# 8.1: Introduction

- Page ID
- 18081

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 review^{1}

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

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

These obey the relations

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

and have the following Taylor series approximations:

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

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

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

^{1}Also see problem 7.