File:Stokes wave max height.svg

Summary

<a href="https://en.wikipedia.org/wiki/Stokes_wave" class="extiw" title="en:Stokes wave">Stokes waves</a> of maximum <a href="https://en.wikipedia.org/wiki/wave_height" class="extiw" title="en:wave height">wave height</a> on deep water, under the action of <a href="https://en.wikipedia.org/wiki/Earth%27s_gravity" class="extiw" title="en:Earth's gravity">gravity</a>. Shown is the simple and accurate approximation of the <a href="https://en.wikipedia.org/wiki/free_surface" class="extiw" title="en:free surface">free surface</a> elevation η(x,t):

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>η<!-- &eta; --></mi><mi>λ<!-- &lambda; --></mi></mfrac></mrow><mo>=</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mi>cosh</mi><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>x</mi><mo>−<!-- &minus; --></mo><mi>c</mi><mi>t</mi></mrow><mi>λ<!-- &lambda; --></mi></mfrac></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="2em"></mspace><mrow class="MJX-TeXAtom-ORD"><mtext>with</mtext></mrow><mspace width="2em"></mspace><mi>A</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>3</mn></msqrt></mrow><mspace width="thinmathspace"></mspace><mi>sinh</mi><mo>⁡<!-- &#8289; --></mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>,</mo><mspace width="2em"></mspace><mrow class="MJX-TeXAtom-ORD"><mtext>for </mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>x</mi><mo>−<!-- &minus; --></mo><mi>c</mi><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>≤<!-- &le; --></mo><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mspace width="thinmathspace"></mspace><mi>λ<!-- &lambda; --></mi><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {\eta }{\lambda }}=A\,\cosh \,\left({\frac {x-ct}{\lambda }}\right),\qquad {\text{with}}\qquad A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac {1}{2}}\right)}},\qquad {\text{for }}|x-ct|\leq {\tfrac {1}{2}}\,\lambda ,}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2fbc893658401f566f3716f8f93dc4c56a3004c" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -4.505ex; width:80.569ex; height:8.176ex;" alt="{\frac {\eta }{\lambda }}=A\,\cosh \,\left({\frac {x-ct}{\lambda }}\right),\qquad {\text{with}}\qquad A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac 12}\right)}},\qquad {\text{for }}|x-ct|\leq {\tfrac 12}\,\lambda ,">

where λ is the <a href="https://en.wikipedia.org/wiki/wavelength" class="extiw" title="en:wavelength">wavelength</a>, H is the wave height and c is the <a href="https://en.wikipedia.org/wiki/phase_speed" class="extiw" title="en:phase speed">phase speed</a>. This approximation – with a maximum error less than 0.7% in the free-surface elevation (relative to the wave height) – was discovered by:
R.C.T. Rainey and <a href="https://en.wikipedia.org/wiki/Michael_S._Longuet-Higgins" class="extiw" title="en:Michael S. Longuet-Higgins">M.S. Longuet-Higgins</a> (2006) "A close one-term approximation to the highest Stokes wave on deep water". Ocean Engineering 33(14–15), pp. 2012–2024.

The maximum wave steepness is H / λ ≈ 0.141. <a href="https://en.wikipedia.org/wiki/surface_gravity_wave" class="extiw" title="en:surface gravity wave">Surface gravity waves</a> of maximum height have a sharp <a href="https://en.wikipedia.org/wiki/crest_(physics)" class="extiw" title="en:crest (physics)">wave crest</a> of 120°.

Licensing

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