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CALTECH PH 136A - Waves and Rotating Flows

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Contents15 Waves and Rotating Flows 115.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.1 Deep Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 Shallow Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.3 Capillary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.4 Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Nonlinear Shallow Water Waves and Solitons . . . . . . . . . . . . . . . . . . 1315.3.1 Korteweg-de Vries (KdV) Equation . . . . . . . . . . . . . . . . . . . 1315.3.2 Physical Effects in the KdV Equation . . . . . . . . . . . . . . . . . . 1515.3.3 Single-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.3.4 Two-Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.3.5 Solitons in Contemporary Physics . . . . . . . . . . . . . . . . . . . . 1815.4 Rotating Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2015.4.1 Equations of Fluid Dynamics in a Rotating Reference Frame . . . . . 2015.4.2 Geostrophic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.4.3 Taylor-Proudman Theorem . . . . . . . . . . . . . . . . . . . . . . . 2 315.4.4 Ekman Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . 2415.4.5 Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.5 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.5.1 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.5.2 Sound Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.5.3 Radiation Reaction, Runaway Solutions, and Matched Asymptotic Ex-pansions1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351Our treatment is based on Burke (1970).0Chapter 15Waves and Rotating FlowsVersion 0615.2.K, 1 7 February 2007. [Same as 0615.1.K, except for addition o f Box 15.3 onImportant Concepts]Please send comments, suggestions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Ca l tech, Pasadena CA 91125Box 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 on fluid mechanics and magnetohydrodynamics ( Chaps.17–18) do not rely significantly on this chapter, nor do any of the remaining chaptersin this book.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 rig orous, stationary solution o f 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 o f their sources. Fluid dynamical waves come in a widevariety of forms. They can be driven by a combination of gravitational, pressure, rotatio na land 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 few examples12of wave modes in fluids, chosen to illustrate general wave properties. As secondary goals ofthis cha pter, we shall also derive some of the principal properties of rotating fluids and ofsurface tension.The most familiar types of wave a re probably gravity waves on a large body of water(Sec. 15.2), e.g. ocean waves and waves on the surfaces of la kes and rivers. We consider thesein 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 he l i oseismology, 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, and 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 nonlinear 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-steepending due t o nonlinearity is stably held in check by dispersion, and par t icularwave profiles are quite robust and can propagate for long intervals of time without breakingor dispersing. We shall demonstrate this by studying flow in a shallow channel. We shallalso explore the remarkable behaviors of such solitons when they pass through each other.Rotating fluids introduce yet more novel properties (Sec. …


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CALTECH PH 136A - Waves and Rotating Flows

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