This paper is the first of a series dealing with the motion of translation waves in open channels. The case treated is that of an arbitrary intumescence with a small height in a uniform channel with a rectangular cross section containing water originally moving with a uniform velocity. Approximate formulas are derived which give the deformation and motion of the intumescence. The case of the intumescence of the form of a rectangular type is also dealt with and for that the form of the rigorous solution is shown. Special emphasis is laid on the dispersive property of intumescences. A method of illustration of the dispersion is presented and both the propagation speed and the dispersion of intumescences of any arbitrary shape are illustrated. The results are all derived with the method of operational calculus on the plane of the complex variables.

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