= k2 Vf cos2 eB , so that the phase velocity is VA cos eB. From equation (9.66), SB is again antiparallel to Su, and the restoring force is solely magnetic tension.

The other two modes have Su lying in the B0-k plane. The field perturbation SB also lies in this plane and, as always, is normal to k. However, Su is neither perpendicular nor parallel to k, resulting in waves that are neither purely transverse nor compressional, but partake of both characteristics. To derive the dispersion relation, we divide equation (9.79) by (9.80), obtaining w4 - w2 k2 Vm;ax + a2T VA k4 cos2 Ob =0 . (9.82)

This gives for the phase velocity

Thus, the two additional modes propagate at distinct phase (and group) velocities, and are designated fast and slow magnetosonic waves. For any angle 0B, the Alfven velocity VA lies between these two speeds. As k approaches B0 in direction, the fast mode becomes an Alfven wave, with Su lying orthogonal to that in the transverse mode. In this same limit, the slow mode degenerates into a sound wave. As k becomes orthogonal to B0, the fast mode becomes a magnetosonic wave, while the slow mode does not propagate.

Which of these three wave modes actually exist within molecular clouds? Inside a hypothetical, nonmagnetized cloud supported only by thermal pressure, any fluid element in supersonic motion eventually forms a shock. Indeed, shocks also arise from ordinary sound waves, through the process of nonlinear steepening. Consider a solitary, low-amplitude pressure disturbance, traveling through the cloud at speed aT. As this wave passes any point, the associated compression raises the gas temperature slightly, an effect we have ignored in our isothermal approximation. The sound speed increases with temperature as T1/2. Hence, any disturbances within the temporarily heated region propagate slightly faster than the original waveform. The resulting pileup increases the pressure gradient until the traveling front becomes a true shock. Heating of matter as it crosses a shock front leads to radiative loss of energy. Thus, both supersonic motion and internal sound waves rapidly dissipate.

This situation changes with the addition of an internal magnetic field. The extra source of pressure makes shocks more difficult to form. For B0 oriented perpendicular to a flow, the criterion is now that the fluid element must be traveling faster than Vmax in equation (9.77). From equation (9.70), larger clouds typically have Vmax greatly exceeding aT, which is only 0.2 km s-1 at a temperature of 10 K. The supersonic velocities deduced from molecular line widths are sometimes sub-Alfvenic, at least if the telescopic beam lies within a single clump (recall equation 3.21). Such motion can survive longer than the characteristic crossing time of L/Va, which is 2 x 106 yr for L =1 pc, nH2 = 103 cm-3, and a mean field of 10 |G. However, both the fast and slow magnetosonic waves undergo steepening, as they involve compression of the gas. If the observed motion indeed represents MHD waves, it is the transverse, Alfven mode that survives, at least until it damps through mode conversion (see below) or ambipolar diffusion (Chapter 10).

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