 29 1211/11/2013
0),,,(11
11
)()()()()()(),,,(
:scoordinate lcylindricaIn
22
2
2
2
2
2
2
2
2
22
tzrkzrr
rrr
zrrr
rr
ezZFrRezZFrRtzr ikctti
ck
tc
:Define 01
2
2
22
![Page 13: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/13.jpg)
PHY 711 Fall 2013 -- Lecture 29 1311/11/2013
0),,,(11 2
2
2
2
2
2
tzrkzrr
rrr
tiezZFrRtzr )()()(),,,(
integer)2()( ;)( mNFFeF im
nsrestrictioother plus real ;)( ziezZ
0)(1 22
2
2
2
2
rRk
r
m
dr
d
rdr
d
![Page 14: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/14.jpg)
PHY 711 Fall 2013 -- Lecture 29 1411/11/2013
a
x'
dx
)(x'dJrJrR
dr
dR
rRr
m
dr
d
rdr
d
kk
mnmn
mnmm
ar
,0for where)()(
0 :conditionsboundary surfaceCylinder
0)(1
define For
22
2
2
2
22222
0)(1 22
2
2
2
2
rRk
r
m
dr
d
rdr
d
![Page 15: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/15.jpg)
PHY 711 Fall 2013 -- Lecture 29 1511/11/2013
)( :functions Bessel xJm
m=0
m=1
m=2
![Page 16: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/16.jpg)
PHY 711 Fall 2013 -- Lecture 29 1611/11/2013
m=0
m=1
m=2
)(
:sderviativefunction Besseldx
xdJm
Zeros of derivatives: m=0: 0.00000, 3.83171, 7.01559 m=1: 1.84118, 5.33144, 8.53632 m=2: 3.05424, 6.70613, 9.96947
![Page 17: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/17.jpg)
PHY 711 Fall 2013 -- Lecture 29 1711/11/2013
Boundary condition for z=0, z=L:
...3,2,1 ,
sin)( 0)()0(
:pipeopen -openFor
pL
p
L
zpzZLZZ
p
22
2
2222
2
'
:sfrequencieResonant
L
p
a
xk
kc
ω
mnmnp
pmn
![Page 18: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/18.jpg)
PHY 711 Fall 2013 -- Lecture 29 1811/11/2013
Example
05.3,84.1,00.0'
....42.9,28.6,14.3
'1
'22222
2
mn
mnmnmnp
x
p
p
x
a
L
L
p
L
p
a
xk
![Page 19: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/19.jpg)
PHY 711 Fall 2013 -- Lecture 29 1911/11/2013
...3,2,1,0 ,
2
12
2
12cos)( 0)(
)0(
:pipe closed-openFor
pL
p
L
zpzZLZ
dz
dZ
p
Alternate boundary condition for z=0, z=L:
22
2
2
12'
L
p
a
xk mn
mnp
![Page 20: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/20.jpg)
PHY 711 Fall 2013 -- Lecture 29 2011/11/2013
01
2
2
22
tc
Other solutions to wave equation:
22 where),(
:solution wavePlane
ckAet tii rkr
![Page 21: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/21.jpg)
PHY 711 Fall 2013 -- Lecture 29 2111/11/2013
)'()'()','(1
where
)','()','(''),(
:function sGreen' of in termsSolution
),(1
2
2
22
3
2
2
22
ttttGtc
tfttGdtrdt
tftc
rrrr
rrrr
r
Wave equation with source:
![Page 22: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/22.jpg)
PHY 711 Fall 2013 -- Lecture 29 2211/11/2013
'4
''
)','(
: thatshowcan We
rr
rr
rr
c
tt
ttG
Wave equation with source -- continued:
![Page 23: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/23.jpg)
PHY 711 Fall 2013 -- Lecture 29 2311/11/2013
Derivation of Green’s function for wave equation
dett
ttttGtc
tti '
2
2
22
2
1)'(
thatRecall
)'()'()','(1
rrrr
![Page 24: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/24.jpg)
PHY 711 Fall 2013 -- Lecture 29 2411/11/2013
Derivation of Green’s function for wave equation -- continued
2
2222 where','
~
:satisfymust ,~
,~
2
1,
,,~
:Define
ckGk
G
deGtG
dtetGG
ti
ti
rrrr
r
rr
rr
![Page 25: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/25.jpg)
PHY 711 Fall 2013 -- Lecture 29 2511/11/2013
Derivation of Green’s function for wave equation -- continued
0,~
,~1
,~
:0for and ' Define --Check
'4,'
~
:'in isotropy assumingSolution
','~
22
222
'
22
RGkRGRdR
d
RRGk
RR
eG
Gk
ik
rr
rrrr
rr
rrrr
rr
![Page 26: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/26.jpg)
PHY 711 Fall 2013 -- Lecture 29 2611/11/2013
Derivation of Green’s function for wave equation -- continued
R
eB
R
e ARG
BeA eRGR
RGRkRGRdR
d
RGkRGRdR
d
RRGk
R
ikRikR
ikRikR
,~
,~
0,~
,~
0,~
,~1
,~
:0For
22
2
22
222
![Page 27: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/27.jpg)
PHY 711 Fall 2013 -- Lecture 29 2711/11/2013
Derivation of Green’s function for wave equation – continued need to find A and B.
4
1
''4
1 : thatNote 2
BA
rrrr
R
e RG
ikR
4,
~
![Page 28: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/28.jpg)
PHY 711 Fall 2013 -- Lecture 29 2811/11/2013
Derivation of Green’s function for wave equation – continued
dee
dee
deGttG
ttii
ttiik
tti
c'
'
''
'
'42
1
'42
1
,'~
2
1','
rr
rr
rrrr
rr
rr
![Page 29: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/29.jpg)
PHY 711 Fall 2013 -- Lecture 29 2911/11/2013
Derivation of Green’s function for wave equation – continued
'4
''
','
2
1 that Noting
'42
1',' '
'
rr
rr
rr
rrrr
rr
ctt
ttG
u δ dω eπ
dee
ttG
ui
ttii c
![Page 30: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/30.jpg)
PHY 711 Fall 2013 -- Lecture 29 3011/11/2013
For time harmonic forcing term we can use the corresponding Green’s function:
'4
,'~ '
rrrr
rr
ike G
In fact, this Green’s function is appropriate for boundary conditions at infinity. For surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.
![Page 31: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/31.jpg)
PHY 711 Fall 2013 -- Lecture 29 3111/11/2013
SV
rdhgghrdhggh
)g()h(2322 ˆ: thatNote
and functions woConsider t
theoremsGreen'
n
rr
)'(),'(~
),(~~~
22
22
rrrr
r
Gk
fk
rdGG
rdfG
Ggh
V
2
S
3
ˆ),(~
,'~
,'~
),(~
,,'~
'),(~
~ ;
~
nrrrrrr
rrrrrr
![Page 32: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/32.jpg)
PHY 711 Fall 2013 -- Lecture 29 3211/11/2013
'ˆ),'(~
,'~
,'~
),'(~
',','~
),(~
2
S
3
rdGG
rdfGV
nrrrrrr
rrrr
![Page 33: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/33.jpg)
PHY 711 Fall 2013 -- Lecture 29 3311/11/2013
),(1
2
2
22 tf
tcr
Wave equation with source:
),( of alueboundary v as drepresente becan
ˆ amplitude , radius ofpiston harmonic time),(
:Example
t
atf
r
zr
z
yx
![Page 34: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/34.jpg)
PHY 711 Fall 2013 -- Lecture 29 3411/11/2013
'ˆ),'(~
','~
,'~
'),'(~
','~
,'~
),(~
2
S
3
rdGG
rdfGV
nrrrrrr
rrrr
0 ;'' re whe'4'4
,'~
:0at gradient ingith vanishfunction w sGreen' Need :Note''
zzzee
G
zikik
rrrrrr
rrrr
Treatment of boundary values for time-harmonic force:
222
222
0for
for 0~
:exampleour for aluesBoundary v
ayxai
ayx
zz
![Page 35: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/35.jpg)
PHY 711 Fall 2013 -- Lecture 29 3511/11/2013
''),'(
~,'
~),(
~
0'
dydxz
GS: z
rrrr
0 ;'2
,'~
0 ;'' re whe'4'4
,'~
0'
'
0'
''
ze
G
zzzee
G
z
ik
z
ikik
rrrr
rrrrrr
rr
rrrr
![Page 36: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/36.jpg)
PHY 711 Fall 2013 -- Lecture 29 3611/11/2013
a
z
ik
S: z
eddrrai
dydxz
G
0
2
0 0'
'
0'
'2'''
''),'(
~,'
~),(
~
rr
rrrr
rr
'sinsin''ˆ'
ˆcosˆsinˆ
plane; yz in the is ˆ Assume
'ˆ' ;For
2
rrr
rar
rrrr
zyr
r
rrrr
'sin''
'cos'' :domainn Integratio
ry
rx
![Page 37: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/37.jpg)
PHY 711 Fall 2013 -- Lecture 29 3711/11/2013
aikr
iu
aikr
ikr
krJdrrr
eai
uJed
eddrrr
eai
0
0
0
2
0
'sin
0
2
0
'sinsin'
)sin'(''),(~
)('2
1 : thatNote
'''2
),(~
r
r
sin
)sin(),(
~
)()(
13
0
10
ka
kaJ
r
eai
wwJuuduJ
ikr
w
r
![Page 38: 11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:](https://reader035.vdocuments.net/reader035/viewer/2022062621/551c2ac2550346a84f8b604f/html5/thumbnails/38.jpg)
PHY 711 Fall 2013 -- Lecture 29 3811/11/2013
2
164320
2
*02
1
*21
sin
)sin(
2
1ˆ
:angle solidper power averaged Time
:average timeTaking
:fluxEnergy
ka
kaJakcr
dΩ
dP
i
p
p
e
e
e
rj
vj
vj