John Fenton's Homepage - last updated 18 September 2017


Cristina and John, May 2010


Home in Vienna



Home address

St-Ulrichs-Platz 2/4
1070 Vienna

Home +43 1 522 7467
John +43 664 7313 1035
Cristina +43 650 762 2417

Family History

John's mother spent many years researching the Family History


A method for approximating, smoothing, differentiating, interpolating, or calculating an envelope to data

While working on river rating curves I developed a program using quadratic or cubic splines. Its primary use was to approximate noisy data, but the method seems quite flexible to approximate more-or-less scattered data, or to smooth, differentiate, interpolate, or calculate an envelope to that data.  An example is Nikuradse's results for resistance in pipes, which shows variable curvature:

The program and a supporting theoretical document are described here.

Collected papers

Full list of papers

Recent publications

  • Darvishi, E., Fenton, J. D. and Kouchakzadeh, S. (2017) Boussinesq equations for flows over steep slopes and structures, Journal of Hydraulic Research 55(3), 324-337. Offprints available
    A finite-slope Boussinesq equation is developed to model curved transcritical flow over spillways and broad-crested weirs, even with large slopes. A number of laboratory experiments were performed with different transcritical flow problems including changes in channel gradients and a trapezoidal weir. The equation and the numerical model were tested using results from those experiments and from those for a steep and sharply-curved weir structure, with good results. They can be used as a computational flume to determine the head-discharge characteristics of proposed structures. A novel feature of the equation and numerical method is that higher derivatives of the bed topography are best ignored, apparently mimicking the effects of flow separation in smoothing it.
  • Fenton, J. D. (2016) Hydraulics: science, knowledge, and culture, Journal of Hydraulic Research 54(5), 485-501. Offprints available
  • The processes of thinking, research, dissemination, and use of research and knowledge in hydraulics are examined, differentiating it from hydrology, and suggesting greater use of scientific methods and theories. At highly technical levels, that is already done, but it is suggested that there is room for a greater simplicity of approach, based on scientific rigour, recognising that much of what is done in hydraulics is modelling. This would make understanding, access to, and participation in research easier for members of the profession. A number of recommendations and conclusions are made. The article has a critical tone, but its main intention is to be helpful to individual hydraulicians and to the profession at large. Suggestions are made as to how the profession might use the Web to give open access to research findings and to create an open resource for hydraulics knowledge, as connection by colleagues in all countries is now possible and feasible.

  • Fenton, J. D. and Darvishi, E. (2016) Discussion of "Minimum specific energy and transcritical flow in unsteady open-channel flow" by Oscar Castro-Orgaz and Hubert Chanson, Journal of Irrigation and Drainage Engineering 142(10),
  • We show that there is no justification for the Singular Point Theory of Massé (1938) for calculating transitional flows between sub- and super-critical flows.

  • Fenton, J. D. (2015) Generating stream rating information from data, Paper 8, Alternative Hydraulics.

    The problem of using measurements of water level and discharge to generate stream rating curves is considered. Current approaches are criticised. Two methods for the automatic processing of data and generation of curves are developed, one based on global polynomials and the other on piecewise-continuous approximating splines. Both are found to work well. They allow the specification of a weight for each data point, enabling the filtering of data, possibly incorporating its age, and allowing the computation of a rating curve for any time in the past up to the present. It is suggested that ephemeral changes in bed resistance in some streams may play a more important role than usually recognised, making results less certain. For such streams, with scattered or loopy data, the approximation methods can be used to generate a rating envelope, allowing the calculation also of maximum and minimum expected flows.

  • Fenton, J. D. (2015) Basic physical processes in rivers, Chapter 1 in Rivers – Physical, Fluvial and Environmental Processes, (eds) Paweł Rowiński and Artur Radecki-Pawlik, Springer.

    The governing long wave equations for a curved channel can be developed with very few limiting approximations, especially using momentum rather than energy. The curvature is then shown to be rarely important and is subsequently ignored. Wave periods, imposed by boundary conditions, are asserted to be fundamental. Long waves have speeds and propagation properties that depend on period, and there is no such thing as a single long wave speed. Examination of dimensionless equations and solution of linearised equations using wave period shows a novel interpretation of terms in the momentum equation: the ”kinematic” approximation and wave are misnomers: the approximation lies not in the neglect of inertial terms but is actually a very long period one. The outstanding problem of river modelling, however, is that of resistance to the flow. A large data set from stream-gaugings is considered and it is shown that the state of the bed, namely the arrangement of bed grains by previous flows, is more important than actual grain size. A formula for resistance is proposed which contains a parameter representing bed state. As that state is usually changing with flow, one can not be sure what the resistance actually will be. This uncertainty may have important implications for modelling. The momentum principle is then applied also to obstacles such as bridge piers, and a simple approximation gives greater understanding and a practical method for incorporation in river models. Finally, river junctions are considered, and the momentum approach with the very long period approximation shows that they can be modelled simpl


Lectures in hydraulics, numerics, and maritime engineering

Introductory page

A First Course in Hydraulics

This is an introduction to hydraulics. True energy conservation is presented, and Bernoulli's equation is left to a subsidiary (and occasionally useful) role. It is an integrated momentum equation which is valid along a streamline and whose "constant" varies across streamlines. It is absurd to expect to apply it in fully-turbulent three-dimensional situations - for example the flow from a reservoir to a tap in a house. It is much more intellectually honest simply to use conservation of energy, which is more easily derived and is a more plausible model of most hydraulic problems.

Coastal and Ocean Engineering

A final year elective subject, dealing with elementary oceanography, water wave theory, tsunami, and coastal engineering

River Engineering

An introduction to waves and disturbances in channels, measurement, and structures

Computations and Open Channel Hydraulics

An introduction to numerical methods, presenting theory and applications largely using the optimisation package in Microsoft Excel.

Coastal and Ocean Engineering - Steady water waves

One wave of a steady wave train,
                showing dimensions, co-ordinates and velocities

A computer program ("Fourier") that solves the problem of steadily-progressing waves over a flat bottom is described and made available here: Fourier

Programs that implement Stokes and cnoidal theories are also available. The instructions file for all is Instructions.pdf., which is also included in The latest changes are shown as highlighted comments.

The three wave programs are

  1. A Fourier approximation method whose only approximation lies in truncating the number of terms in the approximating series: - current version, 23 July 2015.
  2. An implementation of cnoidal theory, which is based on series expansions in shallowness, requiring that the waves be long relative to the water depth : - current version, 20 March 2015 (to be unpacked in a sub-directory of the Fourier one). This is an approximation, and not as applicable to higher waves as the Fourier method. It can be used as a check on that method - for long waves that are not high, both should give the same results.
  3. An implementation of Stokes theory, requiring that the waves be not too long relative to the water depth: - current version 20 March 2015 (to be unpacked in a sub-directory of the Fourier one). This is also an approximation, and not as applicable to higher waves as the Fourier method. It can be used as a check on that method - for waves that are not high or long, both should give the same results.

Maintained and authorised by John Fenton; Last modified: Monday 18 September 2017