Abstract
Sound and water waves are familiar longitudinal and transverse disturbances relative to the direction of propagation in a fluid, respectively. Sound waves arise in a compressible fluid, but water (gravity) waves are well described in the subsonic incompressible approximation. Our main emphasis in this chapter is on linear wave theory, where the disturbances from an equilibrium or steady state are assumed small and solutions may be obtained by superposition as Fourier series or integral forms. Our analysis is extended to superposed fluids, where hydrodynamic instability may occur—and the other topics chosen either consolidate earlier concepts or are relevant to developments in the following two chapters. The additional bibliographic entries at the end of this chapter provide relevant supplementary reading.
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Notes
- 1.
Fluid particles near the water surface actually move forward as a wave peak passes, and then backward as a wave trough passes, so there are local circulations that flatten with depth into horizontal oscillatory motions within a surface layer. However, for small-amplitude waves it is reasonable to assume that the flow remains irrotational outside this narrow layer.
- 2.
The relevant traditional analysis assumed a steadily moving line load corresponding to the forcing function \(f(x,t)={\mathrm {const}}.\, \delta (x-Vt)\), which essentially renders the response in the direction the load moves.
- 3.
The two-dimensional wave patterns were originally predicted by J.W. Davys, R.J. Hosking and A.D. Sneyd (Journal of Fluid Mechanics 158, 269–287, 1985). The corresponding time-dependent asymptotic analysis for an impulsively started concentrated point source demonstrated that a steady state is approached for all load speeds other than \(V=c_\mathrm{min}=c_\mathrm{g}\), including the load speed \(V=\sqrt{gH}\) in the long wavelength limit \(k\rightarrow 0\)—cf. W.S. Nugroho, K. Wang, R.J. Hosking & F. Milinazzo (Journal of Fluid Mechanics 381, 337–355, 1999).
- 4.
K. Wang, R.J. Hosking and F. Milinazzo (Journal of Fluid Mechanics 521, 295–317, 2004).
- 5.
E. Parau and F. Dias (Journal of Fluid Mechanics 460 281–305, 2002); F. Bonnefoy, M.H. Meylan and P. Ferrant (Journal of Fluid Mechanics 621 215–242, 2009); J.-M. Vanden-Broeck and E.I. Parau (Philosophical Transactions of the Royal Society A 369, 2957–2972, 2011); and E.I. Parau and J.-M. Vanden-Broeck (Philosophical Transactions of the Royal Society A 369, 2973–2988, 2011).
- 6.
V.A. Squire (International Journal of Offshore and Polar Engineering, 18, 241–253, 2008).
- 7.
Atmospheric internal gravity waves, where the buoyancy of the air due to a vertical potential temperature gradient rather than gravity alone is the restoring force, are of course quite different—and the alternative term buoyancy waves is actually more appropriate. Buoyancy waves have the remarkable property that their group velocity is perpendicular to the direction of phase propagation, so they can propagate energy high into the atmosphere. The rotation of the Earth provides another restoring force, leading to further important atmospheric wave types (e.g. “inertio-gravity” and Rossby waves) not discussed here, and the interested reader is referred to the book by Holton cited in the previous chapter.
Bibliography
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981). (The classic treatise mentioned in the Preface and referenced in Chapter 2)
R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Springer, New Jersey, 1977). (Reprint of the classic treatise originally published in 1948, but still remarkably suitable advanced textbook treating the dynamics of compressible fluids)
P.G. Drazin, W.H. Reid, Hydrodynamic Stability, 2nd edn. (Cambridge University Press, Cambridge, 2004). (First referenced in Chapter 3)
J.P. Goedbloed, S. Poedts, Principles of Magnetohydrodynamics (Cambridge University Press, Cambridge, 2004). (First referenced in Chapter 2)
H. Lamb, Hydrodynamics, 6th edn. (Cambridge University Press, Cambridge, 1993). (First referenced in Chapter 2)
J. Lighthill, Waves in Fluids (Cambridge University Press, Cambridge, 2001). (Valuable source on sound waves, shock waves and water waves)
P.D. Miller, Applied Asymptotic Analysis (American Mathematical Society, Providence, 2006). (First referenced in Chapter 3)
J.W.S. Rayleigh, The Theory of Sound: Volume One, Unabridged 2nd Rev. edn. (Dover, New York, 1976). (Classic text by the distinguished British scientist)
V.A. Squire, R.J. Hosking, A.D. Kerr, P.J. Langhorne, Moving Loads on Ice Plates (Kluwer, Dordrecht, 1996). (Mathematical and experimental field work undertaken to identify key phenomena for vehicles and aircraft to safely travel over floating ice sheets)
J.J. Stoker, Water Waves: The Mathematical Theory with Applications (Interscience, New York, 1992). (Well known presentation on linear gravity waves in liquids with a free surface)
J.S. Turner, Buoyancy Effects in Fluids (Cambridge University Press, Cambridge, 1979). (On motion due to buoyancy forces in a non-rotating stratified fluid under gravity)
J.-M. Vanden-Broeck, Gravity-Capillary Free-Surface Flows (Cambridge University Press, Cambridge, 2010). (Extensive discussion on surface tension effects in flows and waves)
R. von Mises, Mathematical Theory of Compressible Fluid Flow (Dover, New York, 2004). (A textbook where three chapters by von Mises are supplemented by two more chapters and notes written by others after he died, also directed to supersonic flow and shocks)
G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 2011). (Highly recommended discussion of hyperbolic and dispersive waves first published in 1974, with extensive applications previously investigated by the author—including flood waves in rivers, waves in glaciers, traffic flow, sonic booms, blast waves, and ocean waves from storms)
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Hosking, R.J., Dewar, R.L. (2016). Waves in Fluids. In: Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer, Singapore. https://doi.org/10.1007/978-981-287-600-3_4
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