On the role of nonlinear distortion in the theory of wave-like aquatic propulsion

Recent experimental investigations of small model boats propelled by propagating flexural waves carried out by the present author and his co-workers demonstrated viability of this type of propulsion as an alternative to a well-known screw propeller. Since the amplitudes of propagating flexural waves propelling the model boats are large enough, it is natural to consider the effect of nonlinear distortion of propagating localised flexural waves on generated thrust. This problem is explored in the present work by adding nonlinear harmonics of propulsive flexural waves to the well-known Lighthill's formula for generated thrust, which predicts a zero value of thrust in the case of linear flexural wave of constant amplitude. For simplicity, only the lowest (third) harmonic growing linearly with the distance of propagation is used. The resulting formula for the averaged thrust shows that, due to the effect of the third harmonic, the thrust is no longer zero, thereby demonstrating that nonlinear distortion of the propulsive flexural waves contributes positively to the generated thrust.