Consider a stably stratified fluid at rest with a small (negative) vertical density gradient d/dz. (a) By

Consider a stably stratified fluid at rest with a small (negative) vertical density gradient dρ/dz.

(a) By modifying the analysis in this section, ignoring the effects of viscosity, heat conduction, and concentration gradients, show that small-amplitude linear waves, which propagate in a direction making an angle θ to the vertical, have an angular frequency given by ω = N| sin θ|, where N ≡ [(g · ∇) ln ρ]^1/2 is known as the Brunt-Väisälä frequency. These waves are called internal waves. They can also be found at abrupt discontinuities as well as in the presence of a slow variation in the background medium. They are analogous to the Love and Rayleigh waves we have already met in our discussion of seismology (Sec. 12.4.2). Another type of internal wave is the Kelvin-Helmholtz wave (Sec. 14.6.1).

(b) Show that the group velocity of these waves is orthogonal to the phase velocity, and interpret this result physically.