Analytical Considerations on Cnoidal and Solitary Waves

Iwasa, Yoshiaki

doi:10.2208/jscej1949.1956.43

Cited by 7 publications

(3 citation statements)

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“…This is a generalization for a dry granular material of the water wave equation developed by Serre [], Benjamin and Lighthill [], or Iwasa [] to study cnoidal waves and bores. Note that the water wave formulation is regained from equation simply by setting K = 1, as deduced by Hager and Hutter [, ] using streamline coordinates.…”

Section: Simplified Forms Of Boussinesq Extended Flow Equationsmentioning

confidence: 99%

“…This model was also developed by Benjamin and Lighthill [1954] following a different theoretical treatment based on a third-order expansion of the stream function. Application of Boussinesq-type equations to civil engineering was successful for the accurate description of flow over weirs, undular jumps, sand waves, Favre waves, dam breaks, overfalls, and slope breaks [Fawer, 1937;Iwasa, 1955Iwasa, , 1956Mandrup-Andersen, 1975Marchi, 1963Marchi, , 1992Marchi, , 1993Matthew, 1963Matthew, , 1991Engelund and Hansen, 1966;Basco, 1983;Hager, 1983;Hutter, 1984a, 1984b;Montes, 1986;Soares-Frazão and Zech, 2002;Mohapatra and Chaudhry, 2004;Dey, 2007, 2009;Chaudhry, 2008;Castro-Orgaz and Hager, 2009]. Castro-Orgaz et al [2012 demonstrated that Boussinesq-type enhanced equations also describe free surface groundwater flow.…”

Section: Introductionmentioning

confidence: 99%

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Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

Castro-Orgaz

Hutter

Giráldez

et al. 2015

J. Geophys. Res. Earth Surf.

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Understanding granular mass flow is a basic step in the prediction and control of natural or man-made disasters related to avalanches on the Earth. Savage and Hutter (1989) pioneered the mathematical modeling of these geophysical flows introducing Saint-Venant-type mass and momentum depth-averaged hydrostatic equations using the continuum mechanics approach. However, Denlinger and Iverson (2004) found that vertical accelerations in granular mass flows are of the same order as the gravity acceleration, requiring the consideration of nonhydrostatic modeling of granular mass flows. Although free surface water flow simulations based on nonhydrostatic depth-averaged models are commonly used since the works of Boussinesq (1872, 1877), they have not yet been applied to the modeling of debris flow. Can granular mass flow be described by Boussinesq-type gravity waves? This is a fundamental question to which an answer is required, given the potential to expand the successful Boussinesq-type water theory to granular flow over 3-D terrain. This issue is explored in this work by generalizing the basic Boussinesq-type theory used in civil and coastal engineering for more than a century to an arbitrary granular mass flow using the continuum mechanics approach. Using simple test cases, it is demonstrated that the above question can be answered in the affirmative way, thereby opening a new framework for the physical and mathematical modeling of granular mass flow in geophysics, whereby the effect of vertical motion is mathematically included without the need of ad hoc assumptions.

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Section: Simplified Forms Of Boussinesq Extended Flow Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

Castro-Orgaz

Hutter

Giráldez

et al. 2015

J. Geophys. Res. Earth Surf.

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“…Benjamin & Lighthill (1954) discussed cnoidal waves in connection with the formation of Hydraulic jumps and bores. A study of cnoidal waves was also made by Iwasa (1955), and Keulegan & Patterson (1940). Laiton (1960) while studying higher order approximations of solitary and cnoidal waves, pointed out that the pressure is no longer hydrostatic and the vertical motions are no longer negligible in the higher order approximations.…”

Section: Introductionmentioning

confidence: 99%

Cnoidal waves in a compressible fluid over a toroidal channel

Raman¹

1970

Tellus

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In this paper we have discussed the problem of cnoidal waves at the free surface of a compressible fluid in a toroidal channel. We have utilised the technique of expanding the dependent variables in terms of a parameter, to analyse the problem. The expression for the cnoidal wave profile has been obtained, and also some particular cases have been discussed at the end. The accuracy of the results presented here has been established by correlating them with those of Shen (1967).

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Vertically Integrated Non-hydrostatic Free Surface Flow Equations

Castro-Orgaz

Hager

2017

Advances in Geophysical and Environmental Mechanics and Mathematics

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Analytical Considerations on Cnoidal and Solitary Waves

Cited by 7 publications

References 5 publications

Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

Cnoidal waves in a compressible fluid over a toroidal channel

Vertically Integrated Non-hydrostatic Free Surface Flow Equations

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