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11/04/2013PHY 711 Fall Lecture 263 Review of wave equation in one-dimension – here (x,t) can describe either a longitudinal or transverse wave. Traveling wave solutions --
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PHY 711 Fall 2013 -- Lecture 26 111/04/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 26:Traveling and standing waves in 1 and 2 dimensions
1. Brief review of 1-d from Chap. 72. Waves in 2-d elastic membranes –
Chap. 8
PHY 711 Fall 2013 -- Lecture 26 211/04/2013
PHY 711 Fall 2013 -- Lecture 26 311/04/2013
equation. wave thesatisfies )()(
:or function any for that Note
0 2
22
2
2
ctxgctxfμ(x,t)g(q)f(q)
x c
t
Review of wave equation in one-dimension – here (x,t) can describe either a longitudinal or transverse wave.
Traveling wave solutions --
PHY 711 Fall 2013 -- Lecture 26 411/04/2013
ctx
ctx
x
x
x
dxxc
ctxctxtx
dxxc
xxg
dxxc
xxf
g(x)f(x)x
dxxc
xgxf
dxdg(x)
dxdf(x)cx)(x,
tμ
xgxfx)μ(x,
x)(x,tμx)μ(x,
')'(21)()(
21),(
')'(1)(21)(
')'(1)(21)(
: and find ,each For
')'(1)()(
)(0
)()()(0 : then
)(0 and )(0 :problem valueInitial
PHY 711 Fall 2013 -- Lecture 26 511/04/2013
Standing wave solutions of wave equation:
.00with
0 2
22
2
2
μ(L,t),t)μ(x
c t
c kρ(x)k
dxρ(x)d
ρ(x)etx ti
0 where
),( :Assume
22
2
ckL
k
LxAx
sin)(
PHY 711 Fall 2013 -- Lecture 26 611/04/2013
PHY 711 Fall 2013 -- Lecture 26 711/04/2013
Extension of ideas to wave motion on a two-dimensional surface – elastic membrane (transverse wave; linear regime).
u(x,y,t)
PHY 711 Fall 2013 -- Lecture 26 811/04/2013
0///
0
:principle sHamilton'
,,;,,,
,,;,,, :density Lagrangian
2
1
yuyxuxtutu
Ldtδ
dxdytyxtu
yu
xuuL
tyxtu
yu
xuu
t
t
LLLL
L
L
PHY 711 Fall 2013 -- Lecture 26 911/04/2013
2222
2
22
where 0
0///
21
21,,;,,,
: and constant with membrane elasticfor density Lagrangian
cuctu
yuyxuxtutu
ututyx
tu
yu
xuu
LLLL
L
ckyxk
yxetyxu
cuctu
ti
where 0),(
),(),,(
:solutions waveStanding
where 0
:equation waveldimensiona-Two
22
2222
2
PHY 711 Fall 2013 -- Lecture 26 1011/04/2013
Consider a rectangular boundary:
a
b
mnmnmn
mn
ckωb
na
mk
byn
axmAyx
bxxyay
sinsin),(
0),()0,(),(),0(:conditionsboundary Clamped
222
PHY 711 Fall 2013 -- Lecture 26 1111/04/2013
22
211
bak
2 222
21
bak 22 22
222
bak
2 222
12
bak
PHY 711 Fall 2013 -- Lecture 26 1211/04/2013
More general boundary conditions:
side free"" represents 0
spring with dconstraine side boundard represents
b
bb
u
uu
mnmnmn
mn
ckωb
na
mk
byn
axmAyx
xya
xybxx
sincos),(
0),(),0(),()0,(
:conditionsboundary Mixed
222
222
11
bak
PHY 711 Fall 2013 -- Lecture 26 1311/04/2013
Consider a circular boundary:R
0),(:),(for conditionsboundary Clamped
Rr
c
krk where 0),(22
integer )2()( :Note
)( :Let
)()(),( :Assume
11
system coordinate lcylindricaIn
2
2
22
22
m
e
rfrrrrr
im
PHY 711 Fall 2013 -- Lecture 26 1411/04/2013
)(function Neumann lCylindrica )(function Bessel lCylindrica
solutions ential transcendorder withinteger ofequation Bessel
0)(1
:function radialfor equation alDifferenti
22
2
2
2
zNzJ
rfkrm
drd
rdrd
m
m
Consider circular boundary -- continued
PHY 711 Fall 2013 -- Lecture 26 1511/04/2013
Some properties of Bessel functions
11.620,... 8.417, ,136.5 :210.173,... 7.016, ,832.3 :18.654,... ,520.5 ,406.2 :0
0 functions esselB of Zeros
2
1
0
n
n
n
mnm
zmzmzm
zJ
42cos2)( :form Asymptotic
)(2)()(
)(2)()( :relationsRecursion
2!!1
2)( :series Asending
1
11
11
2
0
mzz
zJ
dzzdJzJzJ
zJzmzJzJ
zmjj
zzJ
zm
mmm
mmm
j
j
jm
m
PHY 711 Fall 2013 -- Lecture 26 1611/04/2013
Some properties of Bessel functions -- continued
'2
1
2
0
'
2
: functions for theidentity
following theprove topossible isIt :Note
nnmnm
Rmn
mmn
m
mnm
zJRrdrrR
zJrR
zJ
rR
zJ
Rz kr
RzAJ(r)f
rfkrm
drd
rdrd
mnmn
mnmmn
;
0)(1
:function radialfor equation aldifferenti toReturning
22
2
2
2
PHY 711 Fall 2013 -- Lecture 26 1711/04/2013
r
RzAJ(r)f)(r 01
00101 ,
r
RzAJ(r)f)(r 02
00202 ,
520.5 406.20201 R
kR
k
PHY 711 Fall 2013 -- Lecture 26 1811/04/2013
cos
cos,
111
1111
rRzAJ
(r)f)(r
cos
cos,
121
1212
rRzAJ
(r)f)(r
016.7 832.31211 R
kR
k
PHY 711 Fall 2013 -- Lecture 26 1911/04/2013
Maple server evaluation project Request from Rick Matthews, Associate Provost
for Technology and Information Systems The University is exploring the possibility of
making Maple available on a server rather than on individual computersNeed students to exercise the software Need students to assess the performancNeed students to identify issues
Entirely voluntary
Possible pizza party or ??? rewards
PHY 711 Fall 2013 -- Lecture 26 2011/04/2013
Introduction to algebraic manipulation software
Available from the webpage: http://help.wfu.edu/public/vcl
If you have any trouble with this installation and setup, please contact Ching-Wan Yip at [email protected]
Note: This is a remote server that the university is testing. If you are off campus you will need to use VPN to gain access.
PHY 711 Fall 2013 -- Lecture 26 2111/04/2013
PHY 711 Fall 2013 -- Lecture 26 2211/04/2013
PHY 711 Fall 2013 -- Lecture 26 2311/04/2013