When waves are required in a lab to study aspects like loads on hydraulic structures, wave dynamics, and sediment and pollutant transport, they are generated by wavemakers. The wavemaker’s motion is prescribed by a control signal. Most commonly, a position-controlled wavemaker is used, for which the control signal is the paddle position. This means that the paddle position is prescribed to the machine. An alternative approach is to prescribe the force measured directly in front of the wavemaker. In this case, the control signal is the demand force and it is called a force-controlled wavemaker. When active absorption is included, the force in front of the wavemaker is measured and compared to the demand force. If the measured force deviates from the demand force, the movement of the wavemaker is adjusted. Force-controlled wavemakers are already in use for over forty years but the theory behind calculating the demand force is still in development. In this thesis, the theory for the control signal of a new force-controlled piston wavemaker is studied.

A control signal can be calculated with different theories, of which two theories have been expanded to force-control wavemakers, referred to as the broad-band and the narrow-band theory. The broad-band theory makes no assumptions about the type of wave and the force equations specifically have been derived in Bayle et al. (2025); Spinneken & Swan (2009a). The broad-band theory has been implemented in a MATLAB code of Edinburgh Designs Ltd. The narrow-band theory assumes a narrow wave spectrum and it neglects the evanescent modes in the force equations, which results in a set of equations that is faster to compute. For a narrow wave spectrum, the narrow-band theory should give the same results as the broad-band theory. However, for a broader wave spectrum, inaccuracies will occur. The force equations of the narrow-band theory have been derived in Bayle et al. (2025). In this thesis it is studied if the force equations of both theories are correct and it is analysed under which conditions the narrow-band theory can be used to calculate the demand force instead
of the broad-band theory.

To reach these objectives, the force equations of the narrow-band theory are evaluated, they are connected to the broad-band theory, one of the missing force terms is derived and all equations are implemented in a MATLAB code. To be consistent with the force equations, the evanescent modes are neglected in all narrow-band equations. Subsequently, the intermediate steps to find the demand force are calculated with both the broad-band and narrow-band codes and the results of the two theories are compared. This analysis is performed for focussed wave groups of a Gaussian spectrum in shallow to deep water (kph = 0.28 to kph = 3.1) in which both the first and second order harmonics are included. In addition, it is verified that the control signal is processed correctly when it is provided directly to the wavemaker.

For a narrow spectrum (ν = 0.02, where ν is the spectral width) and a shallow water wave (kph = 0.28), the narrow-band and broad-band theory give similar results (< 0.8% relative difference for all components) and consequently the narrow-band theory can be used. For waves in intermediate and deep waters (kph = 0.58 to kph = 3.1), some of the intermediate calculation steps give a higher relative difference. This can be caused by the choice to neglect the evanescent modes for the narrow-band theory or by implementation errors and requires further research. A broad spectrum (ν = 0.10) results in a relative difference for the second order harmonics of up to 8% for the shallow water wave (kph = 0.28) and a larger difference of up to 100% for intermediate and deep waters. This suggests that the broad-band theory should be used for a broad spectrum if the second order harmonics play an important role in the studied process. The large relative difference requires research to explain. The results show that for all studied cases, the evanescent modes can be neglected for the shallow water wave (kph = 0.28) but they should be included in intermediate and deep waters (kph = 0.58 to kph = 3.1). Before any definitive conclusions can be drawn, the narrow-band theory should first be completed with respect to the force equations and an analysis of the final demand force calculated with both the broad-band and narrow-band theory is required.