In the special case of a weak and purely radial perturbation, , it is easy to show the dependence of the local oscillation frequency on the position of the particle within the density wave. While this is quite different from the potential arising from a shearing density wave, one can readily verify by numerical integration of the orbits that the oscillation frequency of mass points in a potential of the type of Eq. (2) roughly follows the behaviour predicted here with minor deviations resulting from the time-dependence of the wavelength of the perturbation (Fig. 4).
where is an integration constant. To the author's knowledge, a closed solution of this differential equation does not exist. We therefore chose to investigate its solutions using the method of harmonic balance (e.g., Bogoliubov & Mitropolski 1961). We assume a quasi-harmonic solution of the form
where . Inserting this into Eq. (A1) and Fourier analyzing the nonlinear term leads to
where and denote the Bessel functions of order 0 and 1, respectively. By comparing the 'leading' terms proportional to one obtains
A straightforward application of (A3) to the perturbation used in
our simulations - justified a posteriori by Fig. 4 - can
be done by identifying with the distance of
the guiding center
of a particle from the wave crest. Since the epicycle sizes are small compared to the wavelegth of the perturbation, one may furthermore approximate . This leads to Eq. (3). These approximations obviously break down for epicycle sizes of the order of a typical stellar epicycle size where the amplitude of the oscillation in has to be zero.
© European Southern Observatory (ESO) 1998
Online publication: February 16, 1998