ABSTRACT
Cnoidal wave and its extreme case, the solitary wave, can be described by the KdV equation, which was first derived by Korteweg and de Vires with the first-order accuracy. Subsequently, different authors proposed their derivations and claimed that their equations, sharing similar expressions but different corresponding coefficients, were the same as the original one. After introducing a unified dimensionless frame, this study re-derived the KdV equation with respect to seven existing methods and confirmed that KdV equation indeed refers to a type of first-order equations, rather than a specified one. Differences in equations come from the influence of the second-order quantities associated with the derivation process. Regarding the cnoidal wave and solitary wave, the KdV-type equations obtained using different methods present the same first-order solution for wave profile. Nevertheless, in their directly derived results, different wave celerities and water particle velocities are presented due to the influence of second-order quantities. Additionally, comparing with the second-order solutions, all directly derived wave celerity solutions predict well for the Ursell number between 20 and 100. As for the first-order solution of the water particle velocity, all methods present the same result except Dean’s expression which contains a different coefficient.
Notation
= | Coordinate along channel bed | |
= | Coordinate upward from channel bed | |
= | Time | |
= | Wave height | |
= | Wavelength | |
= | Still water depth | |
= | surface Water surface elevation measured from the still water | |
= | Perturbation component of | |
= |
| |
= |
| |
= | Depth average velocity | |
= | Re-derived first-order water particle velocity following D1965 | |
= | First-order water particle velocity given by K1895, KP1940, K1948, IK1983, M2005, and Z2005 | |
= | Second-order depth average water particle velocity given by Isobe and Kraus (Citation1983) | |
= | Wave celerity | |
= | Re-derived wave celerity following D1965, listed in | |
= | Second-order wave celerity defined by Stokes first and second definitions (Isobe and Kraus Citation1983) | |
= | The limiting phase speed | |
= | Potential function | |
= | Stream function | |
= | Gravity | |
= | Water depth below the wave trough | |
= | Relative wave height | |
= | Depth parameter | |
= | Ursell number | |
= | Flow rate across a vertical cross section | |
= | Modulus of Jacobian elliptic function | |
= | Complete elliptic integral of the first kind and the second kind | |
= | Arbitrary constants from integration | |
= | Arbitrary constants from integration | |
= | Three roots of EquationEquation (10) | |
= | Corresponding coefficients in the dimensional KdV-type equation | |
= | Corresponding coefficients in the dimensionless KdV-type equation |
Disclosure statement
No potential conflict of interest was reported by the authors.