Are all KdV-type shallow water wave equations the same with uniform solutions?

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.

KEYWORDS:

• KdV equations
• solitary wave
• cnoidal wave
• wave celerity
• water particle velocity

Notation

$x$	=	Coordinate along channel bed
$z$	=	Coordinate upward from channel bed
$t$	=	Time
$H$	=	Wave height
$L$	=	Wavelength
$h$	=	Still water depth
$\eta$	=	surface Water surface elevation measured from the still water
${\eta }_i$	=	Perturbation component of $\eta$
$u$	=	$x$-component of velocity
$w$	=	$z$-component of velocity
$\bar{u}$	=	Depth average velocity
$u_{\mathrm{D}\mathrm{e}\mathrm{a}\mathrm{n}}=c_{\mathrm{D}\mathrm{e}\mathrm{a}\mathrm{n}}\frac{\eta }{h}$	=	Re-derived first-order water particle velocity following D1965
$u_{1\mathrm{s}\mathrm{t}-\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}}$	=	First-order water particle velocity given by K1895, KP1940, K1948, IK1983, M2005, and Z2005
$u_{2\mathrm{n}\mathrm{d}-\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}}$	=	Second-order depth average water particle velocity given by Isobe and Kraus (1983)
$c$	=	Wave celerity
$c_{\mathrm{D}\mathrm{e}\mathrm{a}\mathrm{n}}$	=	Re-derived wave celerity following D1965, listed in Table 4
$c_{S1}$, $c_{S2}$	=	Second-order wave celerity defined by Stokes first and second definitions (Isobe and Kraus 1983)
$c^{\ast }$	=	The limiting phase speed
$\varphi$	=	Potential function
$\psi$	=	Stream function
$g$	=	Gravity
$h_{tr}$	=	Water depth below the wave trough
$\epsilon =\frac{H}{h}$	=	Relative wave height
$\mu =\frac{h}{L}$	=	Depth parameter
$Ur=\frac{\epsilon }{{\mu }^2}=\frac{L^{2}H}{h^3}$	=	Ursell number
$Q$	=	Flow rate across a vertical cross section
$\kappa$	=	Modulus of Jacobian elliptic function
$K\left(\kappa \right)$, $E\left(\kappa \right)$	=	Complete elliptic integral of the first kind and the second kind
${\alpha }_i$, ${\beta }_i$	=	Arbitrary constants from integration
$G$, $F$	=	Arbitrary constants from integration
${\xi }_i$	=	Three roots of Equation (10)(10) $\begin{array}{c}{\xi }_1=\frac{H}{{\kappa }^{2}h}\left(1-\frac{E}{K}\right)\\ \\ {\xi }_2=\frac{H}{{\kappa }^{2}h}\left(1-{\kappa }^2-\frac{E}{K}\right)\\ \\ {\xi }_3=-\frac{H}{{\kappa }^{2}h}\frac{E}{K}\end{array}$(10)
$r$, $s$	=	Corresponding coefficients in the dimensional KdV-type equation
$m$, $n$	=	Corresponding coefficients in the dimensionless KdV-type equation

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was financially supported by the National Natural Science Foundation of China (No. 11632012, No. 51761135015), the Natural Science Foundation of Zhejiang Province, China (No. LZ19E090001), and the Fundamental Research Funds for the Central Universities (No. 2020QNA4023).