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CALTECH PH 136A - Waves and Convection

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Contents15 Waves and Convection 115.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Gravity Waves on the Surface of a Fluid . . . . . . . . . . . . . . . . . . . . 315.2.1 Deep Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 Shallow Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.3 Capillary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.4 Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Nonlinear Shallow Water Waves and Solitons . . . . . . . . . . . . . . . . . . 1415.3.1 Korteweg-de Vries (KdV) Equation . . . . . . . . . . . . . . . . . . . 1415.3.2 Physical Effects in the KdV Equation . . . . . . . . . . . . . . . . . . 1615.3.3 Single-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.3.4 Two-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.3.5 Solitons in Contemporary Physics . . . . . . . . . . . . . . . . . . . . 2015.4 Rossby Waves in a Rotating Fluid . . . . . . . . . . . . . . . . . . . . . . . . 2115.5 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.5.1 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2515.5.2 Sound Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.5.3T2 Radiation Reaction, Runaway Solutions, and Matched Asymp-totic Expansions1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2915.6 T2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.6.1T2 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.6.2T2 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . 4015.6.3T2 Rayleigh-B´enard Convection . . . . . . . . . . . . . . . . . . . . 4215.6.4T2 Convection in Stars . . . . . . . . . . . . . . . . . . . . . . . . . 4815.6.5T2 Double Diffusion — Sa lt Fingers . . . . . . . . . . . . . . . . . 521Our treatment is based on Burke (1970).0Chapter 15Waves and ConvectionVersion 0815.3.K, 18 February 2009. Please send comments, suggestions, and errata viaemail to [email protected] or on paper to Kip Thorne, 130-33 Ca l tech, Pasadena CA91125Box 15.1Reader’s Guide• This chapter relies heavily on Chaps. 12 and 13.• Chap. 16 (compressible flows) relies to some extent on Secs. 15.2, 15.3 and 15.5 ofthis chapter.• The remaining chapters of this book do not rely significantly on this chapter.15.1 Ove rviewIn 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 can 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 andcan be studied quite independently of their sources. Fluid dynamical waves come in a widevariety of forms. They can be driven by a combination of gravitational, pressure, rotationaland surface-tension stresses and also by mechanical disturbances, such as water rushing pasta boat or air passing through a larynx. In this chapter, we shall describe a f ew examples ofwave modes in fluids, chosen to illustrate general wave properties.The most familiar types of wave are probably gravity waves on a large body of water(Sec. 15.2), e.g. ocean waves and waves on the surfaces of lakes and r ivers. We consider these12in the linear approximation and find that they are dispersive in general, though they becomenondispersive in the lo ng-wavelength (shallow-water) limit. We shall illustrate gravity wavesby their roles in helioseismology, the study of coherent-wave modes excited within the bodyof the sun by convective overturning motions. We shall also examine the effects of surfacetension on gravity waves, a nd in this connection shall develop a mathematical description ofsurface tension (Box 15.2).In contrast to the elastodynamic waves of Chap. 11, waves in fluids often develop ampli-tudes large enough that no nlinear effects become important (Sec. 15.3). 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, and particularwave profiles are quite robust and can propag ate for long intervals of time without breakingor dispersing. We shall demonstrate this by studying flow in a shallow channel. We shallalso explore the remarka ble behaviors of such solitons when they pass through each other.In a nearly rigidly rota ting 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. 15.4, areimportant in both the oceans and t he atmosphere.The simplest …


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CALTECH PH 136A - Waves and Convection

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