Hankel Functions: Solutions to Cylindrical Wave Equation

[aim1732]

How are Hankel functions the solutions of the cylindrical wave equation and the Bessel functions aren't? My textbook says the the two Bessel functions are like generalized sines and cosines.Now sine and cosine are solution to the wave equation so why do we need the Hankel to do the job in cylindrical system?Is it because they are complex and the wave functions are complex in QM?I seriously doubt that.

 

[SteamKing]

Hold on there, pilgrim! Hankel functions are linear combinations of Bessel functions of the first and second kind:

http://en.wikipedia.org/wiki/Bessel_function

 

[aim1732]

Well I already looked that page up.It does not help.And I know very well what Hankel functions are.That is not what I was asking.

 

[DrDu]

Which boundary conditions?

 

[dextercioby]

Depending on the parameters of the Bessel ODE, the solutions can be either

* Bessel functions of the 1st kind
* Bessel functions of the 2nd kind
* Bessel functions of the 3rd kind

and there are also spherical Bessel functions...

It all dependens on the ODE and the possible values of the parameters. The solutions to these ODE are described in special functions books (starting with the old thick book by Watson).

 

[aim1732]

Yes I am talking about the Bessel functions of the third kind.They are actually two independent linear combinations of the Bessel and the Neumann functions.

The boundary conditions, or rather the physical requirement of the situation is that the solution to the ODE represent outward traveling waves. This is what I am asking.How do the Hankel functions represent traveling wave solutions and Bessel functions of the first and second kind do not?

 

[DrDu]

the Bessel and Neumann functions correspond to sin and cos functions which are also not traveling waves, while the Hankel functions correspond to $\exp(\pm ikx-i\omega t)$ which are traveling waves(the x(t) for which the exponent is constant travel at constant speed; $x(t)=\mp \omega t/k$). This should be evident from the asymptotic expansion of the Hankel functions for large x.

 

[aim1732]

You are right.The sines and cosines aren't traveling wave solutions to the Schrödinger equation(while they may be traveling wave solutions to the string equation).Only the exponential should work and this is reflected similarly in the Bessel and Hankel functions.