The solitary wave on a stream with an arbitrary distribution of vorticity

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Abstract
The theoretical work reported herein makes a departure from the many previous analyses of the solitary wave which have treated the wave as an example of irrotational fluid motion. The present analysis is of more general scope in that it covers the whole category of examples in which the wave may propagate in either direction on a horizontal stream whose primary velocity distribution U(y) is an arbitrary function (i.e. there is no restriction on the extent of the variations of U(y)). An approximate form of the wave profile is found in general to be a sech2 {(x − ct)/b}, as it is according to previous theories applicable to the wave on a uniform stream, but the relationships amongst the wave amplitude a, the length scale b, and the two propagation velocities c (positive downstream and negative upstream) depend in complicated fashion on the form of U(y).

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