The use of dimensionless cumulants to characterize the shape of the linear response of a semi-infinite prismatic channel is extended and simplified by means of the dimensionless coefficient (γR) linking the Rth order shape factor to the 3rd order shape factor raised to the power of (R—2). It is shown that, for the special structure of the cumulants of the linearized St. Venant equations, the dimensionless cumulant coefficient (γR) of any order R > 4 is a single-valued function of γ4. It is also shown that the value of γRfor any Froude number F0combined with any shape of section and any friction law (both characterized by the parameter m) is intermediate between the values for the two limiting cases of the diffusion analogy model (F0= 0) and the rapid flow model (F0= 1). The dimensionless cumulant coefficient γRis related to γ4through a parameter p(m, F0) by a polynomial of order [R/2]—1.