1962
DOI: 10.1029/jz067i004p01555
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Limiting conditions for cnoidal and Stokes waves

Abstract: The second approximation to cnoidal waves is compared with the third approximation for Stokes waves of permanent form in water of finite depth. The comparison clearly indicates that cnoidal wave theory should not be applied to finite amplitude waves if their wavelengths are shorter than 5 times the depth. It is shown how the limiting heights of cnoidal waves are also related to the vanishing of the pressure gradient near the wave crest. The third approximation to Stokes waves in finite water depths is verified… Show more

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Cited by 49 publications
(28 citation statements)
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“…They occur in shallow water just seaward of the breaker zone. Cnoidal wave theory (Boussinesq, 1871;Korteweg and de Vries, 1895) is defined in terms of the Jacobian elliptic function and has been approximated to third order (Keulegan and Patterson, 1940;Keller, 1948;Latoine, 1962). However, because these methods are difficult to apply (Demirbilek and Vincent, 2002), Wiegel (1960Wiegel ( , 1964 summarized the principal results in graphical form.…”
Section: Shallow Water Waves As Modeled By Cnoidal and Cokelet's Theorymentioning
confidence: 99%
See 1 more Smart Citation
“…They occur in shallow water just seaward of the breaker zone. Cnoidal wave theory (Boussinesq, 1871;Korteweg and de Vries, 1895) is defined in terms of the Jacobian elliptic function and has been approximated to third order (Keulegan and Patterson, 1940;Keller, 1948;Latoine, 1962). However, because these methods are difficult to apply (Demirbilek and Vincent, 2002), Wiegel (1960Wiegel ( , 1964 summarized the principal results in graphical form.…”
Section: Shallow Water Waves As Modeled By Cnoidal and Cokelet's Theorymentioning
confidence: 99%
“…The limiting factors for cnoidal waves to be applicable are that U R ≥ 20 (Latoine, 1962) and L wc ≥ 21.5de − 1.87(H w / d) (Fenton and McKee, 1990).…”
Section: Shallow Water Waves As Modeled By Cnoidal and Cokelet's Theorymentioning
confidence: 99%
“…As shown by many comparisons, e.g. see Lairone [1962], this shallow d water expansion method is satisfactory for howavelengths greater than 5 times the water depth. It is important to note that to the approximation given by (a/d)* • 0 the relations for the wave profile (7) and the pressure (Ap) are unaltered by this conversion.…”
mentioning
confidence: 61%
“…wave profile r• which is represented mathematically by the Jacobinn elliptic functions. The nonlinear shallow water theory used by both Chappelear [1962] and Lairone [1960Lairone [ , 1962 is based upon the systematic asymptotic expansion method introduced by Friedrichs [1948]. I-Ie generalized the classical nonlinear shallow water first approximation by a suitable stretching of the vertical dimensions which are considered to be everywhere physically smaller than the wavelength.…”
mentioning
confidence: 99%
“…The characteristics of such shoaling, non-linear waves have been modeled by many authors (e.g. Bretschneider, 1960;Latoine, 1960;Skjelbreia and Hendrickson, 1961;Chappelear, 1962;Latoine, 1962Latoine, , 1965Ippen, 1966;Peregrine, 1972;Cokelet, 1977;Phillips, 1977;Fenton, 1985;Chakrabarti, 1987;Fenton, 1988;Dean and Dalrymple, 1991;Mei, 1991). However, the simplified but fully integrated equations of Le Roux (2007aRoux ( ,b, 2008a are used here, because they eliminate the discrepancies produced when using different wave theories for different water depths.…”
Section: Wave Characteristics and Particle Velocities In Any Water Depthmentioning
confidence: 98%