Analytical study of wave making in a flume with a partially reflecting end-wall

Based on the eigenfunction expansion technique, the wave generation by a piston wave maker in a wave flume with a partially reflecting end-wall is studied. The corresponding velocity potential and wave elevation in the flume are obtained. The present analytical solution is verified by the numerical results obtained from a time-domain higher-order boundary element method in a closed flume. Numerical experiments are further carried out to study the difference between the partial/full reflection boundary and the transmission boundary and the effects of the reflection coefficient and the motion period of the wave maker on the wave height. Meanwhile, the natural frequency of the wave flume can be obtained from the analytical expression. The resonance occurs when the motion frequency is equal to the natural frequency. Even the partial reflection of the end-wall in the wave flume experiments has a great influence on the wave height, therefore, inaccurate measurements would be resulted in long-time simulations, especially when the wave frequency approaches the wave flume natural frequency. The present study can serve as a guidance for the physical experiment in wave flumes.