Contents16 Waves and Convection 116.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2 Gravity Waves on the Surface of a Fluid . . . . . . . . . . . . . . . . . . . . 316.2.1 Deep Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2.2 Shallow Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2.3 Capillary Waves and Surface Tension .................. 916.2.4 Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.3 Nonlinear Shallow-Water Waves and Solitons . . . . . . . . . . . . . . . . . . 1516.3.1 Korteweg-de Vries (KdV) Equation . . . . . . . . . . . . . . . . . . . 1616.3.2 Physical Effects in the KdV Equation . . . . . . . . . . . . . . . . . . 1816.3.3 Single-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.3.4 Two-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.3.5 Solitons in Contemporary Physics . . . . . . . . . . . . . . . . . . . . 2116.4 Rossby Waves in a Rotating Fluid . . . . . . . . . . . . . . . . . . . . . . . . 2316.5 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.5.1 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.5.2 Sound Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.5.3T2 Radiation Reaction, Runaway Solutions, and Matched Asymp-totic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.6T2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3716.6.1T2 Diffusive Heat Conduction ..................... 3716.6.2T2 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . 4216.6.3T2 Rayleigh-B´enard Convection . . . . . . . . . . . . . . . . . . . . 4316.6.4T2 Convection in Stars . . . . . . . . . . . . . . . . . . . . . . . . . 5016.6.5T2 Double Diffusion — Salt Fingers . . . . . . . . . . . . . . . . . 540Chapter 16Waves and ConvectionVersion 1116.1.K, 13 February 2012. Please send comments, suggestions, and errata viaemail to [email protected] or on paper to Kip Thorne, 350-17 Caltech, Pasadena CA 91125Box 16.1Reader’s Guide• This chapter relies hea vily on Chaps. 13 and 14.• Chap. 17 (compressible flows) relies to some extent on Secs. 16.2, 16.3 and 16.5 ofthis chapter.• The remaining chapters of this book do not rely significantly on this chapter.16.1 OverviewIn the preceding chapters, we have derived the basic equations of fluid dynamics and devel-oped a variety of techniques to describe stationary flows. We have also demonstrated how,even if there exists a rigorous, stationary solution of these equations for a time-steady flow,instabilities may develop and the amplitude of oscillatory disturbances will grow with time.These unstable modes of an unstable flow can usually be thought of as waves that interactstrongly with the flow and extract energy from it. Waves, though, are quite general and canbe studied independently of their sources. Fluid dynamical waves come in a wide varietyof forms. They can be driven by a combination of gravitational, pressure, rotational a ndsurface-tension stresses and also b y mec hanical disturbances, such as water rushing past aboat or air passing through a larynx. In this chapter, we shall describe a few examples ofwave modes in fluids, chosen to illustrate general wave properties.The most familiar types of wave are probably gravity waves on the surface of a largebody of water (Sec. 16.2), e.g. o cean waves and waves on lakes and rivers. We considerthese in the linear approximation and find that they are dispersive in general, though they12become nondispersive in the long-wavelength (shallow-water) limit, i.e., when they can feelthe water’s bottom. We shall illustrate gravity waves by their roles in helioseismology,thestudy of coherent-wave modes excited within the body of the sun by convective overturningmotions. We shall also examine the effects of surface tension on gravity waves, and in thisconnection shall develop a mathematical description o f surface tension (Box 16.4).In contrast to the elastodynamic waves of Chap. 12, waves in fluids often develop ampli-tudes large enough that nonlinear effects become important (Sec. 16.3). The nonlinearitiescan cause the front of a wave to steepen and then break—a phenomenon we have all seen atthe sea shore. It turns out that, at least under some restrictive conditions, nonlinear waveshave some very surprising properties. There exist soliton or solitary-wave modes in whichthe front-steepening due to nonlinearity is stably held in check by dispersion, so particularwave profiles are quite robust and propagate for long intervals of time without breaking ordispersing. We shall demonstrate this by studying flow in a shallow channel. We shall alsoexplore the remarkable behaviors of such solitons when they pass through each other.In a nearly rigidly rotating fluid, there is a remarkable type of wave in which the restoringforce is the Coriolis effect, and which have the unusual property that their group and phasevelocities are oppositely directed. These so-called Rossby waves, studied in Sec. 16.4, areimportant in both the oceans and the atmosphere.The simplest fluid w aves of all are small-amplitude sound waves—a paradigm for scalarwaves. These are nondispersive, just like electromagnetic waves, and are therefore sometimesuseful for human communication. We shall study sound waves in Sec.16.5 and shall usethem to explore (i) the radiation reaction force that acts back on a wave-emitting object (afundamental physics issue), and (ii) matched asymptotic expansions …
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