On the Highest and Other Solitary Waves

For solitary waves an expansion parameter which differs from those previously employed permits calculation to extremely high order. The observed behavior of the coefficients entering the power series for the position or velocity field yields much information about the nature of singularities in the solution. The principal conclusions are, (i) the wave of maximum amplitude has a nondimensional amplitude $ \cong 3\sqrt {3}/2\pi $ and a nondimensional speed (Froude number) $ \cong (3\sqrt {3}/\pi)^{1/2} $. (ii) All previous theories employing an expansion parameter are incomplete. (iii) The various series relating Froude number to amplitude, recently advanced to the ninth order, are asymptotic.For undular bores direct numerical calculations show that (i) the relationship between relative elevation and relative velocity given by long wave theory is approached for the “ahead of” and “behind” an undular bore even when the bore is generated in ways which violate the conditions of the long wave theory, (ii) the a...