Note on the propagation of a shallow water wave in water of variable depth

In this technical note, the phenomena of non-linear water-wave propagation above a seabed with variable depth is re-examined. The conventional Korteweg-de Vries (KdV) equation is re-derived for the general case of variable water depth. In the new form of KdV equation, the seabed bottom function is included. Two different bottom profiles are considered in this study; case 1: $b^{\prime }\left(x^{\prime }\right)=c\epsilon \sin \lambda x^{\prime }$ and case 2: $b^{\prime }\left(x^{\prime }\right)=c\epsilon {\mathrm{e}}^{-\lambda {\left(x^{\prime }-x_0^{\prime }\right)}^2}$. The effects of three bottom profile parameters, c, λ and ε on the wave profile are examined. Numerical results indicate that both ε and λ affect the wave profile significantly in case 1, while ε significantly affects the wave profile in case 2.