  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 the laboratory, we demonstrate that the second-order theory gives a
considerably improved fit to experimental results.
