Clapotis

Incoming wave (red) reflected at the wall produces the outgoing wave (blue), both being overlaid resulting in the clapotis (black).

In hydrodynamics, the clapotis (from French: "lapping of water") is a non-breaking standing wave pattern, caused for example, by the reflection of a traveling surface wave train from a near vertical shoreline like a breakwater, seawall or steep cliff.^[1][2][3][4] The resulting clapotic^[5] wave does not travel horizontally, but has a fixed pattern of nodes and antinodes.^[6] These waves promote erosion at the toe of the wall,^[7] and can cause severe damage to shore structures.^[8] The term was coined in 1877 by French mathematician and physicist Joseph Valentin Boussinesq who called these waves ‘le clapotis’ meaning ‘standing waves’.^[9][10]

In the idealized case of "full clapotis" where a purely monotonic incoming wave is completely reflected normal to a solid vertical wall,^[11][12] the standing wave height is twice the height of the incoming waves at a distance of one half wavelength from the wall.^[13] In this case, the circular orbits of the water particles in the deep-water wave are converted to purely linear motion, with vertical velocities at the antinodes, and horizontal velocities at the nodes.^[14] The standing waves alternately rise and fall in a mirror image pattern, as kinetic energy is converted to potential energy, and vice versa.^[15] In his 1907 text, Naval Architecture, Cecil Peabody described this phenomenon:

At any instant the profile of the water surface is like that of a trochoidal wave, but the profile instead of appearing to run to the right or left, will grow from a horizontal surface, attain a maximum development, and then flatten out till the surface is again horizontal; immediately another wave profile will form with its crests where the hollows formerly were, will grow and flatten out, etc. If attention is concentrated on a certain crest, it will be seen to grow to its greatest height, die away, and be succeeded in the same place by a hollow, and the interval of time between the successive formations of crests at a given place will be the same as the time of one of the component waves."^[16]

Related phenomena

True clapotis is very rare, because the depth of the water or the precipitousness of the shore are unlikely to completely satisfy the idealized requirements.^[15] In the more realistic case of partial clapotis, where some of the incoming wave energy is dissipated at the shore,^[17] the incident wave is less than 100% reflected,^[11] and only a partial standing wave is formed where the water particle motions are elliptical.^[18] This may also occur at sea between two different wave trains of near equal wavelength moving in opposite directions, but with unequal amplitudes.^[19] In partial clapotis the wave envelope contains some vertical motion at the nodes.^[19]

When a wave train strikes a wall at an oblique angle, the reflected wave train departs at the supplementary angle causing a cross-hatched wave interference pattern known as the clapotis gaufré ("waffled clapotis").^[8] In this situation, the individual crests formed at the intersection of the incident and reflected wave train crests move parallel to the structure. This wave motion, when combined with the resultant vortices, can erode material from the seabed and transport it along the wall, undermining the structure until it fails.^[8]

Clapotic waves on the sea surface may also radiate infrasonic microbaroms into the atmosphere, and microseismic vibrations called microseisms coupled through the ocean floor to the Earth's crust.^[20]

See also

• Rogue wave
• Seiche
• Soliton

Further reading

• Boussinesq, J. (1872). "Théorie des ondes liquides périodiques". Mémoires présentés par divers savants à l'Académie des Sciences (Paris) 20: 509–616. 
• Boussinesq, J. (1877). "Essai sur la théorie des eaux courantes". Mémoires présentés par divers savants à l'Académie des Sciences (Paris) 23 (1): 1–660. 
• Hires, G. (1960). "Étude du clapotis". La Houille Blanche 15: 153–63. 
• Leméhauté, B.; Collins, J.I. (1961). Clapotis and Wave Reflection: With an Application to Vertical Breakwater Design. Civil Engineering Dept., Queen's University at Kingston, Ontario. 

References

1. ^ "clapotis". Glossary of Meteorology. American Meteorological Society. http://amsglossary.allenpress.com/glossary/browse?s=c&p=37. Retrieved 2007-11-27. 
2. ^ "clapotis". Glossary of Scientific Terms. University of Alberta. http://www.math.ualberta.ca/~bruce/glossary.html. Retrieved 2007-11-27. 
3. ^ Eid, B.M.; Zemell, S.H. (1983). "Dynamic analysis of a suspended pump in a vertical well connected to the ocean". Canadian Journal of Civil Engineering 10 (3): 481–491. doi:10.1139/l83-075. "The standing wave system resulting from the reflection of a progressive wave train from a vertical wall (clapotis)…" Eid, Bassem M.; Zemell, Sheldon H. (1984). "Erratum: Dynamic analysis of a suspended pump in a vertical well connected to the ocean". Canadian Journal of Civil Engineering 11: 137. doi:10.1139/l84-025. 
4. ^ prepared by the Task Committee on Hydrology Handbook of Management Group D of the American Society of Civil Engineers. (1996). Hydrology handbook. New York: ASCE. ISBN 0-7844-0138-1. "This simplification assumes that a standing wave pattern, called clapotis, forms in front of a wall where incident and reflected waves combine." 
5. ^ Carter, Bill (1989). Coastal environments: an introduction to the physical, ecological, and cultural systems of coastlines. Boston: Academic Press. p. 50. ISBN 0-12-161856-0. "…if the wave travels in exactly the opposite direction then a standing, or clapotic, wave can develop." 
6. ^ Matzner, Richard A. (2001). Dictionary of geophysics, astrophysics, and astronomy. Boca Raton: CRC Press. p. 81. ISBN 0-8493-2891-8. http://astropole.ru/astronomy/library/common/Dictionary_of_Geophysics.pdf. "clapotis…denotes a complete standing wave — a wave which does not travel horizontally but instead has distinct nodes and antinodes." 
7. ^ Beer, Tom (1997). Environmental oceanography. Boca Raton: CRC Press. p. 44. ISBN 0-8493-8425-7. "... the reflected wave energy interacted with the incoming waves to produce standing waves known as clapotis, which promote erosion at the toe of the wall." 
8. ^ ^{a b c} Fleming, Christopher; Reeve, Dominic; Chadwick, Andrew (2004). Coastal engineering: processes, theory and design practice. London: Spon Press. pp. 47. ISBN 0-415-26841-9. "Clapotis Gaufre When the incident wave is at an angle a to the normal from a vertical boundary, then the reflected wave will be in a direction a on the opposite side of the normal." 
9. ^ Iooss, G. (2007). "J. Boussinesq and the standing water waves problem". C. R. Mecanique 335 (9–10): 584–589. Bibcode 2007CRMec.335..584I. doi:10.1016/j.crme.2006.11.007. http://math.unice.fr/~iooss/publis1/Iooss-Boussinesq.pdf. Retrieved 2007-11-28. "In this short Note we present the original Boussinesq's contribution to the nonlinear theory of the two dimensional standing gravity water wave problem, which he defined as ‘le clapotis’." 
10. ^ Iooss, G.; Plotnikov, P.I.; Toland, J.F. (2005). "Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity". Archive for Rational Mechanics and Analysis 177 (3): 367–478. Bibcode 2005ArRMA.177..367I. doi:10.1007/s00205-005-0381-6. http://www.maths.bath.ac.uk/~jft/Papers/working6.pdf. Retrieved 2007-11-29. "It was, we believe, Boussinesq in 1877 who was the first to deal with nonlinear standing waves. On pages 332-335 and 348-353 of[7]he refers to ‘le clapotis’, meaning standing waves, and his treatment, which includes the cases of finite and infinite depth, is a nonlinear theory taken to second order in the amplitude." 
11. ^ ^{a b} "D.4.14 Glossary" (pdf). Guidelines and Specifications for Flood Hazard Mapping Partners. Federal Emergency Management Agency. 2004-11. http://www.fema.gov/library/file?type=publishedFile&file=frm_cfd414.pdf&fileid=4de56330-91f7-11db-b057-000bdba87d5b. "CLAPOTIS The French equivalent for a type of STANDING WAVE. In American usage it is usually associated with the standing wave phenomenon caused by the reflection of a nonbreaking wave train from a structure with a face that is vertical or nearly vertical. Full clapotis is one with 100 percent reflection of the incident wave; partial clapotis is one with less than 100 percent reflection." 
12. ^ Mai, S.; Paesler, C. and Zimmermann, C. (2004) (PDF). Wellen und Seegang an Küsten und Küstenbauwerken mit Seegangsatlas der Deutschen Nordseeküste : 2. Seegangstransformation (Waves and Sea State on Coasts and Coastal Structures with Sea State Atlas of the German North Sea Coast : 2. Sea State Transformation). Universität Hannover. http://www.dr-smai.de/Literatur/Geb-4/80_fi_heft_90a_mai_cpa_zi.pdf#page=51. Retrieved 2007-12-02. "Ein typischer extremer Fall von Reflektion tritt an einer starren senkrechten Wand auf. (A typical case of extreme reflection occurs on a rigid vertical wall.)" 
13. ^ Jr, Ben H. Nunnally (2007). Construction of Marine and Offshore Structures, Third Edition. Boca Raton, FL: CRC Press. p. 31. ISBN 0-8493-3052-1. "Waves impacting against the vertical wall of a caisson or against the side of a barge are fully reflected, forming a standing wave or clapotis, almost twice the significant wave height, at a distance from the wall of one-half wavelength." 
14. ^ van Os, Magchiel (2002-01). "4.2 Pressures due to Non-Breaking Waves". Breaker Model for Coastal Structures : Probability of Wave Impacts on Vertical Walls. Technische Universiteit Delft, Hydraulic and Offshore Engineering division. pp. 4–33. http://citeseer.ist.psu.edu/489083.html. Retrieved 2007-11-28. "This phenomenon is also called "Clapotis" and the circular orbits of the particle movements have degenerated into straight lines. This results in only vertical velocities at the antinodes and horizontal velocities at the nodes." 
15. ^ ^{a b} Woodroffe, C. D. (2003). Coasts: form, process, and evolution. Cambridge, UK: Cambridge University Press. p. 174. ISBN 0-521-01183-3. "The standing wave will alternately rise and collapse as kinetic energy is converted into potential energy and back again." 
16. ^ Peabody, Cecil Hobart (1904). Naval architecture. New York: J. Wiley & Sons. p. 287. http://books.google.com/?id=So5BAAAAIAAJ&dq=Clapotis&vq=Clapotis&pg=PA287. "This action is most clearly seen where a wave is reflected from a vertical sea-wall, and is known as the clapotis." 
17. ^ Hirayama, K. (2001). "Numerical Simulation of Nonlinear Partial Standing Waves using the Boussinesq Model with New Reflection Boundary". Report ff the Port and Airport Research Institute 40 (4): 3–48. "The waves in front of actual seawalls and harbor breakwaters, however, are rather partial standing waves such that some incident wave energy is dissipated…" 
18. ^ Leo H. Holthuijsen (2007). Waves in Oceanic and Coastal Waters. Cambridge, UK: Cambridge University Press. p. 224. ISBN 0-521-86028-8. "A partially standing wave due to the (partial) reflection of an incident wave against an obstacle. The ellipses are the trajectories of the water particles as they undergo their motion in one wave period." 
19. ^ ^{a b} Silvester, Richard (1997). Coastal Stabilization. World Scientific Publishing Company. ISBN 981-02-3154-7. "Should one of the opposing progressive waves be smaller in height than the other, as in partial reflection from a wall, the resulting nodes and antinodes will be located in the same position but the water-particle orbits will not be rectilinear in character." ^{[page needed]}
20. ^ Tabulevich, V.N.; Ponomarev, E.A.; Sorokin, A.G.; Drennova, N.N. (2001). "Standing Sea Waves, Microseisms, and Infrasound". Izv. Akad. Nauk, Fiz. Atmos. Okeana 37: 235–244. http://www.maik.ru/cgi-perl/search.pl?type=abstract&name=physatm&number=2&year=1&page=218. Retrieved 2007-11-28. "In this process, the interference of differently directed waves occurs, which forms standing water waves, or the so-called clapotis.…To examine and locate these waves, it is proposed to use their inherent properties to exert (“pump”) a varying pressure on the ocean bottom, which generates microseismic vibrations, and to radiate infrasound into the atmosphere."