16.2: Nonlinear Waves
- Page ID
- 30168
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We derived the properties of an ocean surface wave assuming the wave amplitude was infinitely small \(ka = O(0)\). If \(ka << 1\) but not infinitely small, the wave properties can be expanded in a power series of \(ka\) (Stokes, 1847). He calculated the properties of a wave of finite amplitude and found: \[\zeta = a \cos (kx - \omega t) + \frac{1}{2} ka^{2} \cos 2(kx - \omega t) + \frac{3}{8} k^{2} a^{3} \cos 3(kx - \omega t) + \cdots \nonumber \]
The phases of the components for the Fourier series expansion of \(\zeta\) in \((\PageIndex{1})\) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is \(a_{max} = 0.07L \ (ka = 0.44)\). Such steep waves in deep water are called Stokes waves (See also Lamb, 1945, §250).
Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b, 1966), using the tools of high-energy particle physics, worked out to the 6th order the interactions of three or more waves on the sea surface. He, Phillips (1960), and Longuet-Higgins and Phillips (1962) showed that \(n\) free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero: \[\begin{array}{c} \omega_{1} \pm \omega_{2} \pm \omega_{3} \pm \cdots \omega_{n} = 0 \\ \mathbf{k_{1} \pm k_{2} \pm k_{3} \pm \cdots k_{n}} = 0 \\ \omega_{i}^{2} = g k_{i} \end{array} \nonumber \]
where we allow waves to travel in any direction, and \(k_{i}\) is the vector wave number giving wave length and direction. \((\PageIndex{2})\) are general requirements for any interacting waves. The fewest number of waves that meet the conditions of \((\PageIndex{2})\) are three waves which interact to produce a fourth. The interaction is weak; waves must interact for hundreds of wave lengths and periods to produce a fourth wave with amplitude comparable to the interacting waves. The Stokes wave does not meet the criteria of \((\PageIndex{2})\) and the wave components are not free waves; the higher harmonics are bound to the primary wave.
Wave Momentum
The concept of wave momentum has caused considerable confusion (McIntyre, 1981). In general, waves do not have momentum, a mass flux, but they do have a momentum flux. This is true for waves on the sea surface. Ursell (1950) showed that ocean swell on a rotating Earth has no mass transport. His proof seems to contradict the usual textbook discussions of steep, non-linear waves such as Stokes waves. Water particles in a Stokes wave move along paths that are nearly circular, but the paths fail to close, and the particles move slowly in the direction of wave propagation. This is a mass transport, and the phenomena is called Stokes drift. But the forward transport near the surface is balanced by an equal transport in the opposite direction at depth, and there is no net mass flux.