Asymptotic solutions of transient waves generated by a wavemaker in water of uniform depth

The generation of harmonic waves in the laboratory using a wavemaker is often accompanied by unavoidable transient effects during the start-up process. To further understand these transient effects, this study revisits the transient wavemaker theory, with the aim of extending existing formulations and addressing unresolved issues. Small-amplitude transient waves generated by a wavemaker in uniform-depth wave flumes of both infinite and finite length are considered. Asymptotic analysis is conducted to investigate the leading waves under impulsive and harmonic forcing, both of which can be expressed in Airy-function form. Moreover, under harmonic forcing, the temporal evolution of the wave envelope is studied analytically. An asymptotic expression for the wave envelope, formulated in terms of a Fresnel integral, is obtained. It exhibits a monotonic increase to a local maximum, followed by damped oscillations toward a constant asymptotic value. A new series of experiments under harmonic forcing is conducted to support the findings, and the time-series data show that the two asymptotic solutions can accurately capture different stages of the transient waves. Furthermore, using a sliding-window approach, the temporal evolution of the wave envelope is extracted from the experimental data to validate the asymptotic solutions and the transition toward quasi-steady-state motion. The resulting waiting-time estimate provides a theory-based indicator for determining when wavemaker-generated harmonic waves become sufficiently harmonic and stable for measurement at a fixed location.