226 FOUNDATIONS OF RADIATION HYDRODYNAMICS

fall midway between the nodes of Iw] 1, with two nodes per vertical wavelength, and the amplitude would be exactly zero at each node. Figure 54.7 ill ustrates refraction of acoustic-gravity waves in response to the continuous variation of properties of the solar atmosphere. Because the sound speed changes little with height, high-frequency waves show 1ittle bending of the direction of the phase velocity VP or group velocity v~. Gravity waves, in contrast, show strong refraction from the large changes in co~v, with VP bending away from the vertical as mBv decreases, and v~ bending toward the vertical. Though the direction of v, tends toward the vertical, the magnitudes of both u ~ and WE decrease to zero as ~~v decreases to O. 5.3 Shock Waves The theory developed in $$5.1 and 5.2 applies only to small-amplitude disturbances, which propagate essentially adiabatically and are damped only slowly by dissipative processes. As the wave amp] itude increases, this simple picture breaks down because of the effects of the nonlinear terms in the equations of hydrodynamics. When nonlinear phenomena become important, the character of the flow alters markedly. In particular, in an acoustic disturbance a region of compression tends to overrun a raref actio n that precedes it; thus as an acoustic wave propagates, the leading part of the profile progressively steepens, eventually becoming a near discon-tinuity y, which we identify as a shock. Once a shock forms it moves through the fluid supersonically and therefore outruns preshock acoustic disturbances by which adjustments in local fluid properties might otherwise take place; it can therefore persist as a distinct entity in the flow until it is damped by dissipative mechanisms. The material behind a shock is hotter, denser, and has a higher pressure and entropy than the material in front of it; the stronger the shock (i.e., the higher its velocity) the more pronounced is the change in material proper-ties across the discontinuity. The rise in entropy across a shock front implies that wave energy has been dissipated irreversibly; this process damps, and ultimately destroys, the propagating shock (sometimes rapidly). In contrast to acoustic waves, internal gravity waves do not develop shocks. Instead in the nonlinear regime they break and degenerate into turbulence. We will not discuss these phenomena in this book; see fo