082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.8 Bessel’s equation and Bessel functions 587
19.8 BESSEL’SEQUATIONANDBESSELFUNCTIONS
Alineardifferentialequationthathasimportantapplicationsinthetheoryofwavepropagation,
particularly when using cylindrical or spherical polar coordinates, is known as
Bessel’sequation.InfactthetermBessel’sequationrepresentsawholefamilyofequations
having the form
x 2d2 y
dx +xdy 2 dx +(x2 −v 2 )y=0.
Herexandyaretheindependentvariableanddependentvariablerespectively.Thethird
quantity v,termedaparameter,isknownastheorderoftheequation. 1 Aswevarythe
parameter v we obtain the family of differential equations. Note that v can be any real
number. In Sections 19.8.1 and 19.8.2 we will solve Bessel’s equation when v = 0 and
v = 1.Intheengineeringapplicationswhichfollowlaterinthechapteryouwillseethat
v may be other integers including those that are negative. Observe that the equation is
linearandhomogeneous,butitdoesnothaveconstantcoefficients:thecoefficientof d2 y
dx 2
isx 2 , a function of the independent variable. In what follows we show that solutions of
Bessel’sequationcanbefoundintheformofinfinitepowerserieswhichwecallBessel
functions.
Bessel functions have many applications in physics and engineering. In electrical
engineeringtheyarecommonlyusedinthefieldofcommunications.Theyareimportant
in finding the bandwidth and frequency content of some radio signals. They are also
essentialintheanalysisofpropagatingmodesincylindricalwaveguidesandcylindrical
cavityresonators,bothofwhichareimportantdevicesusedinhighfrequencymicrowave
engineering(seeEngineeringapplication19.12).Besselfunctionsarealsoutilisedinthe
design of filters.
19.8.1 Bessel’sequationoforderzero
To introduce the solution of Bessel’s equation and thereby introduce a Bessel function
the following example deals with the specialcase when v = 0.
Example19.31 By seeking a power series solution in the form
∞∑
a m
x m obtain the first four non-zero
termsinapower series solution ofBessel’s equation inthe specialcase when v = 0.
Solution When v = 0 Bessel’s equation is:
x 2d2 y
dx +xdy 2 dx +x2 y=0.
which fornon-zeroxbecomes
x d2 y
dx + dy +xy=0. (19.25)
2 dx
m=0
1 Note that in this context the word ‘order’ means something different from the order of a differential
equation, which wasdefined in 19.2.1.