Waves, oscillation and optics
Linear
response
Allanodiable
Oman
dynamic
anac
a A
.
ra
↳
cause
Force
Effect
Displacement
I
Tamp.
Heat
Prevure
fluid flow
stres
(Newton]
COhmT
v
flow
strain
Non-linear
NLOC
NS
Hooky
mii=-Rx
↳
linarity
c
spring
+
wix 0w
=
constant
eq").
(Recall Laplace
en:
x
Longitudinal
astien
waves
longitudinal
wor =
m
en
+
w
there
as
To
Trans-sling
a
RCA-90)
=
T x(l
=
-
mi Fn
=
u(1
a)
=
=
-
w,
33)
Transverse
-
small
i ain
-
wj
25a
=
-
-
q)
if 10<1.
2sinc
==
2
oxcillation
i a
=
+
a
=
s)
-
alternative
An
i
E
=
x
p
=-
z
SDS
6.2
z=
E
SHO.
(r)
=
()
(si
↑
Mz
I
=
of
w-x
-
M
analysis
=
n
-1
D
=
=
M
A
TrM
-
Det
(M)
SD5z
=
stE DSE
=
F bu
=
->
Wilt=
ebitwst
Linear
↑
analysis
T
z
312
M
Mz
=
R
=
M S D5
-
D:
=
= ME
T
=
w
05z
Sz
=
=
i
d,W;
For SHO.
312-
eM
WCt
=
=
b
=
d I
,(w 5)
=
ebitWi(s]
W,(t)
"Dies out"
(Note:
diag(),,62)
SDSE
E
-
C021)2: Trcor
=
=
Run-away
Oscillatory
IWo
I+
DWCH)
n+
H...
>
wx
=
R
eP*N(Ol)
=
f(n
+
f(x)
8x)
=
5xf'(x)
+
+
(Sn f"Ine--
LeMon)fMT
=
-
generator of
2) Note
that
(special
3)
wo FAmp.
for SHO
Equation.
Sine-Gordan
ii-u +
1)
since
ii+
sinu =
To
ignore
i
(x)
=
-
0
=
(u(x,t) 0)
=
0
I
[x ... ]
-
1
=
translation.
1x1<>
(l
Tram-gordon
is
no
ear
Damped
mix
3
i
kx
+
=
SHO
+
xx
w
=
c
0
subs.
eft,x ax,x
=
=
wi
25wox +
21 wC 3
-
=
SHO
+
0
f
1)
L NOTE
critical
PAMPING
4
=
happens
what
X(t]
WC+)
x(t)
&+E
Ale
A,e*xt
=
=
=
+
=
damping
9=7
+Aet
at
z
=
*
97/
1?
(t) w(+)X.(t]
*
=
X(t)
co
Coe*+ Citeft
cit
=
of ANSATZ
cx
=
Pumping d
↑
-- I
x(t)
29at
=
Direct
sol Method 1:
-x+
=
...
24Wox + wix
+
x
t
0
=
A,ex+
=
+
(%
+
Pz)y*+
x
HT
4
*
at
for
e
Solution
&
=1
433
Method #2:
(wizquoJ
=
31
(E
E
=
(n)
5
=
+
2
2
wo[-]
=
Point
1) g
ii)
0
=
470
to
sol"
same
note:
61
=
=
=
61
-
XC)
=
e
15Wo
=-woG
WoYE
IWORT
[AeNotT
+
iwotT]
AoE
x
2qwox
+
For
critical
+
zwox
*
(x + wix)
w-x
+
0
=
point:
wix
+
=
0
(Wox +
+
(x + w,x) +
w(x
w,x)
j
For
y
woX)
+
0
=
me
-
y
0
=
y
woy
+
or
y
0
=
-Not
xtWoX=0:
=)
0
=
AetWot
-y:
0
=
X
=
qe
-
↑2
solution
A
"
ANSAT2.
For
critical
of
AerWit
y:
we
at
be
-
and
infinity,
acompond
into
this sol is
But
that
x+Wox-o
tempted
x,x -0
"may" repruent a
to
therefore
integrable sol", if
square
is
our
this
claim
true??
somewhat
It's
that
this sol
sperposition
also
without
-Not
[to
might
derived
K+WoX=Ae
=)
observation:
.
damping
of
can
normal
be
modes,
oscillatory.
check
claim is
true... It's
Fowler transform
in
by
apply
awning
-x(+)
6 x(a)
=
eiwdw
(aculiweintaw +(aculwoeiNdw= Ae-Not
Sacwl ejt(iw +Woldt JA'et s(w-iw.)dt
-
=
n(W) RTW
f
x(t
=
+
A
No)
(xcreiw+dw
=
Gwiwa
=
x(t)
S(w-Two)
g-Not
A
=
giwado
-
Wo-wo
sol" to
:For
trivial sol
our
exist.
finite
A70,
For
any
x + wX
..can not
of
Two
i)
ii)
-
de
=
oscillating
solving
ze-Not
sons
corresponding
integrabl
non-square
into
linear
to
superposition
sols
features
of these
2
point
sol's
critical
discaving
be zero, which is
must
is
decomposed
be
key
iNot
A
about
soM).
nature
page:
without
of
ANSAT2.
solis)
specially
General I-dim
flour
preview
o
⑭== = =
Wo
=
stationary point
i r(1 r]
=
-
↑
↑
*
*
⑧
xE1
n0
=
=
Limitcycle
612
oscillators:
2*
M
I
=
-
nz
=
.
.
-
#
4....
...
4I
*
Atypical
manifold
ast
-
-
trajectory
as
0.
asymptotically
to, and
z rA
-
I,
#
-
approaches
approaches
the
A
the
stable
unstable
manifold
Analysis
Regions
I
e> 0
(
D=
DEIR
,
É)
'
di
_
conclusions
C. Phase space)
↑→
,z=E÷P
c-
I
Rt
Unstable
node
II
2>0
'
Dee
,
,
2=1%101
☆
Unstable
spiral
II
D=
2=0
En
2<0
☒
a. i.
:-
,
=
-1-956
SHO !
D.
C- G-
-
←⑧→
Hunters
I
K1±z
⑧
'
stable
spiral
I
D=
0
Damped
22=41
Kfwb
d "z=
}
s
-
Doanaapig#
Ho
#
DEIRI
2CO,
x1,2
=
121
-
-
D
I
- -
C
2
LO
)
u
A
+xz
0
=
ARED
121
=
dxz (0,2)
=
w,H)
I
I thi
(Unstable (
etwpLOD
=
Stable.
V)
A <0
D:
61
=
EDI,
D
2
Saddles
Superposition of
MÑa¥
F, 4- 7
V1
-
=
Fct ]
-1
94,11-7-1 BEAT
=
=
CY
inhomogeneous Egns ÷
Wiman
4, PER
,
4144-7-1 pYzCt)
Miu
that
check
=
-
satisfied
is
cult F
.
Critically damped
"
exceptional points
"
↳ critically damped
↳ Band
theory
↳
Liouville
supper operator
( open ouantum
Generalised E
MX
=
2)
-
you
✗
=
-
-
DI )ᵈ
.
system 7
Mxs
.
=
a
degeneracy
Generalised
0
"
✗
form
.
o
define
CM XI)ᵈXd
CM
theory
gordon canonical
appears with
d
Then
1)
of
V s
xx
C- d 3)
tf
-
SHO
# 0
"
d
"
Eigenvectors
CGEVI XD
by
het
d=z
( di
,
did ]
1) LM -2372×2--0
2) CM
-
DI )Xz= ✗
,
↳
to
actual
eigenvector
.
this
apply
hits
/ yes
DSHO
)
[ Wi
din
z±F
1
°
M=
-24W
-
=
=
d
=L
,
=
-
Wo
compute
hit 's
(M
-
CM
4=2
Gtvs
gwo±(F two
.
.
AIT ✗ 1=0
-
@+
@
-
xéxz
too #
=
×
,
(two )
=
1×2--1!w )
:
:)
-11=1
÷
:o)
/
I. %
E(9)
.
.
T :( A)
7-
5-
(
1
-
-
w.
F'MT
o
s
=
"
)
two
-
!)
w
XD
Go: :)
É
Forced
it
,
damped
SHO
,
24wixtwix-fg.ws @
↓
É
complex f
-124W
Trial
.
"
Ét
Wiz
=
g.
eiwt
Sol ?
-2--3 @
iwt
-
t)
w
@
Zo
+
a
-12174W
its > Zo
=
◦
=
22-0
WZO two
'
%
Em
@ 5- WI
b
=
24 Wow
zi-r-o-mcaa-IY-r-mr-E-n.EE?Jzm=fF-bT
a-
=
cos
8
bz-m=
-2m
2-◦
=
FMZM
2- G)
=
[ coss
-
¥gmeiwᵗ→
sins
isins
]
]
✗
It >
=
!n=¥coscwt
-
s]
✗ In
24 wow
+
Soil
i.
-87
Fm÷gmwslwt
=
SS
it
is
Wix
Fo_
wswt
=
linear
a
Coslwt
full
a
Im
t
combination
s)
-
not
is
so
cannot
of
+ 97th
MZM
Whare 4cm
VIN
i.
our
forced
is
dies
out
system
damped
Ict )
it 24 Wow % two ✗
som of
.
both
is
doing
steady
0
Transient't -1
✗
p
n
=
-
-
-
so1
✗ sit ]
e-
✗
state
_
É
_
-
.
.
damped and
MHZ ✗ Ct]
=
in
=
damping
.
if V-X.sc?s.t-MltIXssCt)--
✗ft )
of
because
=
0
XTraniemts
Xo (t)
=
XssH]
+
L A,eb+
2)(
t
A
+
-eb.t b
1
=
us
320
Thingstonote:
1
differentfrom
Freg of transientsol"
w.
w,z
is
was
-
=
SS.
w
s.s
moves
at
DRI
frequency.
27 superposition
Handle
3)
Xss(t)
solutions.
FCT) [C(0)
=
F
=
cluz
20
at
-
(wit)
5)
=
Xs(t)
MpCr
=
>M.
awt
to twin
to his
state
sol
d) Steady
was
A(W)
A(W
0)
==
=
K
A(W = W.)
=
-(r
0)
=
=
224W."
wwf
A(W
Amax
+
=
0)
<
-
- 0
-dA
0
=
drco
Amx (W) ACUma)
=
Cmax
=
wo
-.
↳
ACWmex)
I AG
wo
e
[1 + 5]
&-factor
Beats
(BVZ
41
& Feynman. A
cosuct
=
A
coscit
=
Y 4 4z
=
+
cosQUct]cos(mt)
IA
=
Wc
=
WH
ws 3
W-
Wm=
+
-
2
-
Modulated
waves
m
was.
carrier
Wc
mu
B
↳Wm
w
We
-
n100
5
for
radio
waves.
Wm
↳since
C
=
x
+
p
2m f
yf,
4 IACOSWCt
=
coscict 2
COSWmt
Coswet[1+ A(COlwstmit COS(W>-wmSF)
+
=
\
LOWER
④
I
SIDE
->
a
RC D
BLUE
1
↑
t
wawm
Scan
sideband
UPPER
BAND
wc wm
7
+
w
fat
may be
↳penualized
Parseval's
energy.
theorem:
dGl fdwI
(4)
E(m(n)
=
(n
2=
< R14)
=
say
we
need
S.t.U(U) 1bn)
=
STn16u>
14):
(414)
18SIcu
=
a new bain
↑(H,t)
d
7"[f(n3]
7
*
+
Ockeinn
xx(z)
f(-n]
=
9dwIn
=
e
Energy
=
theorem
Wiener-Khinchin
+ (X*(+)
=
Then
jet
=
Sxx(W)
to
related
-
(n)I f()
f
a (n()1
=
v
·
r0>
#
4(n,0)
=
Sxx(w)
X(t
-
z)]
Fw [vxx (z)]
=
spectral density
- >
oxcillators:
Non-linear
Nos
I
->
C
symmetric
IDEAL
·Physical
Approxim
L
systems
Real
i
twoN
=
S
f(aw *)
+
(weX,
↳
f
f(w,w*
e.g.
phase
invariant
i
X
(
e
=
f (w,w*)
1WTW.
=
clNTw
9,w
+
=
PINLO's
we
4jp i9,z
=
aj
+
write
W Reid
=
a
i
3
a
=
IR
R
t
↑(R
C,B503)
=7<in> 0,
for
9,z 1zzR2
=
+
↳w.
-
finiteradius
can
0
Y
4.R N
=
=
-
seeing phased
the
BOGOLIUBOV
-
X(t) a()ca
=
i
But
=
-
x=
-
x X
Demand
-
acoid-acid (1 +i(+)
=
coso-a rsin
o
0
=
asino
asinp-acost(1 4]
i NF(x,x)
a cod
atint
+
=
=
+
at
Demand
-
a(t) sin (6())
=>
x
-
-
ano
=
is
.
-
itis
+1)
((
actcos(PCT)
-
if
averaging
KRYLON
=
=
as
but
it
isn't.
oscillator,
x
like
itlooks
asino
-
i
so
=
NT
. . .
(iis
harmonic
=
-
F
aq
-
NF
..
=
a NF
sino
=
i
=
-
(ii)
Fcso
-
(F(X,x) x(X 1)
=
=
asind(acoid 1)
-
-
BK
a
Averaging
do
N
=
a
sinoCao-1
da a (a se
=
B
Pikofski
-
du
do
anzo
Caoioof the
mari-Harmonic
Vap
coupled
1
mode
Particle
a.
-
Oscillator
spectroscopy
IR
·
Landau
oscillators:
I normal
·
Stuart
)
physics
computing
knin-gordon eg
·
wave
eq"
-
dynamics
Uncoupling
I
x
x
-
1xx
L
tr
#
K
!
O u rO
M
m
w,
.
-
↓
!
noun
fro
=
amnimb
k
M(x =
xx,
-
Mx
=
-
k.(x,
-
xxy
-
=
=
-
Xz)
k'(xu xi)
-
XI x,IX
x
-
w,X
uncouples
x
=
+
mom,
Mx 1 +
=
i
vi
=
-
timar
w"X:
w=X
-
small
ose
=
o
omy
k
-
-
I
1/
+
k
-I
approx
triatomic
molecules
i( 1 otomon:re
·
k,2
112
=
-
k,2
o
MX=iπ
-
123
X (n i)
=
DSTnormal
ma
to Hamiltonians
Relationship
N
v(q)
&
>
v(q) V(%)
=
VCH):
nij
+
8qi
+
q
Vsqsein,
+
*oscillation
of
*continuum
theory
nas
particles
Rin-Gordoneg",
*In:
eq
Wave
Csec
3.s
Berkeley)
D...
1
l
"
·
I
Z
Min
=
-
Mw;Yn +k(Yn
↳w5
Take
the
-
+1
continum defined
limit
=
+
=
OurI(M
as
+
+
P(z -a,t) P(2,t)
=
=
-W (z,t
=
-
K(4n 4n )
-
-
=
Yu (#) P(z a,t) P(2t)
Pm(+)
Yn)
-
Nult)
->
T(z,t)
ar+anc.e
at
auzu.ee
+
parzye
+
↳
equation.
klein-gordon
kaz
a-
=
a
=
(dimensionally)
Ij=-w54+ jpll
Notice:
wo
soi"
.
gives
o
->
wave
equ
arats
Y(z,t) A(Z) COS(WE +9)
=
i=
-
A"(z)
F
w
[w. w") A(z)
-
1
=
22
-1w4lYe
Case 1:w>Wo
n'(z):
COS(qZ)
A(z) a
=
Case 2:W
Bain(qZ)
+
= W0
wFw"
i
12
=
-
-
Attenuation number
v
=
att
skin
-
Reactive
a(2)2
A(z)
Dispositive.
m
WI
z
I
>
=
number
depth.
Tummeting
distance
=
w
w=[R E>
=
0
Dispersive
Reactive
Example:
=
I(((yu)
wp
=
↳ (plasma
=
-
relation]
ionosphere
Earth's
w
[Dispension
momentum
e
sulation relate
Dispension
En
05+9"
frequency)
47
manx=gfnD NGAXar=Nadae
=
dro
dig
=
-
4+Nqr
=-
Nale
↳N."
e
w
fog w.
carical
in
CTunnelling
physics)
adodo
/
1
0000-w
L
200
=
w,:g
11
m
000
W>w
> W,
&
Tunnelling
mu