Higher order corrections for shallow-water solitary waves: elementary derivation and experiments

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first order equation is shown to be equivalent to the Korteweg−de Vries (KdV) equation, while the second order equation is solved numerically. The propagation velocity found for the solitary waves of the second order equation coincides with a known expression, but it is obtained in a simpler way. By measuring the propagation velocity of solitary waves in laboratory, we demonstrate that the second order theory gives a considerably improved fit to experimental results. Higher order corrections for shallow-water solitary waves 2