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8.1: Introduction

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    Hovhannes Aivazovsky - The Ninth Wave - Google Art Project.jpg
    Figure \(\PageIndex{1}\): The Ninth Wave, oil painting by Ivan Aivazovsky (1817- 1900). Image courtesy of Wikimedia Commons.

    Our second example of the application of the Navier-Stokes equation to natural flows is surface gravity waves, as would occur at the surface of an ocean or lake. We will make several simplifying assumptions:

    • The fluid is inviscid: \(ν\) = 0.
    • The fluid is homogeneous: \(\rho=\rho_0\).
    • The flow is confined to the \(x\)−\(z\) plane, so there is no dependence on \(y\) and no motion in the \(y\) direction.
    • The amplitude of the waves is small enough to allow neglect of nonlinearity.

    The last assumption is new, and is tremendously important in the analysis of geophysical flows. By linearizing the equations, we filter out some very interesting phenomena such as large-amplitude breaking waves, but the resulting simplification allows us to understand the dynamics in detail. This provides a foundation for more sophisticated theories that include large-amplitude phenomena. In the next application, hydraulic flows, the restriction to small-amplitude motions will be removed.

    Hyperbolic functions review1

    In this section we will make frequent use of the hyperbolic functions

    \[\sinh x=\frac{e^{x}-e^{-x}}{2} ; \quad \cosh x=\frac{e^{x}+e^{-x}}{2} ; \quad \text { and } \tanh x=\frac{\sinh x}{\cosh x}. \nonumber \]

    These obey the relations

    \[\frac{d}{d x} \sinh x=\cosh x ; \quad \frac{d}{d x} \cosh x=\sinh x ; \quad \frac{d^{2}}{d x^{2}} \sinh x=\sinh x ; \quad \frac{d^{2}}{d x^{2}} \cosh x=\cosh x, \nonumber \]

    and have the following Taylor series approximations:

    \[\sinh x \approx x ; \quad \cosh x \approx 1 ; \quad \tanh x \approx x \nonumber \]

    These are valid for \(|x|\ll 1\) and become exact in the limit \(|x|\rightarrow 0\). As \(x\rightarrow\pm\inf\),

    \[\sinh x \rightarrow \pm \frac{e^{|x|}}{2} ; \quad \cosh x \rightarrow \frac{e^{|x|}}{2} ; \quad \tanh x \rightarrow \pm 1. \nonumber \]

    1Also see problem 7.

    This page titled 8.1: Introduction is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Bill Smyth via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.