Bessel functions small radii

The linear stability characteristics of the jet are specified by Eq. (10.4.32), where we note that (3 alpa, which may be compared with the plane capillary wave result where crlpX. This behavior is not surprising and can be deduced from dimensional arguments. Indeed, for the jet when a 1, that is, when the wavelengths are small compared with the jet radius, we have from the properties of the Bessel function that /(,( )/I (a) = 1. With f3 = io), Eq. (10.4.32) reduces to the dispersion relation o) - k crlp for stable, sustained surface capillary waves on deep water (Eq. 10.4.19). 

Eq. (17.65) detemiines the stability conditions for the jet. It should be noted that fP Llpa, whereas in the plane case involving capillary waves at the surface of a deep reservoir we had aP according to (17.52). These perturbations also could be derived from (17.73). Indeed, consider the case of a = ak 1 or a 2, i.e. perturbations whose wavelength is small in comparison with the jet s radius. It then follows from the properties of the Bessel function that In ct) 1. Putting p = ico into (17.65), one finds that aP = k L/p L/pX, which corresponds to neutrally stable capillary waves on the surface of a deep water reservoir. 

---