Hankel functions are solutions to the scalar Helmholtz-equation $$\Delta\psi + k_e^2\psi = 0$$ in cylindrical and spherical geometry (with respect to a separated angular dependence). Thus, they are very important describing spherical and cylindrical waves. Here is an example of such a propagation in the spherical case taken from Franz Zotter:


I am searching for a reference that states the phase accumulation of Hankel waves of the form $$F_H^{\mathrm{out/in}}(\mathbf{r}) = H_m^{1/2}(k\rho)\ .$$ Assumed is stationarity with an $e^{-\mathrm{i}\omega t}$ time dependence fixing the meaning of the two different Hankel-waves as outgoing/incoming.

For plane waves one finds that the accumulated phase of a wave in $x$-direction, $$F_p=e^{\mathrm{i}kx}$$ is simply related to its argument, $$\phi_{\mathrm{acc}}(x_1,x_2)=\mathrm{Arg}(F_p(x_2))-\mathrm{Arg}(F_p(x_1)) = k(x_2 - x_1)$$ and it is natural to just use this formula in the Hankel-case, e.g. $$\phi_{\mathrm{acc}}(\rho_1,\rho_2)=\mathrm{Arg}(F_H^{\mathrm{out/in}}(\rho_2))-\mathrm{Arg}(F_H^{\mathrm{out/in}}(\rho_1))$$

However, I was not able to find a suitable reference. Hence my question:

Is there a reference defining the phase accumulation of Hankel waves?

Thank you in advance.


A closed form solution (in terms of more elementary function than Hankel functions) does not exist. However, typically one is only interested in the regime where $k\rho \gg1$, i.e., the asymptotic region far away from the source. There one can use the asymptotic form of the Hankel functions $$H^{1/2}_m (x) \sim \sqrt{\frac2{\pi x}} e^{\pm (ix -i \frac\pi2 m - i\frac\pi4)}.$$ Thus, the accumulated phase is given by $$\phi_\text{acc}(\rho_1,\rho_2) = \pm k(\rho_2 - \rho_1),$$ i.e., the same as for a plane wave.

  • $\begingroup$ Thank you for your answer. However, in my case a definition of the phase accumulation for all $\rho > 0$ is required. Greets $\endgroup$ May 14 '11 at 13:37
  • $\begingroup$ @Rober: why would one ever need something like that? $\endgroup$
    – Fabian
    May 14 '11 at 15:06
  • $\begingroup$ For a complete description of the field in the same sense as you need Hankel functions and not only plane waves. $\endgroup$ May 14 '11 at 15:22
  • $\begingroup$ Ok, there seems no interest in an answer to this question for all $\rho$ so I just accept this one. Greets. $\endgroup$ Jun 23 '11 at 8:24

I am not sure how you get outwardly travelling waves from Hanken functions alone without also using Bessel functions. My understanding is that by analogy with plane waves, you have the stading wave solutions eg.

sin(kx)*cos(wt) and


and the travelling wave comes from the sum of these two. Similarly in cylindrical geometry you have the standing waves

Bess(kr)*cos(wt) and


and the travelling waves shown in the applets are really the sum of these. I'm not sure if this point is relative to the question being asked, but this is how I understand the pictures.

  • $\begingroup$ $H^{1/2}_m = J_m \pm N_m$ with respect to an argument. Please have a look at the Wiki sites I was linking. Greets $\endgroup$ May 14 '11 at 18:08
  • $\begingroup$ OK, right. I guess what I called the Hankel function would actually have to be the Neumann function. $\endgroup$ May 14 '11 at 20:50

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