APPLICATIONS APPLICATIONS OF BESSEL OF BESSEL FUNCTIONSFUNCTIONS

ByErebus

INTRODUCTION

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

THREE IMPORTANT CATEGORIES

Em waves in a cylindrical waveguide, heat conduction.

Electronics and signal processing.

Modes of vibration of an artificial membrane; Acoustics.

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

Electromagnetic Waveguide

Electric field inside an x-band hollow metal waveguide.

Cylindrical waveguide

Heat conduction in cylindrical objects

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

APPLICATIONS IN ELECTRONICS AND SIGNAL PROCESSING

BESSEL FILTERKAISER WINDOWFM SYNTHESIS

What is a Bessel filter?

Friedrich bessel, German mathematician.Belongs to Electronics, signal processingLinear filterMaximally flat group delay(linear phase response)Used in audio crossover systems.

4th order bessel filter

SIM965 Bessel and Butterworth Filter

How is it related to Bessel functions?

Low pass filter is characterized by transfer function.Also called network functionIt’s a rational function.Denominator is reverse Bessel polynomial.

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.

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.

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 shapeN=M+1 where N is length of sequence.

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

What is FM synthesis?

Frequency modulation synthesisForm of audio synthesisTimbre is changed by frequency modulatingResults in complex wave form and different tone.

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.

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

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

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.

Micro-tensioned, randomly arranged cylindrical membrane

Example of membrane analysis

Membranophone and acoustic resonance in artificial membrane

CONCLUSION

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

References:

http://online.physics.uiuc.edu/courses/phys193/Lecture_Notes/P193_Lect4_Ch4_Part2.pdf

http://mathworld.wolfram.com/HeatConductionEquationDisk.html

THANK YOU