Solving the Equations of Uniform Flow

All realizable hydraulic profiles associated with the uniform flow of liquid in an open channel are expressible in terms of three simple integrals. The fifteen profiles fall naturally into five classes. Profiles in Classes I and II (M1, M2, M3, S1, S2, S3, C1, C3) are expressible as a combination of the backwater integrals of the first kind; Class III (H2, H3) involves backwater integrals of the second kind; Class IV (A2, A3) involves backwater integrals of the third kind; and Class V (C2, H1, A1) contains only unrealizable or unstable profiles. The profile equation for each of the significant classes is given explicitly in terms of these integrals. Finally, a practical method of computing the backwater integrals is presented. The integral of the second kind is reduced to a simple algebraic function and those of the first and third kinds are expressed in terms of polynomials of moderate order and polylogarithm functions of low order.