Bessel & Hankel Functions

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

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)

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

)()(),,(

4Cylindrical Resonator Signal Source

Cylindrical Resonator Signal Source

5Cylindrical Resonator Signal Source

IMPATT Diode