1. APPLICATIONSAPPLICATIONS
OF BESSELOF BESSEL
FUNCTIONSFUNCTIONS
By
Erebus

2. INTRODUCTION
 Two types: 1st
kind and 2nd
kind.
 Especially important for problems related to
• Wave propagation
• Static potentials.
 Broadly divided into 3 categories.

3. THREE IMPORTANT
CATEGORIES
Em waves in a cylindrical waveguide,
heat conduction.
Electronics and signal processing.
Modes of vibration of an artificial
membrane; Acoustics.

4. Short length of rectangular waveguide (WG17
with UBR120 connection-flanges)

5.  Electromagnetic Waveguide
Electric field inside an x-band
hollow metal waveguide.

6.  Cylindrical waveguide

7. Heat conduction in cylindrical objects

8.  heat flow initiated within
a hollow infinite cylinder
at an initial condition can
be generated from the
roots of any type of
Bessel's functions.
 The heat conduction
equation can be solved as
spherical Bessel
differential equation

9. APPLICATIONS IN ELECTRONICS AND
SIGNAL PROCESSING
BESSEL FILTER
KAISER WINDOW
FM SYNTHESIS

10. What is a Bessel filter?
Friedrich bessel, German
mathematician.
Belongs to Electronics, signal
processing
Linear filter
Maximally flat group delay(linear
phase response)
Used in audio crossover systems.

11. 4th
order bessel filter
SIM965 Bessel and
Butterworth Filter

12. How is it related to Bessel functions?
Low pass filter is characterized by transfer
function.
Also called network function
It’s a rational function.
Denominator is reverse Bessel polynomial.

13. A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that
the transition from the pass band to the stop band is much slower than for other filters,
but the group delay is practically constant in the passband. The Bessel filter maximizes
the flatness of the group delay curve at zero frequency.

14. What is Kaiser window?
Window function:
•Mathematical function
•Zero-valued outside some interval.
Kaiser window:
•One-parameter family of window function.
Used for digital signal processing.

15. How is it related to Bessel functions?
It is denoted by
Io is the zeroth order modified Bessel
function of first kind.
Alpha –an arbitrary real number determining
shape
N=M+1 where N is length of sequence.

16. Kaiser window function for M = 128 and
π = 1, 2, 4, 8, 16.α

17. What is FM synthesis?
Frequency modulation synthesis
Form of audio synthesis
Timbre is changed by frequency
modulating
Results in complex wave form and
different tone.

18. The harmonic distribution of a simple sine
wave signal modulated by another sine wave
signal can be represented with Bessel
functions .
This provides a basis for a simple
mathematical understanding of FM synthesis
and frequency modulation as well.

19. APPLICATIONS IN ACOUSTICS
(such as a drum or other
membranophone)

20. Solutions to the wave
equation in 2 dimensions
(mathematically non-trivial..)
It’s of cylindrical symmetry ⇒
so-called Bessel functions are
solutions to the wave equation.
Boundary condition: circular
membrane (drum head) is
clamped at the edge a⇒
displacement node at the edge
Modes of vibration of a thin circular
artificial membrane

21.  2-D wave equation has Bessel function solutions in the radial (r)
direction: Jm(xmn) = Jm(kmnr), xmn = kmnr (dimensionless quantity),
where kmn = wavenumber = 1/λmn, and m, n = 0, 1, 2, 3, …..
 The index m refers to the so-called order # of the Bessel function, and
the index n = node #.
 The boundary condition that the membrane is attached at its edge
requires that there be a displacement node at r = a = the radius of drum
head – gives rise to distinct modes of vibration of the drum head.

22. Micro-tensioned, randomly arranged
cylindrical membrane
Example of membrane analysis

23. Membranophone and acoustic
resonance in artificial
membrane

27. CONCLUSION
Other applications include:
Diffusion problems on a lattice.
Solutions to the radial Schrödinger equation (in
spherical and cylindrical coordinates) for a free
particle.

28. References:
http://online.physics.uiuc.edu/courses/phys19
http://mathworld.wolfram.com/HeatConductio

29. THANK YOU