On adjustable undular bore profiles based on the modified steady KdV–Burgers equation

Abstract

In this study, a speed parameter is introduced into the steady Korteweg–de Vries (KdV)–Burgers equation which enables the theoretical undular bore profiles to be adjustable with a proper combination of the speed parameter and the viscous damping parameter. A new criterion for identifying the above two bores is then proposed with respect to these two parameters, whose influence on the undular bore profile is then discussed. For the theoretical solution with a small damping, error after introducing the variable speed parameter is limited. A large speed parameter corresponds to a wide range of acceptable dampings. From the energy perspective, it is confirmed that the speed parameter also denotes the nonlinearity effect. In addition, comparison between the theoretical and experimental results shows the superiority of the present model over the traditional model, which also reveals the physical meanings of the present model.

Keywords:

• Damping parameter
• KdV–Burgers equation
• phase-plane analysis
• speed parameter
• undular bore

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notation

a0	=	Matching point of the oscillation part and the perturbation part in the theoretical solution
b0	=	Constant remains to be solved in the matching process of the theoretical solution
Ep	=	Dimensionless dissipated energy up to the particular wave peak
Er	=	Matching error
F	=	Froude number
k	=	Constant parameter
n	=	Speed parameter
R	=	Viscous damping term
S	=	Dimensionless independent horizontal variable
V	=	Dimensionless water surface elevation
Vp	=	Dimensionless wave peak
$\gamma$	=	Damping parameter
${\gamma }_{\mathrm{c}\mathrm{r}}$	=	The critical damping parameter
$\mathrm{\Delta }$	=	Small parameter
$\delta$	=	Shallow water parameter
$ϵ$	=	Relative wave amplitude
$\xi$	=	Dimensionless horizontal coordinate measured along the channel bed
$\eta$	=	Dimensionless water surface elevation measured from the still water level
$\kappa$	=	Modulus of Jacobian elliptic function
$\sigma$	=	Slow time scale
$\tau$	=	Dimensionless time coordinate

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China [grant numbers 52071288 and 52271292] and the Science and Technology Innovation 2025 Major Project of Ningbo City [grant number 2022Z213].