bessel & hankel functions
DESCRIPTION
bessel and hankel functions for grad emag courseTRANSCRIPT
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1Using Bessel Functions to Describe Wave Behavior
Bessel Function Jm(r)
-2 0 2 4 6 8 10 12 14 16-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
J0(r)
J1(r)
J2(r)
J3(r)
Bessel F
unction V
alu
e
r
-2 0 2 4 6 8 10 12 14 16
-1.0
-0.5
0.0
0.5
Y0(r)
Y1(r)
Y2(r)
Y3(r)
Bessel F
unction V
alu
e
r
Bessel Function Ym(r)
By combining our two Bessel Functions, we can write a
general solution for f is:
1 1( ) ( ) ( )m r m rf r A J r BY r
Hankel FunctionsSingle Bessel functions generally represent standing waves. Linear
combinations of Bessel functions, which are also solutions to Bessels equation, are used to describe propagating waves:
rjYrJrHrjYrJrH mmmmmm 21 and
where the Hs are Hankel functions of the first and second kind.
0 20 40 60 80 100 120 140-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4 J0(r)
Y0(r)
|H0
(1)(r)|
1/SQRT(r)
Ma
gn
itu
de
r
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2Reasonability Checking the Hankel FunctionsWhen we discussed conservation of power, we showed that
energy radiated spherically dropped off as one over R squared:
R
2
2
2
2*/
4H
EHEmWatts
R
PradiatedAv
Thus, either field quantity will drop off as 1/R.
For the propagating Cartesian wave (i.e., a
plane wave, we saw no magnitude dependence
with R in a lossless medium. If energy is
radiated isotropically in a cylindrical system:
r
rLdsL
PL
radiatedP
s
radiated
22
L Thus, power density drops off as 1/r, which
implies that the field quantities (E or H) will
drop off as one over the square root of r. Also,
like the Hankel functions, it is singular at r=0.
Example of a Cylindrical Wave
Propagating in the Radial Direction
Diffracted Cylindrical
Wave
When a wave strikes an edge, such as the wedge apex shown
below, the wave energy will diffract and radiate cylindrically in the
radial direction. The amount of energy re-radiated is commonly
calculated using the Geometrical Theory of Diffraction (GTD)
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3Summary of Cylindrical Wave Functions
direction r
direction r
rjYrJrH
rjYrJrH
mmm
mmm
2
1
Traveling waves:
Standing waves:
)()( rYrJ rmrm and
Evanescent waves: defined by modified Bessel functions
)()(2
)( )2(1
rjJjrIrjHjrK mm
mm
m
m and
Attenuating traveling waves direction r
direction r
rjrHrH
rjrHrH
mm
mm
22
11
Attenuating standing waves
rjrYrYrjrJrJ mmmm and )()(
Cylindrical Wave Example
x
y
z
a
For this cylindrical waveguide example (a
wave propagating inside the cylinder), a field
component will be bounded in r, and hence
we will use a bounded form for f :
)()()( 11 rYBrJArf rmrm
The field is bounded cyclically in , so
mDmCg sincos)( 22
And it is unbounded in z: zjzj zz eBeAzh
33)(
Thus, a general expression for this geometry is:
zjzj
rmrm
zz eBeAmDmC
rYBrJAzr
3322
11
sincos
)()(),,(
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4Cylindrical Resonator Signal Source
Cylindrical Resonator Signal Source
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5Cylindrical Resonator Signal Source
IMPATT Diode