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Stokes wave

Stokes wave is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Stokes wave rather than just read about it. In short: In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion.

Stokes wave — main illustration
Stokes wave — illustration

Key takeaways

  • Stokes wave belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Stokes wave to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Stokes wave from memory before moving on to harder problems.

Reference excerpt

In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion. Stokes's wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations. While, in the strict sense, Stokes wave refers to a progressive periodic wave of permanent form, the term is also used in connection with standing waves and even random waves.

Examples The examples below describe Stokes waves under the action of gravity (without surface tension effects) in case of pure wave motion, so without an ambient mean current.

Third-order Stokes wave on deep water

According to Stokes's third-order theory, the free surface elevation η, the velocity potential Φ, the phase speed (or celerity) c and the wave phase θ are, for a progressive surface gravity wave on deep water – i.e. the fluid layer has infinite depth:

… excerpt ends here. Continue reading the full article.

Illustrations

Stokes wave: Model testing with periodic waves in the wave–tow tank of the Jere A. Chase Ocean Engineering Laboratory, University of New Hampshire.
Model testing with periodic waves in the wave–tow tank of the Jere A. Chase Ocean Engineering Laboratory, University of New Hampshire.
Stokes wave: Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude.[1] The undulations following behind the bore front appear as slowly modulated Stokes waves.
Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude.[1] The undulations following behind the bore front appear as slowly modulated Stokes waves.
Stokes wave: Third-order Stokes wave in deep water under the action of gravity. The wave steepness is: ka = 0.3.
Third-order Stokes wave in deep water under the action of gravity. The wave steepness is: ka = 0.3.
Stokes wave: The three harmonics contributing to the surface elevation of a deep water wave, according to Stokes's third-order theory. The wave steepness is: ka = 0.3. For visibility, the vertical scale is distorted by a factor of four, compared to the horizontal scale.Description: * the dark blue line is the surface elevation of the 3rd-order Stokes wave, * the black line is the fundamental wave component, with wavenumber k (wavelength λ, k = 2π / λ), * the light blue line is the harmonic at 2 k (wavelength .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄2 λ), and * the red line is the harmonic at 3 k (wavelength 1⁄3 λ).
The three harmonics contributing to the surface elevation of a deep water wave, according to Stokes's third-order theory. The wave steepness is: ka = 0.3. For visibility, the vertical scale is distorted by a factor of four, compared to the horizontal scale.Description: * the dark blue line is the surface elevation of the 3rd-order Stokes wave, * the black line is the fundamental wave component, with wavenumber k (wavelength λ, k = 2π / λ), * the light blue line is the harmonic at 2 k (wavelength .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄2 λ), and * the red line is the harmonic at 3 k (wavelength 1⁄3 λ).
Stokes wave: The ratio .mw-parser-output .mathcal{font-family:"Lucida Calligraphy","Monotype Corsiva","URW Chancery L","Apple Chancery","Tex Gyre Chorus",cursive,serif}S = a2 / a of the amplitude a2 of the harmonic with twice the wavenumber (2 k), to the amplitude a of the fundamental, according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength, while the vertical axis is the Stokes parameter S divided by the wave steepness ka (with k = 2π / λ).Description: * the blue line is valid for arbitrary water depth, while * the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and * the dash-dot green line is the asymptotic limit for deep water waves.
The ratio .mw-parser-output .mathcal{font-family:"Lucida Calligraphy","Monotype Corsiva","URW Chancery L","Apple Chancery","Tex Gyre Chorus",cursive,serif}S = a2 / a of the amplitude a2 of the harmonic with twice the wavenumber (2 k), to the amplitude a of the fundamental, according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength, while the vertical axis is the Stokes parameter S divided by the wave steepness ka (with k = 2π / λ).Description: * the blue line is valid for arbitrary water depth, while * the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and * the dash-dot green line is the asymptotic limit for deep water waves.

Worked examples

Example 1 — a first encounter with Stokes wave

Start with the simplest possible case. Write down what Stokes wave claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stokes wave before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stokes wave ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stokes wave

In research
Stokes wave appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Stokes wave in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Stokes wave is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fluid dynamics, Water waves, so understanding it makes those chapters shorter.
In everyday life
Look for Stokes wave outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Stokes wave in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Stokes wave means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Stokes wave out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stokes wave in simple terms?

In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained app…

Why does Stokes wave matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stokes wave?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stokes wave.

Tags

  • Fluid dynamics
  • Water waves

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